Sequences
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Infinite Geometric Series
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- The Sigma Notation for Summation
- Introduction to Series
- Introduction to Series (continued)
- Convegent and Divergent Series
- Infinite Geometric Series
- Exercise 1 part a
- Exercise 1 part b
- Exercise 1 part c
- Exercise 1 part d
- Exercise 1 part e
- Exercise 1 part f
- Exercise 1 part g
- Exercise 1 part h
- Exercise 1 part i
- Exercise 2 part a
- Exercise 2 part b
- Exercise 2 part c
- Exercise 2 part d
- Exercise 3

The Harmonic Series and The P-Series
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Algebraic Properties of Series
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The Divergence Test
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The Integral Test
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The Limit Comparison Test
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- Exercise 7 Part a
- Exercise 7 Part b
- Exercise 7 Part c
- Exercise 7 Part d
- Exercise 7 Part e
- Exercise 7 Part f
- Exercise 7 Part g
- Exercise 7 Part h
- Exercise 7 Part i
- Exercise 7 Part j
- Exercise 7 Part k
- Exercise 7 Part l
- Exercise 7 Part m
- Exercise 7 Part n
- Exercise 7 Part o
- Exercise 7 Part p
- Exercise 7 Part q
- Exercise 7 Part r

The Ratio Test
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The Alternating Series Test
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Absolute and Conditional Convergence of Series
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Sequences","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Sequence","Duration":"9m 42s","ChapterTopicVideoID":8181,"CourseChapterTopicPlaylistID":52089,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8181.jpeg","UploadDate":"2016-12-14T00:08:56.8770000","DurationForVideoObject":"PT9M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"In this clip, we\u0027re starting a new topic, sequences."},{"Start":"00:04.725 ","End":"00:09.945","Text":"A sequence is an ordered list of numbers and those numbers are called its terms."},{"Start":"00:09.945 ","End":"00:14.550","Text":"But in some places they\u0027re called members or elements of"},{"Start":"00:14.550 ","End":"00:19.500","Text":"the list and the list could be finite or infinite."},{"Start":"00:19.500 ","End":"00:24.645","Text":"But in this course, we\u0027re only going to be concerned with infinite sequences."},{"Start":"00:24.645 ","End":"00:28.370","Text":"Anyway, let\u0027s look at some examples."},{"Start":"00:28.370 ","End":"00:30.095","Text":"Here\u0027s the 1st one."},{"Start":"00:30.095 ","End":"00:33.605","Text":"The first term is 1 over 1,"},{"Start":"00:33.605 ","End":"00:35.870","Text":"the second term is 1.5,"},{"Start":"00:35.870 ","End":"00:41.410","Text":"then 1/4, 1/4 and you can guess the pattern and so on."},{"Start":"00:41.410 ","End":"00:43.935","Text":"In the second example,"},{"Start":"00:43.935 ","End":"00:47.190","Text":"the first term is 1 over 2,"},{"Start":"00:47.190 ","End":"00:49.160","Text":"2^11, 1 over 2 squared,"},{"Start":"00:49.160 ","End":"00:50.270","Text":"1 over 2 cubed,"},{"Start":"00:50.270 ","End":"00:52.520","Text":"1 over 2^4, and again,"},{"Start":"00:52.520 ","End":"00:53.870","Text":"you can probably guess the pattern,"},{"Start":"00:53.870 ","End":"00:58.740","Text":"the next element term would be 1 over 2 to the fifth."},{"Start":"00:59.170 ","End":"01:03.530","Text":"Third example, just alternating minus 1,"},{"Start":"01:03.530 ","End":"01:05.525","Text":"1, minus 1, 1,"},{"Start":"01:05.525 ","End":"01:08.600","Text":"minus 1, 1, and so on."},{"Start":"01:08.600 ","End":"01:11.555","Text":"Our last example here,"},{"Start":"01:11.555 ","End":"01:17.015","Text":"this is what it looks like and you can probably guess the pattern."},{"Start":"01:17.015 ","End":"01:23.450","Text":"For example, the next one in the sequence would be root 5 over 6."},{"Start":"01:23.450 ","End":"01:25.610","Text":"I think you get the idea."},{"Start":"01:25.610 ","End":"01:29.990","Text":"Now there\u0027s a notation for sequences on the terms of the sequence."},{"Start":"01:29.990 ","End":"01:33.890","Text":"I just want to remind you though that terms sometimes called members or"},{"Start":"01:33.890 ","End":"01:38.480","Text":"elements and we denote them as follows."},{"Start":"01:38.480 ","End":"01:42.350","Text":"The first term of the sequence is usually called a_1,"},{"Start":"01:42.350 ","End":"01:44.435","Text":"the second term a_2,"},{"Start":"01:44.435 ","End":"01:48.410","Text":"the third term a_3, and so on."},{"Start":"01:48.410 ","End":"01:51.185","Text":"But it doesn\u0027t have to be the letter a."},{"Start":"01:51.185 ","End":"01:54.650","Text":"You could have b_1 for the first term,"},{"Start":"01:54.650 ","End":"01:56.240","Text":"b_2, for the second term,"},{"Start":"01:56.240 ","End":"01:59.045","Text":"b_3, and so on."},{"Start":"01:59.045 ","End":"02:07.099","Text":"Some examples, and actually these are the same examples that we had above."},{"Start":"02:07.099 ","End":"02:10.040","Text":"I just added the notation so,"},{"Start":"02:10.040 ","End":"02:11.375","Text":"in the first sequence,"},{"Start":"02:11.375 ","End":"02:13.080","Text":"the first term is a_1,"},{"Start":"02:13.080 ","End":"02:14.340","Text":"next one is a_2,"},{"Start":"02:14.340 ","End":"02:16.395","Text":"a_3, a_4, and so on."},{"Start":"02:16.395 ","End":"02:18.620","Text":"If we have several sequences at once,"},{"Start":"02:18.620 ","End":"02:20.030","Text":"we don\u0027t want to mix the letters up,"},{"Start":"02:20.030 ","End":"02:21.470","Text":"so I\u0027ll give each one a different letter."},{"Start":"02:21.470 ","End":"02:24.969","Text":"This would be say, b_1, b_2, b_3, b_4,"},{"Start":"02:24.969 ","End":"02:27.965","Text":"and then here I used the letter c,"},{"Start":"02:27.965 ","End":"02:34.770","Text":"and here the letter d and the subscript indicates what the position is in the sequence."},{"Start":"02:35.000 ","End":"02:42.020","Text":"Next, I\u0027ll be talking about something called the general term of a sequence."},{"Start":"02:42.020 ","End":"02:44.270","Text":"Now we need to talk about a general term"},{"Start":"02:44.270 ","End":"02:46.220","Text":"because we don\u0027t if we just want to talk about the 1st,"},{"Start":"02:46.220 ","End":"02:47.945","Text":"2nd, 3rd, 4th terms."},{"Start":"02:47.945 ","End":"02:52.490","Text":"I want to talk about the 17th term, the 119th term."},{"Start":"02:52.490 ","End":"02:54.740","Text":"In general, for a number n,"},{"Start":"02:54.740 ","End":"03:00.605","Text":"I want to know what is the element or term in the nth place."},{"Start":"03:00.605 ","End":"03:02.420","Text":"Perhaps to have consistency,"},{"Start":"03:02.420 ","End":"03:10.605","Text":"I should just stick with the same term which is term member element term yeah."},{"Start":"03:10.605 ","End":"03:12.540","Text":"In the nth place,"},{"Start":"03:12.540 ","End":"03:15.110","Text":"and it\u0027s the nth element,"},{"Start":"03:15.110 ","End":"03:20.680","Text":"and it\u0027s usually denoted as a_n."},{"Start":"03:20.680 ","End":"03:25.850","Text":"In the nth place I have a and then a subscript n. Of course,"},{"Start":"03:25.850 ","End":"03:28.100","Text":"if I\u0027m not using a in this sequence,"},{"Start":"03:28.100 ","End":"03:34.190","Text":"so general term would be b_n or c_n or d_n, and so on."},{"Start":"03:34.190 ","End":"03:39.500","Text":"But the subscript denotes what place in the sequence we are."},{"Start":"03:39.500 ","End":"03:44.110","Text":"Now, I\u0027ll give some examples which are actually these examples."},{"Start":"03:44.110 ","End":"03:46.575","Text":"In this example here,"},{"Start":"03:46.575 ","End":"03:49.815","Text":"for example, the 4th term is 1 over 4."},{"Start":"03:49.815 ","End":"03:51.320","Text":"If you look at the pattern,"},{"Start":"03:51.320 ","End":"04:00.030","Text":"the nth term will be 1 over n. So a_n is 1 over n. Then the 2nd example here,"},{"Start":"04:00.030 ","End":"04:07.870","Text":"which you only went up to 4, you can see that in place n in the nth term is 1 over 2^n."},{"Start":"04:07.870 ","End":"04:13.895","Text":"Now given the other couple of examples which correspond to these two."},{"Start":"04:13.895 ","End":"04:17.420","Text":"For this sequence, which is an alternating sequence,"},{"Start":"04:17.420 ","End":"04:20.015","Text":"meaning it goes minus plus minus plus."},{"Start":"04:20.015 ","End":"04:26.240","Text":"The general term would be minus 1^n."},{"Start":"04:26.240 ","End":"04:30.140","Text":"Now, note that if n is odd,"},{"Start":"04:30.140 ","End":"04:35.260","Text":"then minus 1^n is minus 1 and if it\u0027s even,"},{"Start":"04:35.260 ","End":"04:40.480","Text":"it\u0027s plus 1 and luckily turned out right that the 1st place was minus,"},{"Start":"04:40.480 ","End":"04:42.010","Text":"the 2nd place was plus."},{"Start":"04:42.010 ","End":"04:46.430","Text":"On a remark, if you wanted to take it the other way around, 1,"},{"Start":"04:46.430 ","End":"04:51.440","Text":"minus 1,1 minus 1, et cetera."},{"Start":"04:51.440 ","End":"04:57.775","Text":"Then the way you do it is you would say that the general term is minus 1^n."},{"Start":"04:57.775 ","End":"05:03.450","Text":"You would add another minus and then it would reverse the odd and the even."},{"Start":"05:03.450 ","End":"05:06.830","Text":"The odd places would get the plus 1,"},{"Start":"05:06.830 ","End":"05:09.670","Text":"and the even places will get the minus 1."},{"Start":"05:09.670 ","End":"05:12.810","Text":"This happens a lot so often you want to adjust"},{"Start":"05:12.810 ","End":"05:17.895","Text":"minus 1^n to be minus 1^n plus 1 and reverse the sign."},{"Start":"05:17.895 ","End":"05:19.970","Text":"In this example also,"},{"Start":"05:19.970 ","End":"05:23.150","Text":"we can see the pattern under the square root."},{"Start":"05:23.150 ","End":"05:25.730","Text":"We see that we have 1, 2, 3, 4,"},{"Start":"05:25.730 ","End":"05:30.060","Text":"so in place n it would be n. On the denominator,"},{"Start":"05:30.060 ","End":"05:32.820","Text":"in place 1 we have 2, 2nd term is 3,"},{"Start":"05:32.820 ","End":"05:34.615","Text":"3rd term is 4, 4th term is 5,"},{"Start":"05:34.615 ","End":"05:36.995","Text":"so the nth term would be an n plus 1."},{"Start":"05:36.995 ","End":"05:44.630","Text":"This would be the nth term or the general term of this sequence."},{"Start":"05:44.630 ","End":"05:46.520","Text":"N is a convenient letter,"},{"Start":"05:46.520 ","End":"05:51.920","Text":"but sometimes it\u0027s convenient to use i or j or m,"},{"Start":"05:51.920 ","End":"05:53.900","Text":"unless we say otherwise we choose n,"},{"Start":"05:53.900 ","End":"05:58.825","Text":"and usually if a is free we\u0027ll use letter a."},{"Start":"05:58.825 ","End":"06:05.240","Text":"Next, I\u0027ll show you how to use the general term to describe the sequence."},{"Start":"06:05.240 ","End":"06:10.430","Text":"How do we describe a sequence by means of its general element?"},{"Start":"06:10.430 ","End":"06:12.230","Text":"In the previous examples,"},{"Start":"06:12.230 ","End":"06:18.070","Text":"we found the formula for the general term a_n in terms of n and from it,"},{"Start":"06:18.070 ","End":"06:21.020","Text":"you can compute a_1, a_2, a_3,"},{"Start":"06:21.020 ","End":"06:24.590","Text":"or any particular a by just letting n equal 1,"},{"Start":"06:24.590 ","End":"06:26.620","Text":"2, 3, or whatever."},{"Start":"06:26.620 ","End":"06:29.840","Text":"If I just give you the formula or describe"},{"Start":"06:29.840 ","End":"06:34.715","Text":"the general element a_n then I can figure out the whole sequence."},{"Start":"06:34.715 ","End":"06:38.250","Text":"For example, and this was our first example,"},{"Start":"06:38.250 ","End":"06:39.960","Text":"instead of writing 1,"},{"Start":"06:39.960 ","End":"06:42.185","Text":"1/2, 1/3, 1/4 or 1/5, and so on."},{"Start":"06:42.185 ","End":"06:44.750","Text":"All you have to do is say that a_n is"},{"Start":"06:44.750 ","End":"06:47.780","Text":"1 over n. Because then you can figure out all of these."},{"Start":"06:47.780 ","End":"06:51.590","Text":"You would say, the 1st element, I put an equals 1,"},{"Start":"06:51.590 ","End":"06:53.390","Text":"1 over 1, 2nd element,"},{"Start":"06:53.390 ","End":"06:55.535","Text":"n is 2,1 over 2, and so on."},{"Start":"06:55.535 ","End":"06:58.295","Text":"This is much more compact."},{"Start":"06:58.295 ","End":"07:04.525","Text":"Our 2nd example which was this is written more simply as,"},{"Start":"07:04.525 ","End":"07:07.170","Text":"I\u0027ll use a different letter than a is b."},{"Start":"07:07.170 ","End":"07:12.435","Text":"So b_n, the nth element is 1 over 2^n."},{"Start":"07:12.435 ","End":"07:16.530","Text":"Like the 4th element is 1 over 2^4 and so on."},{"Start":"07:16.530 ","End":"07:19.620","Text":"In our 3rd example is minus 1,1,"},{"Start":"07:19.620 ","End":"07:20.960","Text":"minus 1,1, and so on."},{"Start":"07:20.960 ","End":"07:23.059","Text":"We found that the general element,"},{"Start":"07:23.059 ","End":"07:27.415","Text":"general term c_n is minus1^n."},{"Start":"07:27.415 ","End":"07:34.580","Text":"If I just say this is more of a shorthand way in writing this whole thing with a dot,"},{"Start":"07:34.580 ","End":"07:36.575","Text":"dot, dot at the end."},{"Start":"07:36.575 ","End":"07:43.715","Text":"Now there are some other notations that we can use instead of writing a_n is 1 over n,"},{"Start":"07:43.715 ","End":"07:45.980","Text":"there are other possibilities,"},{"Start":"07:45.980 ","End":"07:50.390","Text":"and I\u0027ll illustrate the notations on the series here,"},{"Start":"07:50.390 ","End":"07:53.335","Text":"the 1/5, 1/3, and so on."},{"Start":"07:53.335 ","End":"07:58.235","Text":"What we wrote simply as a_n equals 1 over n,"},{"Start":"07:58.235 ","End":"08:01.775","Text":"can be written in some minor variations."},{"Start":"08:01.775 ","End":"08:06.170","Text":"The most similar one to this is this,"},{"Start":"08:06.170 ","End":"08:12.080","Text":"which is just putting this in a bracket to emphasize that we\u0027re talking about"},{"Start":"08:12.080 ","End":"08:18.720","Text":"the whole sequence and not about the particular general term, the nth term."},{"Start":"08:19.000 ","End":"08:25.310","Text":"This one is a bit more precise because we assume"},{"Start":"08:25.310 ","End":"08:31.130","Text":"always that we start from the first element and keep going to infinity,"},{"Start":"08:31.130 ","End":"08:33.905","Text":"1, 2, 3, 4, and so on."},{"Start":"08:33.905 ","End":"08:40.609","Text":"This just makes it precise that we do start at element 1 and keep going to infinity."},{"Start":"08:40.609 ","End":"08:43.880","Text":"Sometimes we don\u0027t start at number 1,"},{"Start":"08:43.880 ","End":"08:48.390","Text":"sometimes we start at 0 or at any other number."},{"Start":"08:48.390 ","End":"08:52.600","Text":"The 1st one, we could call it a_3, a_4, a_5."},{"Start":"08:53.030 ","End":"08:55.670","Text":"If we want to be really precise,"},{"Start":"08:55.670 ","End":"08:58.390","Text":"then this will tell us where we start and where we end,"},{"Start":"08:58.390 ","End":"09:01.920","Text":"and if up its infinity it means never-ending."},{"Start":"09:02.620 ","End":"09:06.005","Text":"This is the same thing as this one,"},{"Start":"09:06.005 ","End":"09:11.190","Text":"except that here I use curly brackets instead of round brackets."},{"Start":"09:12.350 ","End":"09:18.350","Text":"This is another variation of this one,"},{"Start":"09:18.350 ","End":"09:21.305","Text":"where instead of saying n goes from 1 to infinity,"},{"Start":"09:21.305 ","End":"09:24.605","Text":"I say, n bigger or equal to 1."},{"Start":"09:24.605 ","End":"09:28.550","Text":"We always assume that n is a whole number,"},{"Start":"09:28.550 ","End":"09:32.165","Text":"but it tells us it\u0027s all the whole numbers from 1 onwards."},{"Start":"09:32.165 ","End":"09:35.390","Text":"These are all variations you might see."},{"Start":"09:35.390 ","End":"09:43.080","Text":"In fact, I\u0027ll be using some of these in the rest of this topic."}],"ID":8335},{"Watched":false,"Name":"Examples of Sequences","Duration":"6m 19s","ChapterTopicVideoID":8182,"CourseChapterTopicPlaylistID":52089,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8182.jpeg","UploadDate":"2016-12-14T00:09:44.2000000","DurationForVideoObject":"PT6M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.300","Text":"In this clip, we\u0027ll be solving some example exercises and here\u0027s the first 1."},{"Start":"00:06.300 ","End":"00:11.010","Text":"We have to write the first 5 terms of the sequence, which is this."},{"Start":"00:11.010 ","End":"00:13.349","Text":"Notice I\u0027ve used the curly brace notation,"},{"Start":"00:13.349 ","End":"00:15.525","Text":"that\u0027s 1 of the several notations."},{"Start":"00:15.525 ","End":"00:21.345","Text":"Basically what it says is that the nth term is n plus 4 over n squared."},{"Start":"00:21.345 ","End":"00:25.350","Text":"What I want to do is plug in successively n equals 1, 2,"},{"Start":"00:25.350 ","End":"00:29.085","Text":"3, 4, 5 and here we are."},{"Start":"00:29.085 ","End":"00:31.980","Text":"Instead of n here and here,"},{"Start":"00:31.980 ","End":"00:35.250","Text":"I put 1 or 2 or 3 or 4 or 5,"},{"Start":"00:35.250 ","End":"00:37.620","Text":"then this is what I get for a_1,"},{"Start":"00:37.620 ","End":"00:40.410","Text":"a_2, a_3, a_4, a_5, first 5 terms."},{"Start":"00:40.410 ","End":"00:42.705","Text":"If I just simplify them,"},{"Start":"00:42.705 ","End":"00:47.025","Text":"then here\u0027s what we get like 1 plus 4 is 5 over 1,"},{"Start":"00:47.025 ","End":"00:50.100","Text":"2 plus 4, 6 over 4,"},{"Start":"00:50.100 ","End":"00:52.655","Text":"7 over 9, and so on."},{"Start":"00:52.655 ","End":"00:55.609","Text":"Now let\u0027s go on to the next example."},{"Start":"00:55.609 ","End":"00:57.590","Text":"In this next example,"},{"Start":"00:57.590 ","End":"01:00.560","Text":"we have to add the first 5 terms of this sequence."},{"Start":"01:00.560 ","End":"01:02.120","Text":"It\u0027s not a_n, it\u0027s b_n."},{"Start":"01:02.120 ","End":"01:08.005","Text":"That\u0027s okay and it\u0027s written with the notation n goes from 1 to infinity."},{"Start":"01:08.005 ","End":"01:10.470","Text":"We just want from 1 to 5,"},{"Start":"01:10.470 ","End":"01:14.240","Text":"so we just take this expression and plug in successively 1,"},{"Start":"01:14.240 ","End":"01:17.440","Text":"2, 3, 4, and 5. Here\u0027s what we get."},{"Start":"01:17.440 ","End":"01:19.430","Text":"Instead of the n, which is here and here,"},{"Start":"01:19.430 ","End":"01:21.395","Text":"I put 1 and then 2,"},{"Start":"01:21.395 ","End":"01:23.350","Text":"3, and so on."},{"Start":"01:23.350 ","End":"01:25.685","Text":"Now we just simplify a little bit."},{"Start":"01:25.685 ","End":"01:31.910","Text":"Note that minus 1 to an even power is plus 1."},{"Start":"01:31.910 ","End":"01:33.305","Text":"So we\u0027ll start with plus,"},{"Start":"01:33.305 ","End":"01:35.870","Text":"and minus 1 to an odd power will be minus."},{"Start":"01:35.870 ","End":"01:37.765","Text":"It will alternate."},{"Start":"01:37.765 ","End":"01:39.705","Text":"This is what we get."},{"Start":"01:39.705 ","End":"01:41.540","Text":"Of course the plus is unnecessary,"},{"Start":"01:41.540 ","End":"01:43.550","Text":"but it\u0027s just for emphasis that we have plus,"},{"Start":"01:43.550 ","End":"01:46.805","Text":"minus, plus, minus, plus."},{"Start":"01:46.805 ","End":"01:55.920","Text":"In this example, we\u0027re given a sequence c_n n bigger or equal to 1,"},{"Start":"01:55.920 ","End":"01:58.410","Text":"meaning n goes 1, 2, 3, 4, 5,"},{"Start":"01:58.410 ","End":"02:02.955","Text":"and so on and we\u0027re given c_n descriptively."},{"Start":"02:02.955 ","End":"02:05.760","Text":"We\u0027re told that c_n is the nth prime number."},{"Start":"02:05.760 ","End":"02:07.830","Text":"It\u0027s not a formula, but it\u0027s"},{"Start":"02:07.830 ","End":"02:11.180","Text":"definitely well defined because we know what the prime numbers are,"},{"Start":"02:11.180 ","End":"02:13.380","Text":"at least the first few."},{"Start":"02:13.570 ","End":"02:16.885","Text":"Assuming you know your primes,"},{"Start":"02:16.885 ","End":"02:20.265","Text":"c_1 is the first prime number is 2."},{"Start":"02:20.265 ","End":"02:22.640","Text":"I know some people think that 1 is a prime,"},{"Start":"02:22.640 ","End":"02:24.650","Text":"but it\u0027s not considered to be."},{"Start":"02:24.650 ","End":"02:26.360","Text":"Next prime number is 3,"},{"Start":"02:26.360 ","End":"02:28.825","Text":"then 5, then 7, then 11."},{"Start":"02:28.825 ","End":"02:31.490","Text":"If they ask for the 6th member,"},{"Start":"02:31.490 ","End":"02:35.180","Text":"for example, I could say c_6 is 13,"},{"Start":"02:35.180 ","End":"02:37.630","Text":"c_7 is the 7th prime,"},{"Start":"02:37.630 ","End":"02:42.975","Text":"it is 17, and so on but we\u0027re just asked for 5 so that\u0027ll do."},{"Start":"02:42.975 ","End":"02:47.420","Text":"This exercise is a little bit different to the previous ones."},{"Start":"02:47.420 ","End":"02:51.080","Text":"We also have a sequence and we want to write the first 5 terms."},{"Start":"02:51.080 ","End":"02:55.250","Text":"But this time, the sequence is defined recursively."},{"Start":"02:55.250 ","End":"02:57.725","Text":"I\u0027ll just highlight that term."},{"Start":"02:57.725 ","End":"03:02.840","Text":"What it means is that we don\u0027t have an explicit formula for"},{"Start":"03:02.840 ","End":"03:07.860","Text":"each n. What we do have is 1 explicit 1,"},{"Start":"03:07.860 ","End":"03:11.400","Text":"usually the 1st 1 is given explicitly a_1."},{"Start":"03:11.400 ","End":"03:17.120","Text":"But subsequent a_n\u0027s are given in terms of previous a_n\u0027s,"},{"Start":"03:17.120 ","End":"03:23.325","Text":"so we have to build up the sequence 1 by 1 and you\u0027ll see in a moment."},{"Start":"03:23.325 ","End":"03:26.220","Text":"Okay. As I said, the 1st 1 is just given to us,"},{"Start":"03:26.220 ","End":"03:27.960","Text":"so we just copy that."},{"Start":"03:27.960 ","End":"03:30.460","Text":"Now, for the next 1,"},{"Start":"03:30.460 ","End":"03:32.910","Text":"how do we find a_2?"},{"Start":"03:32.910 ","End":"03:39.525","Text":"We can get a_2 if we put n equals 1 in this formula, and we get."},{"Start":"03:39.525 ","End":"03:43.370","Text":"Notice that everywhere there\u0027s n I\u0027m putting 1 and it\u0027s colored in blue."},{"Start":"03:43.370 ","End":"03:45.785","Text":"a_1 plus 1, which is a_2,"},{"Start":"03:45.785 ","End":"03:51.450","Text":"will be given from a_1 plus 2 times 1."},{"Start":"03:51.450 ","End":"03:55.780","Text":"Now, we know a_1 because it\u0027s written here."},{"Start":"03:55.940 ","End":"04:00.780","Text":"So a_1 is 1 and 2 times 1 is 2,"},{"Start":"04:00.780 ","End":"04:03.825","Text":"so we have 1 plus 2 is 3."},{"Start":"04:03.825 ","End":"04:06.790","Text":"So a_2 is 3."},{"Start":"04:06.790 ","End":"04:08.360","Text":"But we didn\u0027t find it directly."},{"Start":"04:08.360 ","End":"04:12.739","Text":"We found it in terms of the previous n. In fact, this,"},{"Start":"04:12.739 ","End":"04:15.540","Text":"and I\u0027ll highlight it,"},{"Start":"04:16.090 ","End":"04:20.795","Text":"is a special case of a recursive formula."},{"Start":"04:20.795 ","End":"04:25.220","Text":"Sometimes we say recursive formula,"},{"Start":"04:25.220 ","End":"04:29.165","Text":"but sometimes we say recursive rule."},{"Start":"04:29.165 ","End":"04:30.800","Text":"Anyway, don\u0027t worry about the name,"},{"Start":"04:30.800 ","End":"04:33.665","Text":"that\u0027s just learn how to use such a thing."},{"Start":"04:33.665 ","End":"04:35.150","Text":"Let\u0027s go on to the next 1."},{"Start":"04:35.150 ","End":"04:37.415","Text":"If I want a_3,"},{"Start":"04:37.415 ","End":"04:43.025","Text":"what I have to do is let n equals 2 because then 2 plus 1 is 3."},{"Start":"04:43.025 ","End":"04:47.059","Text":"So we get, if n is 2 in the recursive formula,"},{"Start":"04:47.059 ","End":"04:52.595","Text":"a_2 plus 1 equals a_2 plus twice 2."},{"Start":"04:52.595 ","End":"04:56.075","Text":"Now a_2 we found already is 3,"},{"Start":"04:56.075 ","End":"04:57.350","Text":"so we put that here,"},{"Start":"04:57.350 ","End":"04:59.300","Text":"and twice 2 is 4 is 7."},{"Start":"04:59.300 ","End":"05:03.740","Text":"We got a_3 from previous value a_2."},{"Start":"05:03.740 ","End":"05:07.190","Text":"We still have to do 2 more of these to get a_4 and a_5."},{"Start":"05:07.190 ","End":"05:09.590","Text":"To get a_4, I think you get the idea,"},{"Start":"05:09.590 ","End":"05:11.965","Text":"we let n equals 3."},{"Start":"05:11.965 ","End":"05:14.490","Text":"So n equals 3,"},{"Start":"05:14.490 ","End":"05:19.095","Text":"a_3 plus 1 is a_3 plus twice 3."},{"Start":"05:19.095 ","End":"05:22.050","Text":"We have a_3 because it\u0027s here, it\u0027s 7."},{"Start":"05:22.050 ","End":"05:30.280","Text":"I should have really been indicating that I get the 7 from here like a_2 was 3."},{"Start":"05:30.280 ","End":"05:37.890","Text":"That was what gave me this and the 1st 1 is what gave me this here."},{"Start":"05:37.890 ","End":"05:43.200","Text":"Okay. So a_4 is 13,"},{"Start":"05:43.200 ","End":"05:47.410","Text":"here is 7 plus 6, and we have 1 more."},{"Start":"05:47.810 ","End":"05:50.910","Text":"This time we put n equals 4."},{"Start":"05:50.910 ","End":"05:52.605","Text":"If we want to get a_5,"},{"Start":"05:52.605 ","End":"05:56.175","Text":"so a_4 plus 1 is a_n, which is a_4,"},{"Start":"05:56.175 ","End":"06:01.500","Text":"plus twice 4 and a_ 4 is 13,"},{"Start":"06:01.500 ","End":"06:03.240","Text":"which I got from here."},{"Start":"06:03.240 ","End":"06:07.050","Text":"Twice 4 is 8, 13 and 8 is 21."},{"Start":"06:07.050 ","End":"06:12.120","Text":"Now we have all the first 5 elements,"},{"Start":"06:12.120 ","End":"06:16.485","Text":"terms, members of the sequence."},{"Start":"06:16.485 ","End":"06:20.020","Text":"That\u0027s it for this exercise."}],"ID":8336},{"Watched":false,"Name":"Defining a Sequence as a Function","Duration":"3m 21s","ChapterTopicVideoID":8183,"CourseChapterTopicPlaylistID":52089,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8183.jpeg","UploadDate":"2016-12-14T00:10:08.5430000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.750","Text":"Up to now, we\u0027ve been using a sequence as an ordered list of numbers."},{"Start":"00:07.750 ","End":"00:11.485","Text":"There is another mathematical way of defining a sequence."},{"Start":"00:11.485 ","End":"00:16.010","Text":"A sequence can be considered as a function of sorts."},{"Start":"00:16.310 ","End":"00:19.030","Text":"We\u0027ll illustrate it on an example."},{"Start":"00:19.030 ","End":"00:27.120","Text":"Let\u0027s suppose we have this sequence where the nth term a_n is n plus 4 over n squared."},{"Start":"00:27.120 ","End":"00:31.989","Text":"We can actually view it as a function from the natural numbers."},{"Start":"00:31.989 ","End":"00:33.820","Text":"N is a natural number."},{"Start":"00:33.820 ","End":"00:35.760","Text":"After I\u0027ve made the computation,"},{"Start":"00:35.760 ","End":"00:38.280","Text":"I get a real number, so I have a function."},{"Start":"00:38.280 ","End":"00:40.495","Text":"The domain is natural numbers,"},{"Start":"00:40.495 ","End":"00:42.430","Text":"the range is real numbers,"},{"Start":"00:42.430 ","End":"00:46.985","Text":"and we can write it as f of n equals n plus 4 over n squared."},{"Start":"00:46.985 ","End":"00:54.450","Text":"We define instead of a_n as the nth member of the sequence we, in this notation,"},{"Start":"00:54.450 ","End":"00:57.225","Text":"would say f of n."},{"Start":"00:57.225 ","End":"01:03.065","Text":"This interpretation of a sequence as a function will be useful later on."},{"Start":"01:03.065 ","End":"01:07.460","Text":"For example, it will help us to determine if a sequence is"},{"Start":"01:07.460 ","End":"01:12.965","Text":"increasing or decreasing based on the derivative of this function."},{"Start":"01:12.965 ","End":"01:17.059","Text":"Just point out that it\u0027s not necessarily the natural numbers"},{"Start":"01:17.059 ","End":"01:22.445","Text":"because this notation doesn\u0027t tell us where a runs from."},{"Start":"01:22.445 ","End":"01:24.365","Text":"If I said a_n."},{"Start":"01:24.365 ","End":"01:27.955","Text":"N goes from 1 to infinity,"},{"Start":"01:27.955 ","End":"01:30.200","Text":"then it would be the natural numbers."},{"Start":"01:30.200 ","End":"01:31.579","Text":"But as I mentioned,"},{"Start":"01:31.579 ","End":"01:34.850","Text":"the series doesn\u0027t necessarily start at 1 even if it\u0027s infinite,"},{"Start":"01:34.850 ","End":"01:36.469","Text":"it could start at 0,"},{"Start":"01:36.469 ","End":"01:40.560","Text":"or at 3, or something else in which case we just modify it"},{"Start":"01:40.560 ","End":"01:42.720","Text":"slightly and say not the natural numbers but"},{"Start":"01:42.720 ","End":"01:45.390","Text":"the natural numbers from 3 onwards or something."},{"Start":"01:45.390 ","End":"01:46.710","Text":"I don\u0027t want to get too technical,"},{"Start":"01:46.710 ","End":"01:49.620","Text":"we\u0027ll just stick to the natural numbers to the real numbers,"},{"Start":"01:49.620 ","End":"01:56.090","Text":"and I want to move on and talk about the sketch or graph of such a function."},{"Start":"01:56.090 ","End":"01:58.585","Text":"I\u0027ll just show it to you right away."},{"Start":"01:58.585 ","End":"02:01.940","Text":"We see it\u0027s just a set of isolated points."},{"Start":"02:01.940 ","End":"02:04.385","Text":"That\u0027s what the graph of a sequence looks like."},{"Start":"02:04.385 ","End":"02:10.710","Text":"We only have values for n which is a whole number 1, 2, 3, 4."},{"Start":"02:10.710 ","End":"02:12.410","Text":"We don\u0027t have anything in between,"},{"Start":"02:12.410 ","End":"02:16.600","Text":"so we don\u0027t join the points with a curve."},{"Start":"02:16.600 ","End":"02:19.200","Text":"The way we get it, it\u0027s just regular."},{"Start":"02:19.200 ","End":"02:21.180","Text":"We plug in values."},{"Start":"02:21.180 ","End":"02:23.170","Text":"Let\u0027s say when n is 1,"},{"Start":"02:23.170 ","End":"02:27.455","Text":"then we\u0027d get 1 plus 4 over 1 squared is 5."},{"Start":"02:27.455 ","End":"02:30.080","Text":"I\u0027ll just write the y value is 5."},{"Start":"02:30.080 ","End":"02:31.985","Text":"Of course, the point is 1, 5."},{"Start":"02:31.985 ","End":"02:36.155","Text":"I want to plug in 2. 2 plus 6 over 4 that\u0027s 1/2."},{"Start":"02:36.155 ","End":"02:39.950","Text":"Plug in 3 I would get 7/9,"},{"Start":"02:39.950 ","End":"02:43.020","Text":"and so on. That\u0027s what it looks like."},{"Start":"02:43.020 ","End":"02:49.690","Text":"However, if you did join it with a curve that would give you something else."},{"Start":"02:49.690 ","End":"02:53.165","Text":"That would give you a function from the real numbers to the real numbers."},{"Start":"02:53.165 ","End":"02:56.940","Text":"f of x is x plus 4 over x squared,"},{"Start":"02:56.940 ","End":"02:59.705","Text":"and that\u0027s something totally different."},{"Start":"02:59.705 ","End":"03:05.930","Text":"But they are related and we will be using the properties of the derivative,"},{"Start":"03:05.930 ","End":"03:12.965","Text":"for example, to show that a sequence is decreasing if the function is decreasing."},{"Start":"03:12.965 ","End":"03:16.340","Text":"Anyway, you\u0027ll see more about that later."},{"Start":"03:16.340 ","End":"03:19.250","Text":"That\u0027s assuming you\u0027ve studied derivatives."},{"Start":"03:19.250 ","End":"03:22.320","Text":"I\u0027m just going to end the clip here."}],"ID":8337},{"Watched":false,"Name":"Limit of a Sequence","Duration":"3m 5s","ChapterTopicVideoID":8184,"CourseChapterTopicPlaylistID":52089,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8184.jpeg","UploadDate":"2016-12-14T00:10:35.8300000","DurationForVideoObject":"PT3M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this clip, we\u0027ll be talking about the limit of a sequence and it\u0027s"},{"Start":"00:04.440 ","End":"00:08.430","Text":"got some similarity with limits of functions, as we shall see."},{"Start":"00:08.430 ","End":"00:10.665","Text":"Anyway, we\u0027ll start with an example."},{"Start":"00:10.665 ","End":"00:16.440","Text":"The familiar 1 over n sequence in the nth place,"},{"Start":"00:16.440 ","End":"00:18.300","Text":"we\u0027ll write a few terms,"},{"Start":"00:18.300 ","End":"00:21.330","Text":"starts off 1, 1/2, 1/3 and so on,"},{"Start":"00:21.330 ","End":"00:23.370","Text":"and very quickly we get to small numbers,"},{"Start":"00:23.370 ","End":"00:25.950","Text":"1 over 400 ,1 over 10,000,"},{"Start":"00:25.950 ","End":"00:28.215","Text":"1 over a billion."},{"Start":"00:28.215 ","End":"00:30.180","Text":"If we keep writing them,"},{"Start":"00:30.180 ","End":"00:31.230","Text":"keep adding more terms."},{"Start":"00:31.230 ","End":"00:38.340","Text":"I think it\u0027s intuitively obvious anyway that they approach 0,"},{"Start":"00:38.340 ","End":"00:41.230","Text":"get closer and closer to 0."},{"Start":"00:42.110 ","End":"00:46.730","Text":"We say that the limit of this sequence is 0,"},{"Start":"00:46.730 ","End":"00:51.725","Text":"and we use a notation similar to functions."},{"Start":"00:51.725 ","End":"00:57.880","Text":"We say that the limit as n goes to infinity of 1 over n is 0,"},{"Start":"00:57.880 ","End":"01:01.880","Text":"and there\u0027s an alternative notation."},{"Start":"01:01.880 ","End":"01:06.050","Text":"Sorry, this arrow doesn\u0027t have to be quite so long."},{"Start":"01:06.050 ","End":"01:11.555","Text":"1 over n tends to 0 as n tends to infinity."},{"Start":"01:11.555 ","End":"01:14.940","Text":"This notation, you don\u0027t need the word lim."},{"Start":"01:15.330 ","End":"01:19.940","Text":"I want to introduce another term, converges."},{"Start":"01:19.940 ","End":"01:23.570","Text":"We say that the sequence converges to 0."},{"Start":"01:23.570 ","End":"01:26.540","Text":"If it has a limit of 0, then it converges to 0."},{"Start":"01:26.540 ","End":"01:28.520","Text":"Nothing special about 0."},{"Start":"01:28.520 ","End":"01:30.200","Text":"We could say if the limit was 3,"},{"Start":"01:30.200 ","End":"01:32.030","Text":"then it would converge to 3."},{"Start":"01:32.030 ","End":"01:34.850","Text":"But if we don\u0027t care what it converges to,"},{"Start":"01:34.850 ","End":"01:38.705","Text":"we just say that the sequence is convergent."},{"Start":"01:38.705 ","End":"01:41.040","Text":"It has a limit."},{"Start":"01:41.160 ","End":"01:44.710","Text":"Now, not every sequence has a limit,"},{"Start":"01:44.710 ","End":"01:48.920","Text":"this 1 did, but there are examples which don\u0027t."},{"Start":"01:48.920 ","End":"01:52.450","Text":"For example, look at the following sequence,"},{"Start":"01:52.450 ","End":"01:54.430","Text":"the 1 that alternates 1 minus 1,"},{"Start":"01:54.430 ","End":"01:56.965","Text":"1 minus 1, and so on."},{"Start":"01:56.965 ","End":"02:00.835","Text":"I claim that it does not have a limit,"},{"Start":"02:00.835 ","End":"02:02.920","Text":"and why it doesn\u0027t it have a limit, although,"},{"Start":"02:02.920 ","End":"02:05.875","Text":"I think it\u0027s fairly intuitive?"},{"Start":"02:05.875 ","End":"02:09.640","Text":"Well, it keeps alternating between 1 and minus 1,"},{"Start":"02:09.640 ","End":"02:13.420","Text":"so it doesn\u0027t approach any 1 single number."},{"Start":"02:13.420 ","End":"02:21.070","Text":"It approaches 2 different numbers so it doesn\u0027t have a limit."},{"Start":"02:21.070 ","End":"02:23.980","Text":"If it doesn\u0027t have a limit, It\u0027s not convergent."},{"Start":"02:23.980 ","End":"02:26.360","Text":"There\u0027s a name for not convergent."},{"Start":"02:26.360 ","End":"02:29.600","Text":"Instead of that, we say divergent."},{"Start":"02:29.600 ","End":"02:32.240","Text":"If it\u0027s not convergent, it\u0027s divergent,"},{"Start":"02:32.240 ","End":"02:36.170","Text":"I\u0027d like to mention that there are a whole bunch of techniques you"},{"Start":"02:36.170 ","End":"02:40.160","Text":"learned for computing the limit of a function at infinity."},{"Start":"02:40.160 ","End":"02:41.600","Text":"In the case of functions,"},{"Start":"02:41.600 ","End":"02:44.560","Text":"you would have x goes to infinity."},{"Start":"02:44.560 ","End":"02:48.170","Text":"Pretty much with only minor modifications,"},{"Start":"02:48.170 ","End":"02:51.965","Text":"all that theory applies for limit of sequences,"},{"Start":"02:51.965 ","End":"02:54.040","Text":"so we\u0027re not going to repeat it."},{"Start":"02:54.040 ","End":"02:59.929","Text":"But basically, limits of sequences are like limits of functions when x goes to infinity."},{"Start":"02:59.929 ","End":"03:05.520","Text":"That\u0027s all I want to say about limits of sequences of this stage, so we\u0027re done."}],"ID":8338},{"Watched":false,"Name":"Increasing and Decreasing Sequences","Duration":"7m 40s","ChapterTopicVideoID":8185,"CourseChapterTopicPlaylistID":52089,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8185.jpeg","UploadDate":"2016-12-14T00:11:37.8570000","DurationForVideoObject":"PT7M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.035","Text":"Our next topic as far as sequences go,"},{"Start":"00:04.035 ","End":"00:07.455","Text":"is increasing and decreasing sequences."},{"Start":"00:07.455 ","End":"00:11.100","Text":"These occur quite often and they\u0027re particularly"},{"Start":"00:11.100 ","End":"00:14.490","Text":"useful in theorems on limits of sequences."},{"Start":"00:14.490 ","End":"00:18.210","Text":"Many theorems require a sequence to be increasing or decreasing,"},{"Start":"00:18.210 ","End":"00:20.265","Text":"then we can deduce a lot from it."},{"Start":"00:20.265 ","End":"00:22.560","Text":"Let\u0027s give some examples."},{"Start":"00:22.560 ","End":"00:28.845","Text":"Here are 3 different sequences to consider."},{"Start":"00:28.845 ","End":"00:32.010","Text":"In the first sequence, 2, 4, 6,"},{"Start":"00:32.010 ","End":"00:34.925","Text":"8 and assuming this pattern continues,"},{"Start":"00:34.925 ","End":"00:43.780","Text":"each term in the sequence is bigger than the previous 1 and so in this case,"},{"Start":"00:43.780 ","End":"00:50.320","Text":"we say that the sequence is increasing and in the second 1,"},{"Start":"00:50.320 ","End":"00:51.350","Text":"it\u0027s going to be the opposite."},{"Start":"00:51.350 ","End":"00:59.345","Text":"If you look at it, then each term is smaller than the previous 1 and so it\u0027s decreasing."},{"Start":"00:59.345 ","End":"01:00.950","Text":"But in the third case,"},{"Start":"01:00.950 ","End":"01:03.170","Text":"it\u0027s going down then up."},{"Start":"01:03.170 ","End":"01:06.950","Text":"It\u0027s neither increasing nor decreasing."},{"Start":"01:06.950 ","End":"01:12.825","Text":"Now let\u0027s see if we can write something a little more formal than that."},{"Start":"01:12.825 ","End":"01:16.775","Text":"Suppose we have sequence a_n,"},{"Start":"01:16.775 ","End":"01:21.094","Text":"it\u0027s called increasing if each term"},{"Start":"01:21.094 ","End":"01:26.435","Text":"is less than or equal to the following term or if you look at the other way around,"},{"Start":"01:26.435 ","End":"01:30.860","Text":"the following term is bigger or equal to the current term."},{"Start":"01:30.860 ","End":"01:33.055","Text":"If this is true for all n,"},{"Start":"01:33.055 ","End":"01:36.145","Text":"then it\u0027s increasing and"},{"Start":"01:36.145 ","End":"01:41.060","Text":"decreasing means that each term is bigger than the following term."},{"Start":"01:41.060 ","End":"01:47.510","Text":"Or I\u0027d rather say the following term is always less than or equal to the current term."},{"Start":"01:47.510 ","End":"01:50.460","Text":"If it\u0027s increasing, if n is 1,"},{"Start":"01:50.460 ","End":"01:55.350","Text":"we have here a_1 and less than or equal to a_2."},{"Start":"01:55.350 ","End":"01:56.790","Text":"If put n equals 2,"},{"Start":"01:56.790 ","End":"02:04.425","Text":"we get that a_2 is less than or equal to a_3, and so on."},{"Start":"02:04.425 ","End":"02:07.700","Text":"Here the decreasing, if it at n equal 1,"},{"Start":"02:07.700 ","End":"02:10.625","Text":"we get that a_1 bigger or equal to a_2,"},{"Start":"02:10.625 ","End":"02:18.300","Text":"put an equals 2 you get a_2 bigger or equal to a_3, and so on."},{"Start":"02:18.650 ","End":"02:24.140","Text":"Note that in each of these inequalities are used less than or equal,"},{"Start":"02:24.140 ","End":"02:26.030","Text":"greater than or equal."},{"Start":"02:26.030 ","End":"02:30.080","Text":"We\u0027re going to say that a sequence is"},{"Start":"02:30.080 ","End":"02:35.340","Text":"strictly increasing if we have the strict inequality here."},{"Start":"02:35.340 ","End":"02:40.924","Text":"Similarly, if it\u0027s a strict inequality that each successive term gets smaller,"},{"Start":"02:40.924 ","End":"02:43.070","Text":"then we say strictly decreasing."},{"Start":"02:43.070 ","End":"02:48.499","Text":"That\u0027s the word strictly if the inequalities are strict."},{"Start":"02:48.499 ","End":"02:53.960","Text":"If a sequence is either increasing or decreasing,"},{"Start":"02:53.960 ","End":"02:55.865","Text":"there\u0027s a common name for that."},{"Start":"02:55.865 ","End":"02:59.170","Text":"It\u0027s called a monotonic sequence."},{"Start":"02:59.170 ","End":"03:02.000","Text":"I think you can even say strictly"},{"Start":"03:02.000 ","End":"03:05.675","Text":"monotonic if it\u0027s strictly increasing or strictly decreasing,"},{"Start":"03:05.675 ","End":"03:08.510","Text":"but this is getting too subtle."},{"Start":"03:08.510 ","End":"03:13.040","Text":"Now I want to know if you\u0027ve learned differentiation already,"},{"Start":"03:13.040 ","End":"03:17.870","Text":"then there are quite a few tools to decide if"},{"Start":"03:17.870 ","End":"03:20.855","Text":"a sequence is increasing or decreasing"},{"Start":"03:20.855 ","End":"03:24.395","Text":"using the derivative or what is called differentiation,"},{"Start":"03:24.395 ","End":"03:27.580","Text":"this will turn up in the following example."},{"Start":"03:27.580 ","End":"03:30.695","Text":"Here\u0027s our example exercise."},{"Start":"03:30.695 ","End":"03:32.240","Text":"We have to determine if"},{"Start":"03:32.240 ","End":"03:37.955","Text":"the following sequence is increasing or decreasing or it could be neither."},{"Start":"03:37.955 ","End":"03:40.885","Text":"Let\u0027s write a few terms and see what\u0027s going on."},{"Start":"03:40.885 ","End":"03:44.300","Text":"The numerator is always 1 more than the denominator,"},{"Start":"03:44.300 ","End":"03:48.025","Text":"2/1, 3/2, 4/3, 5/4,"},{"Start":"03:48.025 ","End":"03:50.300","Text":"looks like it\u0027s decreasing."},{"Start":"03:50.300 ","End":"03:53.465","Text":"We\u0027ll prove it in 2 ways."},{"Start":"03:53.465 ","End":"03:58.740","Text":"Let\u0027s first of all do the first way which is straight from the definition."},{"Start":"04:01.820 ","End":"04:07.445","Text":"Actually I didn\u0027t say strictly so we can have bigger or equal to."},{"Start":"04:07.445 ","End":"04:10.490","Text":"This is actually going to be strictly decreasing."},{"Start":"04:10.490 ","End":"04:14.965","Text":"Anyway, don\u0027t worry about the greater than or greater than or equal to."},{"Start":"04:14.965 ","End":"04:20.239","Text":"This is what we have to prove and then just rewrite it in terms of the definition."},{"Start":"04:20.239 ","End":"04:22.805","Text":"A_n is this, I just copied it,"},{"Start":"04:22.805 ","End":"04:27.155","Text":"A_n plus 1 means replace n by n plus 1."},{"Start":"04:27.155 ","End":"04:29.580","Text":"We have n plus 1 plus 1,"},{"Start":"04:29.580 ","End":"04:32.490","Text":"which is I\u0027m plus 2 and here m plus 1."},{"Start":"04:32.490 ","End":"04:34.215","Text":"If we can show this,"},{"Start":"04:34.215 ","End":"04:36.315","Text":"then this is true."},{"Start":"04:36.315 ","End":"04:41.160","Text":"All these quantities are positive, especially the denominators."},{"Start":"04:41.160 ","End":"04:45.935","Text":"We can say that this is bigger than this if and only if"},{"Start":"04:45.935 ","End":"04:50.960","Text":"the cross multiplication this times this is bigger than this times this,"},{"Start":"04:50.960 ","End":"04:57.190","Text":"it\u0027s equivalent and let\u0027s see if we can show that."},{"Start":"04:57.190 ","End":"05:02.450","Text":"Expanding this is equivalent to showing that this is bigger than this."},{"Start":"05:02.450 ","End":"05:07.475","Text":"Subtract m squared plus 2n from both sides and get 1 is bigger than 0."},{"Start":"05:07.475 ","End":"05:10.945","Text":"Anyway, it\u0027s obvious and that concludes the proof."},{"Start":"05:10.945 ","End":"05:12.230","Text":"Because 1 is bigger than 0,"},{"Start":"05:12.230 ","End":"05:14.600","Text":"this is bigger than this, so this is bigger than this."},{"Start":"05:14.600 ","End":"05:19.925","Text":"We trace it backwards we\u0027ve got a_n is actually strictly bigger than a_n plus 1."},{"Start":"05:19.925 ","End":"05:24.875","Text":"We\u0027ve even proved that the sequence is strictly decreasing."},{"Start":"05:24.875 ","End":"05:27.995","Text":"Now let\u0027s do the solution another way,"},{"Start":"05:27.995 ","End":"05:31.760","Text":"but only if you\u0027ve learned derivatives and differentiation."},{"Start":"05:31.760 ","End":"05:37.250","Text":"If you haven\u0027t learned derivatives yet then skip the rest of this clip."},{"Start":"05:37.250 ","End":"05:47.510","Text":"I just want to remind you which is scrolled off that a_n was n plus 1."},{"Start":"05:47.510 ","End":"05:52.205","Text":"Now remember we talked about sequences of functions."},{"Start":"05:52.205 ","End":"05:57.200","Text":"This n plus 1 is a function of n. What we\u0027re going to do"},{"Start":"05:57.200 ","End":"06:02.820","Text":"is extend it to a function of real numbers."},{"Start":"06:02.820 ","End":"06:07.335","Text":"All x\u0027s or at least x bigger or equal to 1,"},{"Start":"06:07.335 ","End":"06:11.270","Text":"we\u0027ll define this function and if"},{"Start":"06:11.270 ","End":"06:15.515","Text":"x is bigger or equal to 1 and we won\u0027t have any problem with the denominator either,"},{"Start":"06:15.515 ","End":"06:22.390","Text":"and then a_n is just f of n. Now we\u0027re going to use derivatives to"},{"Start":"06:22.390 ","End":"06:29.210","Text":"show that f is decreasing as a function of x for x bigger or equal to 1."},{"Start":"06:29.210 ","End":"06:31.045","Text":"To show it\u0027s decreasing,"},{"Start":"06:31.045 ","End":"06:35.705","Text":"we show that the derivative is negative."},{"Start":"06:35.705 ","End":"06:39.745","Text":"Here\u0027s a computation of the derivative."},{"Start":"06:39.745 ","End":"06:42.805","Text":"Hope you remember the quotient rule."},{"Start":"06:42.805 ","End":"06:44.859","Text":"This is a quotient,"},{"Start":"06:44.859 ","End":"06:49.675","Text":"so we take the derivative of the numerator times denominator"},{"Start":"06:49.675 ","End":"06:56.935","Text":"minus the derivative of the denominator times the numerator over the denominator squared."},{"Start":"06:56.935 ","End":"07:00.235","Text":"Anyway, it comes out to be minus 1/x squared."},{"Start":"07:00.235 ","End":"07:02.430","Text":"This thing is positive."},{"Start":"07:02.430 ","End":"07:04.850","Text":"Minus 1 is negative,"},{"Start":"07:04.850 ","End":"07:06.410","Text":"negative and positive is negative."},{"Start":"07:06.410 ","End":"07:08.330","Text":"It\u0027s always negative."},{"Start":"07:08.330 ","End":"07:13.125","Text":"So the derivative of the function is negative."},{"Start":"07:13.125 ","End":"07:15.800","Text":"The function of x is decreasing."},{"Start":"07:15.800 ","End":"07:20.600","Text":"If f is decreasing for all x bigger or equal to 1,"},{"Start":"07:20.600 ","End":"07:22.955","Text":"it\u0027s also decreasing if I just take"},{"Start":"07:22.955 ","End":"07:29.360","Text":"whole numbers and the sequence is decreasing for n equals 1,"},{"Start":"07:29.360 ","End":"07:31.710","Text":"2, 3, and so on."},{"Start":"07:33.040 ","End":"07:41.290","Text":"That\u0027s that for increasing and decreasing and that concludes this clip."}],"ID":8339},{"Watched":false,"Name":"Bounded Sequences","Duration":"15m 38s","ChapterTopicVideoID":8186,"CourseChapterTopicPlaylistID":52089,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8186.jpeg","UploadDate":"2016-12-14T00:13:44.2400000","DurationForVideoObject":"PT15M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this clip, we\u0027ll talk about bounded sequences."},{"Start":"00:03.720 ","End":"00:06.050","Text":"We\u0027ve already learned increasing sequences,"},{"Start":"00:06.050 ","End":"00:09.090","Text":"decreasing sequences, limits of the sequence."},{"Start":"00:09.090 ","End":"00:12.195","Text":"Now it\u0027s time for bounded sequences."},{"Start":"00:12.195 ","End":"00:16.440","Text":"First I need to talk about upper and lower bounds."},{"Start":"00:16.440 ","End":"00:18.480","Text":"To help me illustrate this concept,"},{"Start":"00:18.480 ","End":"00:20.670","Text":"let\u0027s consider the following 2 sequences."},{"Start":"00:20.670 ","End":"00:21.990","Text":"This one, 2, 4, 6,"},{"Start":"00:21.990 ","End":"00:23.940","Text":"8, the even numbers,"},{"Start":"00:23.940 ","End":"00:25.560","Text":"and here 1, 1/2,"},{"Start":"00:25.560 ","End":"00:28.600","Text":"1/3, 1/4, we\u0027ve seen this before."},{"Start":"00:29.870 ","End":"00:33.420","Text":"We\u0027ll consider the first sequence."},{"Start":"00:33.420 ","End":"00:37.535","Text":"First, I\u0027m going to use it to illustrate the concept of a lower bound."},{"Start":"00:37.535 ","End":"00:40.460","Text":"Notice that all the elements,"},{"Start":"00:40.460 ","End":"00:42.320","Text":"no matter how far I go,"},{"Start":"00:42.320 ","End":"00:44.840","Text":"they\u0027re all bigger or equal to 2,"},{"Start":"00:44.840 ","End":"00:47.010","Text":"there are at least 2."},{"Start":"00:47.260 ","End":"00:49.340","Text":"This being the case,"},{"Start":"00:49.340 ","End":"00:56.850","Text":"we say that the sequence is bounded below, that\u0027s general."},{"Start":"00:58.640 ","End":"01:03.980","Text":"Specifically, 2 is a lower bound of the sequence."},{"Start":"01:03.980 ","End":"01:10.355","Text":"Because 2 is less than or equal to all the elements of the sequence, all the terms."},{"Start":"01:10.355 ","End":"01:13.130","Text":"Now, lower bounds are not unique."},{"Start":"01:13.130 ","End":"01:15.775","Text":"If I have a lower bound in this case 2"},{"Start":"01:15.775 ","End":"01:18.800","Text":"anything smaller than 2 will also be a lower bound."},{"Start":"01:18.800 ","End":"01:21.740","Text":"For example, 1 is also a lower bound because it\u0027s"},{"Start":"01:21.740 ","End":"01:25.340","Text":"also less than or equal to all the elements in the sequence."},{"Start":"01:25.340 ","End":"01:27.595","Text":"That\u0027s just something to note."},{"Start":"01:27.595 ","End":"01:36.080","Text":"I\u0027m going to use the second sequence to illustrate the concept of an upper bound."},{"Start":"01:36.080 ","End":"01:40.895","Text":"Notice that in the second sequence,"},{"Start":"01:40.895 ","End":"01:44.270","Text":"all the elements are less than or equal to 1,"},{"Start":"01:44.270 ","End":"01:47.880","Text":"they\u0027re all at most 1."},{"Start":"01:47.990 ","End":"01:52.235","Text":"In this case, we say the sequence is bounded from above"},{"Start":"01:52.235 ","End":"01:56.590","Text":"and 1 is an upper bound of the sequence."},{"Start":"01:56.590 ","End":"02:00.380","Text":"Here too, we don\u0027t have uniqueness because anything"},{"Start":"02:00.380 ","End":"02:04.400","Text":"that\u0027s bigger than an upper bound is also going to be an upper bound."},{"Start":"02:04.400 ","End":"02:06.620","Text":"7 is bigger than 1."},{"Start":"02:06.620 ","End":"02:08.180","Text":"It\u0027s also going to be an upper bound."},{"Start":"02:08.180 ","End":"02:12.840","Text":"7 is bigger or equal to all the elements of the sequence."},{"Start":"02:13.120 ","End":"02:20.630","Text":"Now it just so happens that this second sequence also has a lower bound."},{"Start":"02:20.630 ","End":"02:22.190","Text":"It\u0027s bounded from below."},{"Start":"02:22.190 ","End":"02:24.605","Text":"Because if I take 0,"},{"Start":"02:24.605 ","End":"02:32.390","Text":"all these terms are positive and they\u0027re all bigger than or equal to 0."},{"Start":"02:32.390 ","End":"02:35.590","Text":"It also has a lower bound."},{"Start":"02:35.590 ","End":"02:40.450","Text":"This has an upper bound and a lower bound."},{"Start":"02:40.450 ","End":"02:44.405","Text":"Because this sequence has both an upper and a lower bound,"},{"Start":"02:44.405 ","End":"02:45.620","Text":"we say it\u0027s bounded."},{"Start":"02:45.620 ","End":"02:49.655","Text":"In general, the sequence is bounded above and below,"},{"Start":"02:49.655 ","End":"02:51.650","Text":"then we just say it\u0027s bounded."},{"Start":"02:51.650 ","End":"02:53.930","Text":"Here we had an example that was bounded,"},{"Start":"02:53.930 ","End":"02:56.420","Text":"but the previous one, what was it?"},{"Start":"02:56.420 ","End":"03:01.415","Text":"It was 2, 4, 6, 8, etc."},{"Start":"03:01.415 ","End":"03:04.370","Text":"This one isn\u0027t bounded."},{"Start":"03:04.370 ","End":"03:07.685","Text":"We showed that it\u0027s bounded from below,"},{"Start":"03:07.685 ","End":"03:10.160","Text":"but it has no bound from above,"},{"Start":"03:10.160 ","End":"03:11.944","Text":"and no upper bound."},{"Start":"03:11.944 ","End":"03:15.170","Text":"It can\u0027t be just bounded."},{"Start":"03:15.170 ","End":"03:19.195","Text":"It\u0027s half bounded, if you want it bounded from below."},{"Start":"03:19.195 ","End":"03:24.230","Text":"Now I\u0027m going to introduce 2 more concepts."},{"Start":"03:24.230 ","End":"03:29.600","Text":"One of them is the greatest lower bound and the other one is the least upper bound."},{"Start":"03:29.600 ","End":"03:31.610","Text":"They have alternative names,"},{"Start":"03:31.610 ","End":"03:35.705","Text":"Latin-sounding names, more international."},{"Start":"03:35.705 ","End":"03:41.270","Text":"The greatest lower bound is called the infimum."},{"Start":"03:41.270 ","End":"03:45.515","Text":"The least upper bound will also be called the supremum."},{"Start":"03:45.515 ","End":"03:47.525","Text":"Let\u0027s start with one of them,"},{"Start":"03:47.525 ","End":"03:49.675","Text":"the greatest lower bound."},{"Start":"03:49.675 ","End":"03:52.240","Text":"What is this?"},{"Start":"03:52.240 ","End":"03:54.530","Text":"Let\u0027s go back to that example."},{"Start":"03:54.530 ","End":"03:55.790","Text":"We had the sequence 2,"},{"Start":"03:55.790 ","End":"03:58.540","Text":"4, 6, 8 and so on."},{"Start":"03:58.540 ","End":"04:02.390","Text":"We found that 2 was a lower bound,"},{"Start":"04:02.390 ","End":"04:07.310","Text":"but that wasn\u0027t the only one that anything less than 2 is also a lower bound."},{"Start":"04:07.310 ","End":"04:09.680","Text":"It has lots of lower bounds."},{"Start":"04:09.680 ","End":"04:11.515","Text":"In fact, infinitely many."},{"Start":"04:11.515 ","End":"04:16.215","Text":"For example, 2, 1, minus 17,"},{"Start":"04:16.215 ","End":"04:22.770","Text":"minus a million, 0, lots of them."},{"Start":"04:22.770 ","End":"04:25.440","Text":"But there\u0027s 1 special 1."},{"Start":"04:25.440 ","End":"04:31.640","Text":"In some sense 2 is a special lower bound out of all the lower bounds."},{"Start":"04:31.640 ","End":"04:35.350","Text":"You might ask what\u0027s so special about it?"},{"Start":"04:35.350 ","End":"04:40.405","Text":"Well, it\u0027s the greatest from all the lower bounds of this sequence,"},{"Start":"04:40.405 ","End":"04:43.285","Text":"2 is actually the greatest."},{"Start":"04:43.285 ","End":"04:48.830","Text":"There\u0027s no other lower bound that\u0027s greater than 2."},{"Start":"04:49.140 ","End":"04:53.920","Text":"Because anything greater than 2"},{"Start":"04:53.920 ","End":"04:59.255","Text":"will no longer be greater than all the terms of the sequence."},{"Start":"04:59.255 ","End":"05:06.940","Text":"It\u0027s not surprising that we give it the name greatest lower bound and often abbreviated."},{"Start":"05:06.940 ","End":"05:12.655","Text":"In fact, I\u0027ll typically be abbreviating it as GLB in this context."},{"Start":"05:12.655 ","End":"05:21.030","Text":"As I mentioned, it\u0027s also called the infimum of the sequence."},{"Start":"05:21.030 ","End":"05:23.280","Text":"For our sequence, it was 2,"},{"Start":"05:23.280 ","End":"05:26.740","Text":"but this property exists in general."},{"Start":"05:26.770 ","End":"05:32.285","Text":"Very similar to the greatest lower bound will be the least upper bound."},{"Start":"05:32.285 ","End":"05:36.140","Text":"It\u0027s just going to work on the opposite side."},{"Start":"05:36.140 ","End":"05:40.650","Text":"Instead of taking lower bounds,"},{"Start":"05:40.650 ","End":"05:42.495","Text":"we\u0027re going to take upper bound."},{"Start":"05:42.495 ","End":"05:45.980","Text":"Let\u0027s return to the sequence we had before."},{"Start":"05:45.980 ","End":"05:47.510","Text":"This one, 1, 1/2, 1/3,"},{"Start":"05:47.510 ","End":"05:49.405","Text":"1/4, and so on."},{"Start":"05:49.405 ","End":"05:53.525","Text":"We said that 1 is an upper bound,"},{"Start":"05:53.525 ","End":"05:57.710","Text":"but anything greater than 1 is also an upper bound."},{"Start":"05:57.710 ","End":"06:00.415","Text":"There\u0027s infinitely many."},{"Start":"06:00.415 ","End":"06:06.225","Text":"For example, 1, 2, 13, 100,"},{"Start":"06:06.225 ","End":"06:09.660","Text":"245, they are all upper bounds,"},{"Start":"06:09.660 ","End":"06:14.035","Text":"they are all bigger or equal to everything in the sequence."},{"Start":"06:14.035 ","End":"06:20.285","Text":"Amongst all these infinitely many upper bounds is 1 special 1,"},{"Start":"06:20.285 ","End":"06:22.805","Text":"and that is the number 1."},{"Start":"06:22.805 ","End":"06:25.475","Text":"What\u0027s special about it?"},{"Start":"06:25.475 ","End":"06:33.365","Text":"Well, earlier we talked about the greatest lower bound and it was the greatest."},{"Start":"06:33.365 ","End":"06:40.530","Text":"This 1 has the opposite property that it\u0027s the smallest or the least."},{"Start":"06:40.530 ","End":"06:43.845","Text":"There\u0027s no other upper bound less than it."},{"Start":"06:43.845 ","End":"06:52.909","Text":"Anything that\u0027s less than 1 is not going to be an upper bound."},{"Start":"06:52.909 ","End":"06:57.650","Text":"Because an upper bound has to be bigger or equal to all the elements of the sequence,"},{"Start":"06:57.650 ","End":"07:00.390","Text":"so has to be at least 1."},{"Start":"07:01.710 ","End":"07:08.770","Text":"Such an upper bound that\u0027s the least 1 is just called the least upper bound,"},{"Start":"07:08.770 ","End":"07:12.230","Text":"and the abbreviated LUB,"},{"Start":"07:12.570 ","End":"07:16.705","Text":"also called the supremum of the sequence."},{"Start":"07:16.705 ","End":"07:26.590","Text":"Now we also noted earlier that this sequence also has a greatest lower bound."},{"Start":"07:26.590 ","End":"07:29.380","Text":"Well, we noted that it has a lower bound,"},{"Start":"07:29.380 ","End":"07:34.705","Text":"that 0 is a lower bound because all of these are positive and are big or equal to 0."},{"Start":"07:34.705 ","End":"07:41.980","Text":"In fact, 0 is the greatest lower bound."},{"Start":"07:41.980 ","End":"07:46.315","Text":"Nothing greater than 0 will be less than or equal to all of these."},{"Start":"07:46.315 ","End":"07:48.865","Text":"Can you think why?"},{"Start":"07:48.865 ","End":"07:51.625","Text":"Well, I\u0027ll tell you."},{"Start":"07:51.625 ","End":"07:54.475","Text":"Let\u0027s see. I want to keep it inside."},{"Start":"07:54.475 ","End":"07:58.690","Text":"Well, if something is greater than 0,"},{"Start":"07:58.690 ","End":"08:02.350","Text":"then at some point any number that\u0027s larger than 0,"},{"Start":"08:02.350 ","End":"08:08.125","Text":"there\u0027s going to be some n such that 1 over n is less than any positive number."},{"Start":"08:08.125 ","End":"08:11.770","Text":"There\u0027ll be some members of the sequence that will be smaller than it,"},{"Start":"08:11.770 ","End":"08:17.660","Text":"so it can\u0027t possibly be a lower bound,"},{"Start":"08:17.660 ","End":"08:20.080","Text":"so that\u0027s a proof by contradiction."},{"Start":"08:20.080 ","End":"08:24.025","Text":"Anyway, I don\u0027t want to dwell on that too much."},{"Start":"08:24.025 ","End":"08:30.685","Text":"What I want to do is make things a bit more formal."},{"Start":"08:30.685 ","End":"08:33.969","Text":"We define things pretty vaguely."},{"Start":"08:33.969 ","End":"08:38.365","Text":"Let\u0027s just go over these concepts again."},{"Start":"08:38.365 ","End":"08:43.430","Text":"We start off in general with the sequence a_n."},{"Start":"08:43.680 ","End":"08:54.400","Text":"If we have a number capital M such that a_n is less than or equal to M for all n,"},{"Start":"08:54.400 ","End":"09:00.610","Text":"in other words, M is bigger or equal to all the terms in the sequence,"},{"Start":"09:00.610 ","End":"09:04.460","Text":"then it\u0027s called an upper bound of the sequence."},{"Start":"09:04.500 ","End":"09:09.655","Text":"If the sequence has an upper bound and not all sequences do,"},{"Start":"09:09.655 ","End":"09:14.590","Text":"then it said to be bounded above or bounded from above,"},{"Start":"09:14.590 ","End":"09:17.500","Text":"optional the word from."},{"Start":"09:17.500 ","End":"09:20.365","Text":"Some upper bound are special."},{"Start":"09:20.365 ","End":"09:26.035","Text":"An upper bound of a sequence is called the least upper bound, LUB,"},{"Start":"09:26.035 ","End":"09:33.400","Text":"or supremum of the sequence if there is no smaller upper bound of the sequence."},{"Start":"09:33.400 ","End":"09:34.600","Text":"In other words, it\u0027s the smallest."},{"Start":"09:34.600 ","End":"09:36.740","Text":"There\u0027s nothing smaller than it."},{"Start":"09:37.440 ","End":"09:42.699","Text":"All this is pretty much repeated for the lower bounds."},{"Start":"09:42.699 ","End":"09:48.460","Text":"We take a little m such that a_n is bigger or equal to m for all n,"},{"Start":"09:48.460 ","End":"09:53.589","Text":"meaning that little m is less than or equal to all the members of the sequence,"},{"Start":"09:53.589 ","End":"09:55.810","Text":"so it\u0027s called the lower bound."},{"Start":"09:55.810 ","End":"09:58.030","Text":"If the sequence has a lower bound,"},{"Start":"09:58.030 ","End":"10:00.280","Text":"it\u0027s bounded from below."},{"Start":"10:00.280 ","End":"10:02.530","Text":"A lower bound might be special."},{"Start":"10:02.530 ","End":"10:10.224","Text":"It could be the greatest lower bound or infimum if it\u0027s the greatest,"},{"Start":"10:10.224 ","End":"10:14.275","Text":"if there\u0027s no other lower bound that\u0027s greater than it."},{"Start":"10:14.275 ","End":"10:17.020","Text":"Those are the formal definitions."},{"Start":"10:17.020 ","End":"10:20.210","Text":"Next, let\u0027s move on to an example."},{"Start":"10:20.280 ","End":"10:23.890","Text":"In this example exercise,"},{"Start":"10:23.890 ","End":"10:25.780","Text":"we\u0027re given the sequence a_n,"},{"Start":"10:25.780 ","End":"10:29.500","Text":"by this formula, minus 1^n plus 1 over n squared,"},{"Start":"10:29.500 ","End":"10:31.255","Text":"and these 3 parts."},{"Start":"10:31.255 ","End":"10:38.875","Text":"First of all, write a few upper bounds of the sequence and write its least upper bound."},{"Start":"10:38.875 ","End":"10:43.060","Text":"Then part b is to write a few lower bounds of"},{"Start":"10:43.060 ","End":"10:48.190","Text":"the sequence and what is its greatest lower bound."},{"Start":"10:48.190 ","End":"10:49.990","Text":"Finally, the question is,"},{"Start":"10:49.990 ","End":"10:52.580","Text":"is the sequence bounded?"},{"Start":"10:53.010 ","End":"10:55.675","Text":"For the solution."},{"Start":"10:55.675 ","End":"10:57.610","Text":"Before we get properly started,"},{"Start":"10:57.610 ","End":"10:59.379","Text":"let\u0027s just write a few terms,"},{"Start":"10:59.379 ","End":"11:00.880","Text":"see what\u0027s going on."},{"Start":"11:00.880 ","End":"11:03.850","Text":"Let n equal 1,"},{"Start":"11:03.850 ","End":"11:05.485","Text":"then 2, then 3,"},{"Start":"11:05.485 ","End":"11:11.170","Text":"and 4, 5, 6."},{"Start":"11:11.170 ","End":"11:14.560","Text":"You\u0027ll see that these are what we get."},{"Start":"11:14.560 ","End":"11:17.920","Text":"There\u0027s something that happens alternately."},{"Start":"11:17.920 ","End":"11:20.890","Text":"We have a minus 1, a minus 1, a minus 1,"},{"Start":"11:20.890 ","End":"11:25.340","Text":"and here we have a 1, a 1, and a 1."},{"Start":"11:25.340 ","End":"11:28.290","Text":"Now, if you just stare at it a while,"},{"Start":"11:28.290 ","End":"11:33.975","Text":"you\u0027ll see that this term is the least upper bound."},{"Start":"11:33.975 ","End":"11:36.435","Text":"First of all, it\u0027s an upper bound"},{"Start":"11:36.435 ","End":"11:43.540","Text":"because all the terms in the odd places are negative or 0."},{"Start":"11:43.540 ","End":"11:45.910","Text":"Well, this 1 is 0, but this 1 is negative,"},{"Start":"11:45.910 ","End":"11:48.370","Text":"negative, and so on."},{"Start":"11:48.370 ","End":"11:51.100","Text":"Then in the 2nd, 4th,"},{"Start":"11:51.100 ","End":"11:53.890","Text":"and 6th terms, it keeps getting smaller."},{"Start":"11:53.890 ","End":"11:58.810","Text":"We have 1 1/4, 1 1/16, 1 and 1/36."},{"Start":"11:58.810 ","End":"12:01.870","Text":"This is going to be bigger than all of them."},{"Start":"12:01.870 ","End":"12:04.900","Text":"Since it\u0027s 1 of the terms in the sequence,"},{"Start":"12:04.900 ","End":"12:08.365","Text":"it has to be the least upper bound."},{"Start":"12:08.365 ","End":"12:12.910","Text":"Because any upper bound has to be at least 1 1/4,"},{"Start":"12:12.910 ","End":"12:17.080","Text":"it\u0027s going to be bigger than all the terms in the sequence."},{"Start":"12:17.080 ","End":"12:19.180","Text":"Once we have the least upper bound,"},{"Start":"12:19.180 ","End":"12:25.225","Text":"we can write a few more by just taking any few numbers larger than this."},{"Start":"12:25.225 ","End":"12:28.150","Text":"For example, 2 is an upper bound,"},{"Start":"12:28.150 ","End":"12:34.435","Text":"then 100 and 13 1/2 and Pi and whatever."},{"Start":"12:34.435 ","End":"12:38.620","Text":"Just as long as it\u0027s bigger than 1 1/4."},{"Start":"12:38.620 ","End":"12:41.755","Text":"Now, in part b,"},{"Start":"12:41.755 ","End":"12:44.965","Text":"we want some lower bounds."},{"Start":"12:44.965 ","End":"12:49.764","Text":"Once again, it\u0027s easiest to start with the greatest lower bound."},{"Start":"12:49.764 ","End":"12:57.055","Text":"Notice that the terms in the even places are all positive."},{"Start":"12:57.055 ","End":"13:00.385","Text":"They\u0027re all 1 and something: 1 plus this 1, plus this 1, plus this."},{"Start":"13:00.385 ","End":"13:02.155","Text":"So let\u0027s leave those out."},{"Start":"13:02.155 ","End":"13:04.690","Text":"If we take the terms in odd places,"},{"Start":"13:04.690 ","End":"13:06.985","Text":"we have here minus 1 plus 1 is 0,"},{"Start":"13:06.985 ","End":"13:08.485","Text":"minus 1 plus 1/9,"},{"Start":"13:08.485 ","End":"13:11.125","Text":"minus 1 plus 1/25."},{"Start":"13:11.125 ","End":"13:16.240","Text":"It\u0027s always minus 1 plus a bit and that bit keeps getting smaller."},{"Start":"13:16.240 ","End":"13:21.580","Text":"It looks like that minus 1 is going to be the greatest lower bound."},{"Start":"13:21.580 ","End":"13:26.125","Text":"First of all, it\u0027s a lower bound because as I said,"},{"Start":"13:26.125 ","End":"13:28.060","Text":"these terms, this 1,"},{"Start":"13:28.060 ","End":"13:30.340","Text":"and this 1, and this 1 are positive."},{"Start":"13:30.340 ","End":"13:32.170","Text":"In the odd places,"},{"Start":"13:32.170 ","End":"13:35.470","Text":"it\u0027s minus 1 plus something positive,"},{"Start":"13:35.470 ","End":"13:41.300","Text":"so they\u0027re all bigger than minus 1."},{"Start":"13:42.480 ","End":"13:49.810","Text":"Now, why is it the greatest lower bound?"},{"Start":"13:49.810 ","End":"13:53.590","Text":"Because if I take something bigger than minus 1,"},{"Start":"13:53.590 ","End":"13:56.995","Text":"it\u0027s going to be minus 1 plus something positive."},{"Start":"13:56.995 ","End":"14:01.630","Text":"Sooner or later, 1/4,"},{"Start":"14:01.630 ","End":"14:04.465","Text":"1/16, 1/36, or get to 1 over something,"},{"Start":"14:04.465 ","End":"14:07.525","Text":"that\u0027s less than that something positive,"},{"Start":"14:07.525 ","End":"14:12.860","Text":"so it\u0027ll be less than anything that\u0027s bigger than minus 1."},{"Start":"14:12.860 ","End":"14:20.760","Text":"I think it\u0027s intuitively clear that minus 1 is the greatest lower bound."},{"Start":"14:21.310 ","End":"14:26.745","Text":"We were asked to write a few more lower bounds."},{"Start":"14:26.745 ","End":"14:32.950","Text":"Is there anything smaller than minus 1?"},{"Start":"14:32.950 ","End":"14:36.070","Text":"Minus 1 1/4, minus 2,"},{"Start":"14:36.070 ","End":"14:40.700","Text":"minus 17, minus 100, minus a zillion."},{"Start":"14:43.140 ","End":"14:46.915","Text":"Part c, we were asked if the sequence is bounded."},{"Start":"14:46.915 ","End":"14:49.780","Text":"The answer of course is yes,"},{"Start":"14:49.780 ","End":"14:55.570","Text":"because it has some upper bound,"},{"Start":"14:55.570 ","End":"14:58.970","Text":"it\u0027s bounded above, and it has lower bound,"},{"Start":"14:58.970 ","End":"15:00.545","Text":"so it\u0027s bounded below,"},{"Start":"15:00.545 ","End":"15:03.125","Text":"and if it\u0027s bounded above and below,"},{"Start":"15:03.125 ","End":"15:05.565","Text":"then it\u0027s just bounded."},{"Start":"15:05.565 ","End":"15:10.175","Text":"That answers that example exercise."},{"Start":"15:10.175 ","End":"15:13.775","Text":"I want to end this clip with a useful theorem,"},{"Start":"15:13.775 ","End":"15:16.865","Text":"turns out to be very useful, in fact."},{"Start":"15:16.865 ","End":"15:23.410","Text":"If you have a sequence that\u0027s monotonic and it\u0027s bounded above and below,"},{"Start":"15:23.410 ","End":"15:26.135","Text":"that means of course, then it converges,"},{"Start":"15:26.135 ","End":"15:28.530","Text":"meaning it has a limit."},{"Start":"15:28.710 ","End":"15:33.620","Text":"If a sequence is increasing or decreasing and is also bounded,"},{"Start":"15:33.620 ","End":"15:35.870","Text":"then it has a limit."},{"Start":"15:35.870 ","End":"15:39.360","Text":"I\u0027m ending this clip here."}],"ID":8340}],"Thumbnail":null,"ID":52089},{"Name":"Infinite Geometric Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Sigma Notation for Summation","Duration":"6m 55s","ChapterTopicVideoID":6470,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6470.jpeg","UploadDate":"2020-01-16T06:55:43.7970000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"In this clip, I\u0027m going to introduce"},{"Start":"00:03.045 ","End":"00:08.055","Text":"the Sigma notation for summation in case you haven\u0027t seen it already."},{"Start":"00:08.055 ","End":"00:11.475","Text":"This is the Greek letter Sigma,"},{"Start":"00:11.475 ","End":"00:13.860","Text":"It\u0027s capital Sigma, actually,"},{"Start":"00:13.860 ","End":"00:18.790","Text":"there\u0027s also a small Sigma used in statistics, for example."},{"Start":"00:18.800 ","End":"00:21.105","Text":"What is this thing?"},{"Start":"00:21.105 ","End":"00:22.470","Text":"I\u0027ll give you an example."},{"Start":"00:22.470 ","End":"00:27.000","Text":"Suppose you have a sum of several terms of the form,"},{"Start":"00:27.000 ","End":"00:31.590","Text":"1 squared plus 2 squared plus 3 squared plus"},{"Start":"00:31.590 ","End":"00:36.780","Text":"4 squared plus 5 squared plus 6 squared plus 7 squared."},{"Start":"00:36.780 ","End":"00:39.735","Text":"I\u0027ll stop at 7, but it could have even been longer."},{"Start":"00:39.735 ","End":"00:45.470","Text":"Now this sum is quite tiresome to write out in full,"},{"Start":"00:45.470 ","End":"00:49.055","Text":"so we want a shorthand way of writing this."},{"Start":"00:49.055 ","End":"00:54.035","Text":"If you notice all of them are of the form n squared,"},{"Start":"00:54.035 ","End":"00:58.145","Text":"where n is some number,"},{"Start":"00:58.145 ","End":"01:00.605","Text":"1, 2, 3, 4, 5, 6, or 7."},{"Start":"01:00.605 ","End":"01:02.495","Text":"In fact, they\u0027re even in sequence,"},{"Start":"01:02.495 ","End":"01:07.255","Text":"so the way we write this is a convention that we write it as Sigma,"},{"Start":"01:07.255 ","End":"01:11.655","Text":"and we say the sum of n squared,"},{"Start":"01:11.655 ","End":"01:16.165","Text":"and here and here we write where n goes from and to,"},{"Start":"01:16.165 ","End":"01:20.850","Text":"from n equals 1, to n equals 7,"},{"Start":"01:20.850 ","End":"01:23.325","Text":"but we don\u0027t write the n equals at the top."},{"Start":"01:23.325 ","End":"01:27.695","Text":"N goes from 1 to 7, the sum of n squared,"},{"Start":"01:27.695 ","End":"01:29.840","Text":"which means you let n equals 1,"},{"Start":"01:29.840 ","End":"01:32.820","Text":"n equals 2, and so on."},{"Start":"01:33.560 ","End":"01:42.365","Text":"Now another example, 1 cubed plus 2 cubed plus 3 cubed."},{"Start":"01:42.365 ","End":"01:44.675","Text":"I think you\u0027ve probably got the idea,"},{"Start":"01:44.675 ","End":"01:49.250","Text":"the general pattern is n cubed and n goes from 1 to 3,"},{"Start":"01:49.250 ","End":"01:57.165","Text":"so we write this as Sigma n equals 1 to 3 of n cubed."},{"Start":"01:57.165 ","End":"02:00.150","Text":"Now let\u0027s try a reverse example,"},{"Start":"02:00.150 ","End":"02:03.645","Text":"we\u0027ll start with the Sigma n"},{"Start":"02:03.645 ","End":"02:13.170","Text":"equals 4 to 8 of 2n."},{"Start":"02:13.170 ","End":"02:17.080","Text":"What we have to do is substitute n equals 4, then 5,"},{"Start":"02:17.080 ","End":"02:19.600","Text":"then 6, then 7, then 8 in this expression,"},{"Start":"02:19.600 ","End":"02:21.410","Text":"and for each of these,"},{"Start":"02:21.410 ","End":"02:24.250","Text":"we put an addition sign between what I\u0027m saying is,"},{"Start":"02:24.250 ","End":"02:27.205","Text":"it\u0027s 2 times 4, where n equals 4,"},{"Start":"02:27.205 ","End":"02:29.540","Text":"then I let n equals 5,"},{"Start":"02:29.540 ","End":"02:33.155","Text":"and all along I\u0027m adding the Sigma is the sum,"},{"Start":"02:33.155 ","End":"02:35.105","Text":"and then 2 times 6,"},{"Start":"02:35.105 ","End":"02:38.795","Text":"2 times 7, and 2 times 8."},{"Start":"02:38.795 ","End":"02:40.520","Text":"There also is a numerical answer,"},{"Start":"02:40.520 ","End":"02:43.760","Text":"but I\u0027m not bothered to actually compute it."},{"Start":"02:43.760 ","End":"02:50.420","Text":"Another example, the sum from n equals"},{"Start":"02:50.420 ","End":"02:59.435","Text":"2 to 5 of minus 1^n times n plus 1."},{"Start":"02:59.435 ","End":"03:04.580","Text":"Well, let\u0027s see what this equals. I first let n equals 2"},{"Start":"03:04.580 ","End":"03:10.540","Text":"minus 1 squared is 1 and 2 plus 1 is 3,"},{"Start":"03:10.540 ","End":"03:12.830","Text":"so this becomes 3."},{"Start":"03:12.830 ","End":"03:19.080","Text":"Next term, n equals 3 minus 1^3 is minus 1,"},{"Start":"03:19.080 ","End":"03:22.740","Text":"and here it\u0027s 4, so we get minus 4."},{"Start":"03:22.740 ","End":"03:24.925","Text":"Then when n is 4,"},{"Start":"03:24.925 ","End":"03:28.160","Text":"again, we get plus here and here we get 5,"},{"Start":"03:28.160 ","End":"03:34.835","Text":"and when n is 5, we get minus 6."},{"Start":"03:34.835 ","End":"03:37.660","Text":"The thing that makes this go plus, minus, plus,"},{"Start":"03:37.660 ","End":"03:41.040","Text":"minus is the minus 1^n,"},{"Start":"03:41.040 ","End":"03:42.480","Text":"and we\u0027ll see this a lot."},{"Start":"03:42.480 ","End":"03:46.620","Text":"In fact, this thing is called an alternating series."},{"Start":"03:46.620 ","End":"03:48.880","Text":"In case you don\u0027t know what a series is,"},{"Start":"03:48.880 ","End":"03:50.830","Text":"each of these things is a series."},{"Start":"03:50.830 ","End":"03:52.705","Text":"It\u0027s a bunch of numbers,"},{"Start":"03:52.705 ","End":"03:55.120","Text":"quantity with pluses in the middle."},{"Start":"03:55.120 ","End":"03:58.330","Text":"If it\u0027s commas, it\u0027s a sequence and it has pluses,"},{"Start":"03:58.330 ","End":"04:00.760","Text":"it\u0027s a series, so this is a series,"},{"Start":"04:00.760 ","End":"04:02.455","Text":"this is a series, this is a series."},{"Start":"04:02.455 ","End":"04:04.900","Text":"This is a series and this 1 is alternating."},{"Start":"04:04.900 ","End":"04:09.715","Text":"I like you to note that a series is allowed to be infinite."},{"Start":"04:09.715 ","End":"04:16.460","Text":"For example, we could have the sum from n equals"},{"Start":"04:16.460 ","End":"04:23.960","Text":"1 to infinity of 1 over n. What this is equal to,"},{"Start":"04:23.960 ","End":"04:28.040","Text":"if n equals 1, then it\u0027s 1 over 1,"},{"Start":"04:28.040 ","End":"04:31.715","Text":"If n equals 2, it\u0027s 1 over 2."},{"Start":"04:31.715 ","End":"04:37.035","Text":"If n is 3, it\u0027s 1 over 3,"},{"Start":"04:37.035 ","End":"04:41.830","Text":"1 over 4, and so on."},{"Start":"04:41.960 ","End":"04:44.920","Text":"Sometimes I put the general term,"},{"Start":"04:44.920 ","End":"04:48.970","Text":"1 over n dot, dot, dot."},{"Start":"04:48.970 ","End":"04:54.215","Text":"Now let\u0027s look at 1 final example."},{"Start":"04:54.215 ","End":"05:02.730","Text":"There is a mathematical theorem that says that e^x is equal to"},{"Start":"05:02.730 ","End":"05:08.580","Text":"the sum n goes from 0 to infinity"},{"Start":"05:08.580 ","End":"05:15.815","Text":"of x^n over n factorial."},{"Start":"05:15.815 ","End":"05:20.460","Text":"Let\u0027s expand this and see what this is,"},{"Start":"05:20.460 ","End":"05:24.845","Text":"and this will give us what we call the infinite series for e^x."},{"Start":"05:24.845 ","End":"05:29.860","Text":"First n equals 0, x^0 is 1,"},{"Start":"05:29.860 ","End":"05:34.950","Text":"and 0 factorial by convention is 1 also,"},{"Start":"05:34.950 ","End":"05:37.230","Text":"so this gives us 1."},{"Start":"05:37.230 ","End":"05:41.175","Text":"When n equals 1, this gives us x here,"},{"Start":"05:41.175 ","End":"05:45.939","Text":"1 factorial is 1, so this is x."},{"Start":"05:46.620 ","End":"05:48.940","Text":"Then if n equals 2,"},{"Start":"05:48.940 ","End":"05:55.659","Text":"we get x squared over 2 factorial and I\u0027ll leave it as 2 factorial,"},{"Start":"05:55.659 ","End":"05:57.955","Text":"even though I could compute that,"},{"Start":"05:57.955 ","End":"06:04.150","Text":"and you get the idea, x^3 over 3 factorial,"},{"Start":"06:04.150 ","End":"06:07.105","Text":"and so on and so on."},{"Start":"06:07.105 ","End":"06:12.004","Text":"The general term is x^n over n factorial,"},{"Start":"06:12.004 ","End":"06:13.555","Text":"I\u0027ll just copy that from here."},{"Start":"06:13.555 ","End":"06:15.845","Text":"But it goes on forever."},{"Start":"06:15.845 ","End":"06:22.739","Text":"Some people even write e^x in this form instead of the short Sigma notation,"},{"Start":"06:22.739 ","End":"06:26.630","Text":"and this is somewhat cumbersome and tedious."},{"Start":"06:26.630 ","End":"06:30.020","Text":"For example, if you wanted e^4x you\u0027d have to say 1 plus"},{"Start":"06:30.020 ","End":"06:33.970","Text":"4x plus 4x all squared and so on."},{"Start":"06:33.970 ","End":"06:37.365","Text":"The next notation does have its advantages,"},{"Start":"06:37.365 ","End":"06:41.270","Text":"but in general, we\u0027ll be using the Sigma notation,"},{"Start":"06:41.270 ","End":"06:46.610","Text":"which is widely accepted in mathematics everywhere in the world,"},{"Start":"06:46.610 ","End":"06:51.930","Text":"and now you know what it is."},{"Start":"06:51.930 ","End":"06:55.750","Text":"Done with this introduction."}],"ID":6497},{"Watched":false,"Name":"Introduction to Series","Duration":"10m 39s","ChapterTopicVideoID":6471,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6471.jpeg","UploadDate":"2020-01-16T06:56:12.5970000","DurationForVideoObject":"PT10M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.290 ","End":"00:07.410","Text":"In this clip, I\u0027ll be talking about infinite series and I\u0027ll start with an introduction."},{"Start":"00:07.410 ","End":"00:11.440","Text":"Let\u0027s first of all take an infinite sequence."},{"Start":"00:11.720 ","End":"00:22.610","Text":"An infinite sequence might"},{"Start":"00:22.610 ","End":"00:24.890","Text":"be something like 1,"},{"Start":"00:24.890 ","End":"00:34.070","Text":"1/2, 1/3, 1/4, 1/5, and so on."},{"Start":"00:34.070 ","End":"00:42.800","Text":"A sequence is just a set of numbers in order,"},{"Start":"00:42.800 ","End":"00:46.850","Text":"and we write them with commas in between and if it\u0027s infinite,"},{"Start":"00:46.850 ","End":"00:47.990","Text":"just put dot, dot,"},{"Start":"00:47.990 ","End":"00:49.550","Text":"dot at the end."},{"Start":"00:49.550 ","End":"00:52.520","Text":"This is not quite precise."},{"Start":"00:52.520 ","End":"00:56.180","Text":"A person might not see the pattern,"},{"Start":"00:56.180 ","End":"01:00.935","Text":"so we usually make it more explicit by saying something like,"},{"Start":"01:00.935 ","End":"01:05.525","Text":"if a_n is the general term at position n,"},{"Start":"01:05.525 ","End":"01:08.745","Text":"its value is 1."},{"Start":"01:08.745 ","End":"01:12.530","Text":"For example, the 3rd term is a_3,"},{"Start":"01:12.530 ","End":"01:18.755","Text":"which is 1/3, the first term when n is 1, 1/1, and so on."},{"Start":"01:18.755 ","End":"01:22.730","Text":"Often we say the sequence a_n is 1,"},{"Start":"01:22.730 ","End":"01:26.745","Text":"but sometimes to distinguish a sequence from a specific element,"},{"Start":"01:26.745 ","End":"01:30.530","Text":"we\u0027ll write the sequence in a bracket like"},{"Start":"01:30.530 ","End":"01:38.220","Text":"this or even more specifically a_n,"},{"Start":"01:38.220 ","End":"01:43.290","Text":"where the index n goes from 1 to infinity."},{"Start":"01:43.290 ","End":"01:50.810","Text":"It\u0027s also possible to write it in brackets with spelling out some of the terms a_1,"},{"Start":"01:50.810 ","End":"01:53.150","Text":"a_2, a_3, and so on,"},{"Start":"01:53.150 ","End":"01:57.030","Text":"general term a_n, and so on."},{"Start":"01:57.710 ","End":"02:04.760","Text":"But mostly it won\u0027t be any confusion if we just say sequence a_n and then we\u0027ll"},{"Start":"02:04.760 ","End":"02:10.850","Text":"know that n is an index which runs from usually 1 to infinity or 0 to infinity."},{"Start":"02:10.850 ","End":"02:13.700","Text":"But you\u0027d have to use this form to make it"},{"Start":"02:13.700 ","End":"02:17.610","Text":"very explicit of where we\u0027re going from and to."},{"Start":"02:19.600 ","End":"02:23.570","Text":"Now, if we have an infinite sequence,"},{"Start":"02:23.570 ","End":"02:27.485","Text":"we can create from it an infinite series."},{"Start":"02:27.485 ","End":"02:31.880","Text":"Now I\u0027ll get to with concept of infinite series,"},{"Start":"02:31.880 ","End":"02:35.450","Text":"which is similar or related but different."},{"Start":"02:35.450 ","End":"02:41.750","Text":"An infinite series would be like this,"},{"Start":"02:41.750 ","End":"02:42.830","Text":"but instead of the commas,"},{"Start":"02:42.830 ","End":"02:45.980","Text":"we put a plus 1 plus 1/2,"},{"Start":"02:45.980 ","End":"02:50.450","Text":"plus 1/3, plus 1/4 plus 1/5,"},{"Start":"02:50.450 ","End":"02:53.250","Text":"plus, and so on."},{"Start":"02:53.800 ","End":"03:00.800","Text":"We can call that the series 1,"},{"Start":"03:00.800 ","End":"03:04.290","Text":"or the series a_n."},{"Start":"03:06.080 ","End":"03:08.500","Text":"We can write it,"},{"Start":"03:08.500 ","End":"03:16.485","Text":"and we do write it as the sum from n goes from"},{"Start":"03:16.485 ","End":"03:24.945","Text":"1 to infinity of a_n or we can be specific and write 1 instead;"},{"Start":"03:24.945 ","End":"03:31.575","Text":"it\u0027s the same thing if we know that a_n is 1,"},{"Start":"03:31.575 ","End":"03:36.195","Text":"and this thing holds for here also."},{"Start":"03:36.195 ","End":"03:39.350","Text":"That\u0027s the sequence, just a bunch of numbers"},{"Start":"03:39.350 ","End":"03:43.920","Text":"in order and a series is when we sum them up."},{"Start":"03:43.960 ","End":"03:47.690","Text":"Let\u0027s go to another example now."},{"Start":"03:47.690 ","End":"03:51.035","Text":"Let\u0027s take a different sequence."},{"Start":"03:51.035 ","End":"03:53.525","Text":"Let\u0027s take the sequence."},{"Start":"03:53.525 ","End":"03:59.490","Text":"Just write the word sequence, 2,"},{"Start":"03:59.490 ","End":"04:06.190","Text":"4, 8, 16, 32, and so on."},{"Start":"04:06.190 ","End":"04:09.560","Text":"This is the sequence where I can describe"},{"Start":"04:09.560 ","End":"04:17.840","Text":"the general term a_n as equal to 2^n."},{"Start":"04:17.840 ","End":"04:20.469","Text":"Because look, this is just to 2 squared,"},{"Start":"04:20.469 ","End":"04:22.970","Text":"2 cubed, 2^4, 2^5."},{"Start":"04:22.970 ","End":"04:24.995","Text":"Each term is double the previous,"},{"Start":"04:24.995 ","End":"04:31.740","Text":"and we can say that this is the sequence or as I prefer to write it a_n,"},{"Start":"04:31.740 ","End":"04:38.530","Text":"but from n goes from 1 to infinity."},{"Start":"04:38.780 ","End":"04:42.070","Text":"I\u0027ll also fix this."},{"Start":"04:43.480 ","End":"04:46.265","Text":"This is the general term."},{"Start":"04:46.265 ","End":"04:48.410","Text":"This is the series although we sometimes,"},{"Start":"04:48.410 ","End":"04:49.940","Text":"when there\u0027s no confusion,"},{"Start":"04:49.940 ","End":"04:56.200","Text":"just say that this is the series a_n equals 2^n."},{"Start":"04:56.200 ","End":"04:58.935","Text":"Now from the sequence,"},{"Start":"04:58.935 ","End":"05:01.440","Text":"we\u0027ll get a sequence and"},{"Start":"05:01.440 ","End":"05:09.930","Text":"the sequence will be 2 plus 4,"},{"Start":"05:09.930 ","End":"05:12.840","Text":"plus 8, plus 16,"},{"Start":"05:12.840 ","End":"05:16.810","Text":"plus 32 plus, and so on."},{"Start":"05:16.810 ","End":"05:23.760","Text":"This we write as the sum where n goes from"},{"Start":"05:23.760 ","End":"05:30.515","Text":"1 to infinity of a_n or you want to be explicit,"},{"Start":"05:30.515 ","End":"05:37.670","Text":"it\u0027s the series n goes from 1 infinity of 2 to the power of n. Basically,"},{"Start":"05:37.670 ","End":"05:39.415","Text":"when you have a sequence;"},{"Start":"05:39.415 ","End":"05:45.740","Text":"a sequence as in these curly brackets and also 1 to infinity,"},{"Start":"05:45.740 ","End":"05:52.380","Text":"and same with this series."},{"Start":"05:56.200 ","End":"05:58.430","Text":"Along with the sequence,"},{"Start":"05:58.430 ","End":"06:05.130","Text":"we also have a series and the series is just by replacing the commas with pluses."},{"Start":"06:05.130 ","End":"06:10.040","Text":"Instead of just being an ordered list of numbers, it\u0027s a summation."},{"Start":"06:10.040 ","End":"06:13.980","Text":"2 plus 4 plus 8 plus 16;"},{"Start":"06:13.980 ","End":"06:17.620","Text":"I\u0027ll fix that in a second, plus 32."},{"Start":"06:19.540 ","End":"06:21.725","Text":"That\u0027s the series."},{"Start":"06:21.725 ","End":"06:27.204","Text":"We can also write it as the sum."},{"Start":"06:27.204 ","End":"06:29.775","Text":"Basically we take this notation."},{"Start":"06:29.775 ","End":"06:31.140","Text":"Instead of the curly brackets,"},{"Start":"06:31.140 ","End":"06:40.160","Text":"we put the capital Sigma and same a_n and n goes from 1 to infinity."},{"Start":"06:40.160 ","End":"06:42.754","Text":"That\u0027s how we can write it in Sigma notation."},{"Start":"06:42.754 ","End":"06:46.580","Text":"If you want to be explicit about the a_n you can write it explicitly as"},{"Start":"06:46.580 ","End":"06:51.250","Text":"n equals 1 to infinity of 2^n."},{"Start":"06:51.250 ","End":"06:56.545","Text":"Same here I could write 2^n instead of a^n, I mean they\u0027re equal."},{"Start":"06:56.545 ","End":"07:02.845","Text":"Sometimes you can just get lazy and say the series 2^n,"},{"Start":"07:02.845 ","End":"07:06.965","Text":"but I think that\u0027s a bit too lazy."},{"Start":"07:06.965 ","End":"07:11.670","Text":"But this is okay as long as you know what you mean."},{"Start":"07:12.460 ","End":"07:16.110","Text":"Let\u0027s go on to another example."},{"Start":"07:17.630 ","End":"07:20.815","Text":"Next example."},{"Start":"07:20.815 ","End":"07:24.610","Text":"Choose a different color for each 1."},{"Start":"07:24.610 ","End":"07:31.625","Text":"Sorry, sequence first; we start with a sequence and let the sequence be,"},{"Start":"07:31.625 ","End":"07:37.350","Text":"1 minus 1, 1 minus 1, 1."},{"Start":"07:38.050 ","End":"07:42.470","Text":"You get the idea of the pattern alternating 1 and minus 1,"},{"Start":"07:42.470 ","End":"07:45.320","Text":"1 and minus 1, and so on, keeps alternating."},{"Start":"07:45.320 ","End":"07:48.905","Text":"The general term, if we start counting at 1,"},{"Start":"07:48.905 ","End":"07:57.405","Text":"would be a_n is equal to minus 1 to the power of,"},{"Start":"07:57.405 ","End":"08:01.965","Text":"let\u0027s see, we try n and if we are out of sync, we\u0027ll alter it."},{"Start":"08:01.965 ","End":"08:08.055","Text":"Now minus 1 to the n when n is 1 is minus 1 so if I add the plus 1 here, I\u0027ll be okay."},{"Start":"08:08.055 ","End":"08:11.730","Text":"When n is 1, I get minus 1 squared which is 1,"},{"Start":"08:11.730 ","End":"08:15.300","Text":"when n is 2,"},{"Start":"08:15.300 ","End":"08:16.785","Text":"we get minus 1^3,"},{"Start":"08:16.785 ","End":"08:20.130","Text":"which is minus 1, so it\u0027s going to work out."},{"Start":"08:20.130 ","End":"08:22.370","Text":"Now, when we have a sequence,"},{"Start":"08:22.370 ","End":"08:27.090","Text":"we can generate the corresponding series."},{"Start":"08:27.090 ","End":"08:33.480","Text":"The series corresponding to this sequence is just taking classes in between."},{"Start":"08:33.480 ","End":"08:36.270","Text":"1 plus minus 1,"},{"Start":"08:36.270 ","End":"08:37.800","Text":"well you don\u0027t put plus minus,"},{"Start":"08:37.800 ","End":"08:42.135","Text":"we can put a minus also if a term is negative."},{"Start":"08:42.135 ","End":"08:43.920","Text":"1 minus 1, plus 1,"},{"Start":"08:43.920 ","End":"08:48.070","Text":"minus 1 plus 1 etc."},{"Start":"08:49.490 ","End":"09:00.045","Text":"You could call this series the sum n goes from 1 to infinity of a_n,"},{"Start":"09:00.045 ","End":"09:01.845","Text":"given that a_n is this,"},{"Start":"09:01.845 ","End":"09:10.600","Text":"or the sum n goes from 1 to infinity minus 1 to the n plus 1."},{"Start":"09:10.640 ","End":"09:15.735","Text":"Sometimes you just write the series."},{"Start":"09:15.735 ","End":"09:18.155","Text":"As I said, there\u0027s a lazy way."},{"Start":"09:18.155 ","End":"09:22.055","Text":"You just say the series minus"},{"Start":"09:22.055 ","End":"09:27.790","Text":"1^n plus 1 when it\u0027s understood that it starts from 1 and it goes to infinity,"},{"Start":"09:27.790 ","End":"09:31.340","Text":"if that\u0027s understood, you can just write it like that,"},{"Start":"09:31.340 ","End":"09:34.740","Text":"but I prefer to write it in full."},{"Start":"09:34.930 ","End":"09:38.600","Text":"Anyway, don\u0027t worry about the notation,"},{"Start":"09:38.600 ","End":"09:42.440","Text":"the concept is what matters."},{"Start":"09:42.440 ","End":"09:47.990","Text":"You make a series out of a sequence and you write it with the Sigma notation,"},{"Start":"09:47.990 ","End":"09:51.830","Text":"and that\u0027s still part of the introduction."},{"Start":"09:51.830 ","End":"09:53.660","Text":"We haven\u0027t yet talked about what to do with it,"},{"Start":"09:53.660 ","End":"09:54.680","Text":"how to find the answer,"},{"Start":"09:54.680 ","End":"09:56.030","Text":"how to figure out the sum,"},{"Start":"09:56.030 ","End":"09:59.780","Text":"if it has a sum, that will come soon."},{"Start":"09:59.780 ","End":"10:04.070","Text":"I\u0027m heading for more formal definition of an infinite series,"},{"Start":"10:04.070 ","End":"10:07.855","Text":"but I just want to go over 1 more time sequence and series."},{"Start":"10:07.855 ","End":"10:12.825","Text":"In a sequence, for example here,"},{"Start":"10:12.825 ","End":"10:15.540","Text":"it\u0027s just numbers, 1 following the other,"},{"Start":"10:15.540 ","End":"10:16.805","Text":"we don\u0027t do anything with them."},{"Start":"10:16.805 ","End":"10:19.580","Text":"It\u0027s just a list of numbers."},{"Start":"10:19.580 ","End":"10:23.554","Text":"A series uses the same numbers,"},{"Start":"10:23.554 ","End":"10:26.210","Text":"but tries to keep adding them to infinity."},{"Start":"10:26.210 ","End":"10:32.470","Text":"It\u0027s a sum, an addition problem of how to add the terms of the sequence."},{"Start":"10:32.470 ","End":"10:34.790","Text":"That\u0027s sequence and that\u0027s series."},{"Start":"10:34.790 ","End":"10:39.480","Text":"Let\u0027s get on to the definition."}],"ID":6498},{"Watched":false,"Name":"Introduction to Series (continued)","Duration":"10m 53s","ChapterTopicVideoID":6472,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6472.jpeg","UploadDate":"2016-07-04T14:00:25.2270000","DurationForVideoObject":"PT10M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"Here I\u0027m going to define what an infinite series is."},{"Start":"00:03.540 ","End":"00:07.890","Text":"An infinite series is an expression of"},{"Start":"00:07.890 ","End":"00:18.420","Text":"the form where a_n are numbers."},{"Start":"00:18.420 ","End":"00:26.565","Text":"It\u0027s also written as a_1 plus a_2 plus a_3 plus dot-dot-dot."},{"Start":"00:26.565 ","End":"00:32.140","Text":"Sometimes we include the general term."},{"Start":"00:33.110 ","End":"00:38.845","Text":"This is what I call the general term or the nth term."},{"Start":"00:38.845 ","End":"00:44.465","Text":"This would be the nth term,"},{"Start":"00:44.465 ","End":"00:46.350","Text":"and not just with n,"},{"Start":"00:46.350 ","End":"00:48.120","Text":"I mean this is the first term,"},{"Start":"00:48.120 ","End":"00:50.205","Text":"second term, third term."},{"Start":"00:50.205 ","End":"00:52.680","Text":"If we don\u0027t know what n is, it\u0027s the nth term,"},{"Start":"00:52.680 ","End":"00:56.615","Text":"and that\u0027s also called the general term."},{"Start":"00:56.615 ","End":"01:01.230","Text":"The general term is the nth term and this is a series."},{"Start":"01:01.230 ","End":"01:04.305","Text":"Without giving any meaning to it,"},{"Start":"01:04.305 ","End":"01:06.905","Text":"more precise, this is what it is."},{"Start":"01:06.905 ","End":"01:16.230","Text":"An example would be the sum n goes from 1 to infinity of 1"},{"Start":"01:16.230 ","End":"01:25.370","Text":"over n and this is what we had before was 1 plus 1/2 plus 1/3 plus,"},{"Start":"01:25.370 ","End":"01:26.780","Text":"and so on and so on."},{"Start":"01:26.780 ","End":"01:36.765","Text":"The general term is 1 over n and that would be our a_n if we call this series a_n."},{"Start":"01:36.765 ","End":"01:39.705","Text":"Another example."},{"Start":"01:39.705 ","End":"01:49.905","Text":"The sum n goes from 1 to infinity of minus 1^n."},{"Start":"01:49.905 ","End":"01:57.495","Text":"This would be minus 1 plus 1 minus 1 plus 1 minus 1 plus and so on,"},{"Start":"01:57.495 ","End":"01:59.865","Text":"and that\u0027s our general a_n."},{"Start":"01:59.865 ","End":"02:08.804","Text":"I should write here also in general minus 1^n just so we can get the pattern."},{"Start":"02:08.804 ","End":"02:14.034","Text":"The sum n goes from 1 to infinity of 1."},{"Start":"02:14.034 ","End":"02:16.219","Text":"This looks a bit confusing."},{"Start":"02:16.219 ","End":"02:18.450","Text":"There\u0027s no n here."},{"Start":"02:18.590 ","End":"02:21.275","Text":"It\u0027s like a constant function."},{"Start":"02:21.275 ","End":"02:24.140","Text":"Whatever I substitute n is I\u0027ll get 1 because there\u0027s"},{"Start":"02:24.140 ","End":"02:27.080","Text":"nowhere to put the n. When n is 1, it\u0027s 1."},{"Start":"02:27.080 ","End":"02:33.390","Text":"When n is 2, it\u0027s still 1 plus 1 plus 1 plus 1 plus, and so on."},{"Start":"02:33.390 ","End":"02:36.565","Text":"The general term is just 1,"},{"Start":"02:36.565 ","End":"02:41.680","Text":"so I\u0027m just summing 1s. That\u0027s all I\u0027m doing."},{"Start":"02:44.030 ","End":"02:47.220","Text":"Now, another example."},{"Start":"02:47.220 ","End":"02:55.260","Text":"The sum n goes from 1 to infinity of n. Let\u0027s see."},{"Start":"02:55.260 ","End":"02:58.125","Text":"When n is 1, this is 1."},{"Start":"02:58.125 ","End":"03:01.110","Text":"When n is 2, n is 2,"},{"Start":"03:01.110 ","End":"03:04.035","Text":"and when n is 3, n is 3."},{"Start":"03:04.035 ","End":"03:13.260","Text":"It looks like this is the series we get general term n. Yet another example."},{"Start":"03:13.260 ","End":"03:21.030","Text":"The sum n goes from 1 to infinity of n"},{"Start":"03:21.030 ","End":"03:29.550","Text":"over log n. Actually it\u0027s from 2,"},{"Start":"03:29.550 ","End":"03:35.410","Text":"not from 1, so it\u0027s equal to 2 over natural log of 2 plus"},{"Start":"03:35.410 ","End":"03:41.320","Text":"3 over natural log 3 plus 4 over log 4 plus,"},{"Start":"03:41.320 ","End":"03:46.815","Text":"and so on, n over log n plus, etc."},{"Start":"03:46.815 ","End":"03:50.925","Text":"Can you see why I started the n from 2 and not from 1?"},{"Start":"03:50.925 ","End":"03:52.845","Text":"If you look at it, you\u0027ll see why."},{"Start":"03:52.845 ","End":"03:55.725","Text":"I think that\u0027s enough examples,"},{"Start":"03:55.725 ","End":"03:58.020","Text":"but what do we do with series?"},{"Start":"03:58.020 ","End":"03:59.820","Text":"I mean you might ask,"},{"Start":"03:59.820 ","End":"04:01.120","Text":"what does it interest me?"},{"Start":"04:01.120 ","End":"04:04.630","Text":"What are they going to ask on an exam question and that sort of thing."},{"Start":"04:04.630 ","End":"04:06.580","Text":"Let me tell you what\u0027s interesting."},{"Start":"04:06.580 ","End":"04:10.100","Text":"What\u0027s important about series."},{"Start":"04:11.120 ","End":"04:17.570","Text":"Well what interests us about infinite series basically is"},{"Start":"04:17.570 ","End":"04:23.665","Text":"does it the series converge or diverge?"},{"Start":"04:23.665 ","End":"04:30.885","Text":"I\u0027ll go over these terms in a moment and let the answer that it converges."},{"Start":"04:30.885 ","End":"04:36.165","Text":"If it converges, what is the sum?"},{"Start":"04:36.165 ","End":"04:37.890","Text":"To what does it converge?"},{"Start":"04:37.890 ","End":"04:39.810","Text":"Also known as the sum of the series."},{"Start":"04:39.810 ","End":"04:41.850","Text":"What is the sum?"},{"Start":"04:41.850 ","End":"04:44.940","Text":"I\u0027ll informally explain now."},{"Start":"04:44.940 ","End":"04:47.445","Text":"We\u0027ll do it more precisely later,"},{"Start":"04:47.445 ","End":"04:50.465","Text":"but let\u0027s just look at these examples."},{"Start":"04:50.465 ","End":"04:56.345","Text":"Essentially what we want to do with this series is to take it sum,"},{"Start":"04:56.345 ","End":"04:58.995","Text":"but what is an infinite sum?"},{"Start":"04:58.995 ","End":"05:03.380","Text":"What we do basically is just to keep adding terms one at"},{"Start":"05:03.380 ","End":"05:07.580","Text":"a time and as we get further and further along,"},{"Start":"05:07.580 ","End":"05:11.255","Text":"we want to see if it gets closer and closer to some specific number."},{"Start":"05:11.255 ","End":"05:16.700","Text":"For example, funny thing is that now that I\u0027m looking at all these series,"},{"Start":"05:16.700 ","End":"05:18.010","Text":"not one of them converges,"},{"Start":"05:18.010 ","End":"05:21.710","Text":"so I think I\u0027d like to give yet another example of"},{"Start":"05:21.710 ","End":"05:26.255","Text":"something that converges so you\u0027ll have an idea of what I\u0027m looking at."},{"Start":"05:26.255 ","End":"05:32.870","Text":"Let\u0027s take the series 1/2 plus"},{"Start":"05:32.870 ","End":"05:39.680","Text":"1/4 plus 1/8 plus 1/16 plus 1 over 32 plus, and so on."},{"Start":"05:39.680 ","End":"05:43.375","Text":"The general term would be 1 over 2^n,"},{"Start":"05:43.375 ","End":"05:47.040","Text":"and I\u0027m taking the series from 1 to infinity."},{"Start":"05:47.040 ","End":"05:51.040","Text":"If we keep track of the sum,"},{"Start":"05:51.040 ","End":"05:53.420","Text":"if we get up to here,"},{"Start":"05:53.420 ","End":"06:00.555","Text":"and let\u0027s write the accumulating sums."},{"Start":"06:00.555 ","End":"06:02.950","Text":"Up to here, we\u0027ve got 1/2."},{"Start":"06:02.950 ","End":"06:04.865","Text":"After we add a quarter,"},{"Start":"06:04.865 ","End":"06:07.875","Text":"we\u0027ve got 3/4: 1/2 plus a quarter."},{"Start":"06:07.875 ","End":"06:10.800","Text":"If I add another eighth, I now have 7/8."},{"Start":"06:10.800 ","End":"06:13.845","Text":"Next, I get 15/16."},{"Start":"06:13.845 ","End":"06:18.780","Text":"Then I have 31 over 32, and so on."},{"Start":"06:18.780 ","End":"06:20.150","Text":"If you just look at these,"},{"Start":"06:20.150 ","End":"06:22.880","Text":"you see it\u0027s getting closer and closer to 1."},{"Start":"06:22.880 ","End":"06:26.585","Text":"I mean the distance remaining to go is 1/2, a 1/4, an 1/8,"},{"Start":"06:26.585 ","End":"06:30.480","Text":"a 1/16, a 30-second, a 64th."},{"Start":"06:30.480 ","End":"06:33.870","Text":"It\u0027s getting closer and closer to 1,"},{"Start":"06:33.870 ","End":"06:38.915","Text":"so this sum actually does have a limit."},{"Start":"06:38.915 ","End":"06:43.385","Text":"What we\u0027re really looking at is the limit of these things."},{"Start":"06:43.385 ","End":"06:48.770","Text":"If I took this as partial sums of the sequence,"},{"Start":"06:48.770 ","End":"06:50.495","Text":"they get closer and closer,"},{"Start":"06:50.495 ","End":"06:56.005","Text":"and this thing actually turns out to equal"},{"Start":"06:56.005 ","End":"06:59.450","Text":"1 by which I mean that as I get further"},{"Start":"06:59.450 ","End":"07:03.650","Text":"and further along and getting closer and closer and closer to this number."},{"Start":"07:03.650 ","End":"07:07.235","Text":"Now, when it doesn\u0027t converge,"},{"Start":"07:07.235 ","End":"07:09.610","Text":"all sorts of things could happen."},{"Start":"07:09.610 ","End":"07:12.735","Text":"Well, let me illustrate."},{"Start":"07:12.735 ","End":"07:16.800","Text":"In this case, 1 plus 1 plus 1 plus 1 plus 1,"},{"Start":"07:16.800 ","End":"07:18.960","Text":"if I keep taking the partial sums,"},{"Start":"07:18.960 ","End":"07:20.190","Text":"I\u0027ll get 1 and I get 2,"},{"Start":"07:20.190 ","End":"07:23.640","Text":"3, 4, 5, and so on."},{"Start":"07:23.640 ","End":"07:27.230","Text":"The sums are getting larger and larger and they\u0027ll get to infinity."},{"Start":"07:27.230 ","End":"07:28.430","Text":"If I take enough terms,"},{"Start":"07:28.430 ","End":"07:30.525","Text":"I can get larger and larger,"},{"Start":"07:30.525 ","End":"07:32.405","Text":"as large as any number that I want."},{"Start":"07:32.405 ","End":"07:34.160","Text":"If I\u0027m larger than any number I want,"},{"Start":"07:34.160 ","End":"07:38.280","Text":"that means it\u0027s infinity. So this is infinity."},{"Start":"07:38.280 ","End":"07:42.880","Text":"Now, this is also very clearly infinity because these are even larger than these."},{"Start":"07:42.880 ","End":"07:46.870","Text":"We get 1 and we get 3, 6, 10, 15."},{"Start":"07:46.870 ","End":"07:49.490","Text":"This will be infinity also."},{"Start":"07:49.490 ","End":"07:52.840","Text":"Here\u0027s an example where it doesn\u0027t converge,"},{"Start":"07:52.840 ","End":"07:54.940","Text":"but it doesn\u0027t go to infinity either."},{"Start":"07:54.940 ","End":"07:57.715","Text":"Because if I take the partial sums here,"},{"Start":"07:57.715 ","End":"08:01.695","Text":"I start off with minus 1."},{"Start":"08:01.695 ","End":"08:04.860","Text":"After I\u0027ve added the 1, I\u0027m up to 0,"},{"Start":"08:04.860 ","End":"08:06.330","Text":"and then I take away 1 again,"},{"Start":"08:06.330 ","End":"08:07.605","Text":"then I add 1 again,"},{"Start":"08:07.605 ","End":"08:10.140","Text":"and I subtract 1, then I add 1."},{"Start":"08:10.140 ","End":"08:13.270","Text":"Whatever n is, depending whether it\u0027s odd or even,"},{"Start":"08:13.270 ","End":"08:15.700","Text":"I\u0027ll either get minus 1 or 0,"},{"Start":"08:15.700 ","End":"08:19.235","Text":"but I\u0027ll keep getting minus 1 and 0 no matter how far along."},{"Start":"08:19.235 ","End":"08:20.510","Text":"After a million terms,"},{"Start":"08:20.510 ","End":"08:21.770","Text":"it still doesn\u0027t settle down."},{"Start":"08:21.770 ","End":"08:24.110","Text":"It\u0027s still alternating minus 1, 0,"},{"Start":"08:24.110 ","End":"08:27.245","Text":"so it doesn\u0027t get close to any one particular number."},{"Start":"08:27.245 ","End":"08:29.750","Text":"It keeps jumping from minus 1-0."},{"Start":"08:29.750 ","End":"08:34.565","Text":"It doesn\u0027t get close to 1/2 or something or minus 1/2 in the middle."},{"Start":"08:34.565 ","End":"08:36.620","Text":"It just doesn\u0027t get close to anything."},{"Start":"08:36.620 ","End":"08:42.469","Text":"This also diverges, but it doesn\u0027t go to infinity or minus infinity."},{"Start":"08:42.469 ","End":"08:46.470","Text":"As for this, well I happen to know,"},{"Start":"08:46.470 ","End":"08:51.060","Text":"but I\u0027m not going to tell you or maybe I will, 1,"},{"Start":"08:51.060 ","End":"08:55.335","Text":"1/2 and 1/3 if you start summing it,"},{"Start":"08:55.335 ","End":"08:58.990","Text":"plus that 1, 1/2, 1,"},{"Start":"08:59.030 ","End":"09:02.785","Text":"5/6 plus a 1/4, it\u0027s a bit over 2."},{"Start":"09:02.785 ","End":"09:07.490","Text":"You might think it reaches some limit like I don\u0027t know,"},{"Start":"09:07.490 ","End":"09:11.345","Text":"100 or 495 or something,"},{"Start":"09:11.345 ","End":"09:13.775","Text":"but the answer is no."},{"Start":"09:13.775 ","End":"09:15.215","Text":"If I go far enough,"},{"Start":"09:15.215 ","End":"09:17.225","Text":"I can get larger than any number I want."},{"Start":"09:17.225 ","End":"09:19.550","Text":"This in fact goes to infinity"},{"Start":"09:19.550 ","End":"09:22.000","Text":"which is hard to believe because it\u0027s such a small number: the 1/2,"},{"Start":"09:22.000 ","End":"09:24.620","Text":"1/3, a 1/4, a 1/5, and so on."},{"Start":"09:24.620 ","End":"09:26.480","Text":"If you keep summing long enough,"},{"Start":"09:26.480 ","End":"09:28.430","Text":"you\u0027ll get larger than any number you like."},{"Start":"09:28.430 ","End":"09:32.750","Text":"It will actually exceed a billion or any other number you care to throw at me,"},{"Start":"09:32.750 ","End":"09:36.070","Text":"this actually diverges to infinity."},{"Start":"09:36.070 ","End":"09:41.670","Text":"So this one diverges to infinity."},{"Start":"09:41.670 ","End":"09:46.185","Text":"This one diverges to infinity."},{"Start":"09:46.185 ","End":"09:49.770","Text":"This one just diverges,"},{"Start":"09:49.780 ","End":"09:52.010","Text":"but not to infinity,"},{"Start":"09:52.010 ","End":"09:55.550","Text":"not to plus or minus infinity."},{"Start":"09:55.550 ","End":"09:59.690","Text":"This one I just told you it diverges to infinity,"},{"Start":"09:59.690 ","End":"10:07.525","Text":"but you\u0027d be probably surprised diverges also to plus infinity."},{"Start":"10:07.525 ","End":"10:10.085","Text":"This one, I don\u0027t know."},{"Start":"10:10.085 ","End":"10:12.845","Text":"I guess this diverges to infinity also."},{"Start":"10:12.845 ","End":"10:19.705","Text":"Anyway that\u0027s the main thing that exam questions will ask you about series,"},{"Start":"10:19.705 ","End":"10:22.685","Text":"does it converge or diverge and you\u0027d have to give reasons,"},{"Start":"10:22.685 ","End":"10:24.049","Text":"and if it converges,"},{"Start":"10:24.049 ","End":"10:26.320","Text":"what is the sum?"},{"Start":"10:26.320 ","End":"10:28.970","Text":"They could also ask you about diverges."},{"Start":"10:28.970 ","End":"10:32.810","Text":"Does diverge aimlessly and not go to anything like with"},{"Start":"10:32.810 ","End":"10:36.845","Text":"the 1 minus 1 or does it go to infinity or does it go to minus infinity."},{"Start":"10:36.845 ","End":"10:41.180","Text":"The kind of question you might get asked, but so far,"},{"Start":"10:41.180 ","End":"10:47.690","Text":"the only convergent series I\u0027ve shown you is this one is the 1 over 2^n."},{"Start":"10:47.690 ","End":"10:52.950","Text":"Anyway, we\u0027ll continue with this in the next clip."}],"ID":6499},{"Watched":false,"Name":"Convegent and Divergent Series","Duration":"12m 40s","ChapterTopicVideoID":6473,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6473.jpeg","UploadDate":"2020-01-16T06:56:35.3270000","DurationForVideoObject":"PT12M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"In the previous clip,"},{"Start":"00:01.950 ","End":"00:06.285","Text":"we mentioned the terms convergent series and divergent series,"},{"Start":"00:06.285 ","End":"00:11.895","Text":"and here I\u0027d like to make it a little bit more formal and precise,"},{"Start":"00:11.895 ","End":"00:15.270","Text":"but I have to tell you first that although we\u0027ll"},{"Start":"00:15.270 ","End":"00:18.765","Text":"give a definition of convergent and divergent,"},{"Start":"00:18.765 ","End":"00:22.080","Text":"they\u0027re not very practical definitions."},{"Start":"00:22.080 ","End":"00:27.090","Text":"The are more theoretical and in practice we generally don\u0027t use the definition,"},{"Start":"00:27.090 ","End":"00:29.775","Text":"but rather a whole bunch of other tools."},{"Start":"00:29.775 ","End":"00:33.575","Text":"It\u0027s similar to the case of differentiation."},{"Start":"00:33.575 ","End":"00:37.120","Text":"There is a formal definition of what a derivative is,"},{"Start":"00:37.120 ","End":"00:40.790","Text":"but in practice we don\u0027t go back to the definition."},{"Start":"00:40.790 ","End":"00:45.410","Text":"Rather we use various rules and theorems and so on."},{"Start":"00:45.410 ","End":"00:47.045","Text":"So the sign will be here."},{"Start":"00:47.045 ","End":"00:51.690","Text":"Anyway, I as for the definition, here it goes."},{"Start":"00:51.690 ","End":"00:53.540","Text":"We say that the series,"},{"Start":"00:53.540 ","End":"00:57.650","Text":"the sum from n goes from 1 to infinity of a_n"},{"Start":"00:57.650 ","End":"01:05.070","Text":"converges if the limit of the partial sums,"},{"Start":"01:05.070 ","End":"01:10.855","Text":"a_1 plus a_2 plus a_3 and so on, up to a_n."},{"Start":"01:10.855 ","End":"01:12.589","Text":"In other words, not to infinity,"},{"Start":"01:12.589 ","End":"01:14.695","Text":"but just to a_n,"},{"Start":"01:14.695 ","End":"01:18.520","Text":"and goes to infinity here."},{"Start":"01:18.520 ","End":"01:23.180","Text":"If this equals some finite number S,"},{"Start":"01:23.180 ","End":"01:27.360","Text":"so this is a finite number not infinity."},{"Start":"01:27.650 ","End":"01:30.965","Text":"That\u0027s the definition and I\u0027ll just go over it."},{"Start":"01:30.965 ","End":"01:34.279","Text":"We wanted the sum of the series from 1 to infinity,"},{"Start":"01:34.279 ","End":"01:37.640","Text":"so we take the sum up to some number n,"},{"Start":"01:37.640 ","End":"01:39.825","Text":"like up to 3,"},{"Start":"01:39.825 ","End":"01:42.090","Text":"up to 50, up to a 100,"},{"Start":"01:42.090 ","End":"01:44.220","Text":"up to a 1000 terms,"},{"Start":"01:44.220 ","End":"01:47.585","Text":"and then we let the number of terms go to infinity."},{"Start":"01:47.585 ","End":"01:48.950","Text":"We take a million terms,"},{"Start":"01:48.950 ","End":"01:51.350","Text":"a billion terms and so on and keep going and if"},{"Start":"01:51.350 ","End":"01:55.505","Text":"these sums have a limit when n gets large goes to infinity,"},{"Start":"01:55.505 ","End":"01:58.755","Text":"then we say that the series converges,"},{"Start":"01:58.755 ","End":"02:00.710","Text":"and more than that, if this happens,"},{"Start":"02:00.710 ","End":"02:04.450","Text":"then we say that S is the sum of the series."},{"Start":"02:04.450 ","End":"02:07.170","Text":"Because this goes against normal arithmetic."},{"Start":"02:07.170 ","End":"02:09.590","Text":"In arithmetic you can\u0027t take an infinite sum,"},{"Start":"02:09.590 ","End":"02:11.360","Text":"and that\u0027s why we need the limit."},{"Start":"02:11.360 ","End":"02:21.875","Text":"But if the above limit does not exist or is plus or minus infinity,"},{"Start":"02:21.875 ","End":"02:24.935","Text":"that\u0027s a way of saying it doesn\u0027t exist,"},{"Start":"02:24.935 ","End":"02:33.250","Text":"then we say that the series is divergent."},{"Start":"02:35.680 ","End":"02:40.740","Text":"I\u0027ll highlight this term to all the important terms."},{"Start":"02:41.560 ","End":"02:46.160","Text":"We also say that the series diverges."},{"Start":"02:46.160 ","End":"02:48.920","Text":"I made it consistent with converges,"},{"Start":"02:48.920 ","End":"02:52.820","Text":"converges or is convergent diverges or is divergent."},{"Start":"02:52.820 ","End":"02:58.160","Text":"We also sometimes say diverges to plus or minus infinity if that\u0027s the case."},{"Start":"02:58.160 ","End":"03:00.980","Text":"Let\u0027s do some examples and see if we can"},{"Start":"03:00.980 ","End":"03:05.720","Text":"compute whether the series converges or diverges from the definition."},{"Start":"03:05.720 ","End":"03:10.165","Text":"I\u0027ll start with the series, the sum,"},{"Start":"03:10.165 ","End":"03:15.105","Text":"n goes from 1 to infinity of"},{"Start":"03:15.105 ","End":"03:20.860","Text":"1 over n. According to our definition,"},{"Start":"03:20.860 ","End":"03:25.270","Text":"this will be convergent or divergent depending on"},{"Start":"03:25.270 ","End":"03:31.655","Text":"the limit as n goes to infinity of the partial sums."},{"Start":"03:31.655 ","End":"03:36.095","Text":"This series is like 1 plus a 1/2 plus a 1/3."},{"Start":"03:36.095 ","End":"03:38.590","Text":"So we take it up to n terms."},{"Start":"03:38.590 ","End":"03:39.850","Text":"We\u0027ll get 1 over n,"},{"Start":"03:39.850 ","End":"03:43.810","Text":"and stop there and then take the limit as n goes to infinity."},{"Start":"03:43.810 ","End":"03:47.829","Text":"So depending on if this limit exists and is finite,"},{"Start":"03:47.829 ","End":"03:53.520","Text":"convergent, or doesn\u0027t exist or is infinite, divergent."},{"Start":"03:53.520 ","End":"03:56.035","Text":"It\u0027s quite hard to do."},{"Start":"03:56.035 ","End":"04:00.190","Text":"I don\u0027t know any simple formula that will calculate the sum from 1 to"},{"Start":"04:00.190 ","End":"04:04.699","Text":"1 over n. So it really won\u0027t work by using the definition,"},{"Start":"04:04.699 ","End":"04:09.140","Text":"we\u0027ll have to use some other techniques to find out what goes on here."},{"Start":"04:09.140 ","End":"04:12.995","Text":"But I can tell you now that this comes out to be divergent."},{"Start":"04:12.995 ","End":"04:15.545","Text":"In fact, it diverges to infinity."},{"Start":"04:15.545 ","End":"04:17.040","Text":"Doesn\u0027t look like it,"},{"Start":"04:17.040 ","End":"04:20.285","Text":"but if you take a 1/2 plus a 1/3 plus a 1/4 plus a 1/5,"},{"Start":"04:20.285 ","End":"04:23.780","Text":"a 1/6 and so on, it actually goes to infinity very,"},{"Start":"04:23.780 ","End":"04:25.925","Text":"very slowly, keeps getting slower and slower,"},{"Start":"04:25.925 ","End":"04:27.680","Text":"but it gets larger than any number,"},{"Start":"04:27.680 ","End":"04:32.130","Text":"gets larger than a billion or anything if we take enough terms."},{"Start":"04:32.130 ","End":"04:36.265","Text":"Surprising. We\u0027ll prove this sometime later."},{"Start":"04:36.265 ","End":"04:40.160","Text":"This is one we can\u0027t do from the definition and we use other techniques,"},{"Start":"04:40.160 ","End":"04:42.260","Text":"which is actually what happens most of the time."},{"Start":"04:42.260 ","End":"04:44.585","Text":"Let\u0027s take another example."},{"Start":"04:44.585 ","End":"04:54.900","Text":"The sum when n goes from 1 to infinity of 1 over 2 to the n. Let me rewrite that."},{"Start":"04:55.060 ","End":"04:59.000","Text":"Now we can identify it as a geometric series."},{"Start":"04:59.000 ","End":"05:03.530","Text":"Some of you may have studied geometric series or even infinite geometric series,"},{"Start":"05:03.530 ","End":"05:05.285","Text":"and it\u0027s very easy to compute."},{"Start":"05:05.285 ","End":"05:09.290","Text":"But for the sake of those who haven\u0027t studied this,"},{"Start":"05:09.290 ","End":"05:13.445","Text":"I\u0027m going to do it using a trick."},{"Start":"05:13.445 ","End":"05:18.890","Text":"Anyway, this is convergent or divergent according to"},{"Start":"05:18.890 ","End":"05:24.200","Text":"the limit as n goes to infinity of the partial sums,"},{"Start":"05:24.200 ","End":"05:32.365","Text":"1/2 plus 1/2 squared plus 1/2 cubed,"},{"Start":"05:32.365 ","End":"05:37.490","Text":"and so on up to 1/2 to the n. Now, as I said,"},{"Start":"05:37.490 ","End":"05:41.750","Text":"there is a simple formula for this if you know about geometric series,"},{"Start":"05:41.750 ","End":"05:46.835","Text":"but if not, we\u0027re going to do a trick to figure out how much is this."},{"Start":"05:46.835 ","End":"05:51.470","Text":"What I propose is to call this sum A, capital A."},{"Start":"05:51.470 ","End":"05:56.190","Text":"Let\u0027s say that A is equal to"},{"Start":"05:56.190 ","End":"06:02.580","Text":"1/2 plus 1/2 squared plus 1/2 cubed plus,"},{"Start":"06:02.580 ","End":"06:06.425","Text":"and so on, up to 1/2 to the n."},{"Start":"06:06.425 ","End":"06:11.155","Text":"Now I can multiply both sides of an equation by a 1/2, so let\u0027s do that."},{"Start":"06:11.155 ","End":"06:15.195","Text":"We\u0027ll get the 1/2 A equals,"},{"Start":"06:15.195 ","End":"06:18.195","Text":"I\u0027m going to multiply each of these terms by a 1/2."},{"Start":"06:18.195 ","End":"06:22.035","Text":"So 1/2 times a 1/2 is a 1/2 squared."},{"Start":"06:22.035 ","End":"06:28.800","Text":"Let me offset it a bit just so it\u0027ll match the exponents."},{"Start":"06:28.800 ","End":"06:33.060","Text":"A 1/2squared times a 1/2 is a 1/2 cubed."},{"Start":"06:33.060 ","End":"06:40.845","Text":"A 1/2 cubed times a 1/2 is 1/2^ fourth, and so on,"},{"Start":"06:40.845 ","End":"06:50.700","Text":"until we get to 1/2^ n plus 1."},{"Start":"06:50.700 ","End":"06:54.720","Text":"Now what I propose doing is subtracting the 2 equations."},{"Start":"06:54.720 ","End":"06:59.330","Text":"You can subtract 1 equation from the other, this minus this."},{"Start":"06:59.510 ","End":"07:06.300","Text":"What I\u0027ll get is A minus a 1/2 A is a 1/2 A."},{"Start":"07:06.300 ","End":"07:11.940","Text":"A 1/2 minus nothing is 1/2."},{"Start":"07:11.940 ","End":"07:14.835","Text":"This minus this, it cancels."},{"Start":"07:14.835 ","End":"07:18.315","Text":"This minus this cancels."},{"Start":"07:18.315 ","End":"07:21.270","Text":"Here we\u0027ve got to have 1/2^fourth."},{"Start":"07:21.270 ","End":"07:23.280","Text":"It\u0027s all consecutive indices."},{"Start":"07:23.280 ","End":"07:25.500","Text":"This will cancel with this."},{"Start":"07:25.500 ","End":"07:27.480","Text":"They\u0027ll be a 1/2^ n here."},{"Start":"07:27.480 ","End":"07:30.455","Text":"This will cancel with this and we\u0027re only left with this,"},{"Start":"07:30.455 ","End":"07:32.180","Text":"which means that at the end,"},{"Start":"07:32.180 ","End":"07:37.680","Text":"when I subtract, I get minus /2 to the n plus 1,"},{"Start":"07:37.680 ","End":"07:39.240","Text":"which I\u0027ll write over here."},{"Start":"07:39.240 ","End":"07:42.110","Text":"Minus 1/2 to the n plus 1."},{"Start":"07:42.110 ","End":"07:47.660","Text":"Now, we\u0027ll multiply both sides by 2 to get rid of some of the fractions."},{"Start":"07:47.660 ","End":"07:52.640","Text":"So I\u0027ll get that A is equal to 1."},{"Start":"07:52.640 ","End":"07:56.055","Text":"Just multiplying by 2 minus this,"},{"Start":"07:56.055 ","End":"07:58.240","Text":"I\u0027m multiplying by 2."},{"Start":"07:58.240 ","End":"08:03.950","Text":"I claim that it becomes to the power of n because this is like 1/2,"},{"Start":"08:03.950 ","End":"08:06.390","Text":"I\u0027ll do it at the side,"},{"Start":"08:06.390 ","End":"08:13.050","Text":"1/2 to the n plus 1 is 1/2 to the n times 1/2."},{"Start":"08:13.050 ","End":"08:16.050","Text":"If I multiply this by 2,"},{"Start":"08:16.050 ","End":"08:22.560","Text":"2-and-a-half cancel, and so I just get left with the power of n if I multiply this by 2."},{"Start":"08:22.760 ","End":"08:27.575","Text":"Now I can see what the limit of A is."},{"Start":"08:27.575 ","End":"08:32.870","Text":"Actually I should have called this A_n because for each n it\u0027s a different sum."},{"Start":"08:32.870 ","End":"08:37.970","Text":"It depends on n. So each value of A_n in the series is 1 minus 1/2 to"},{"Start":"08:37.970 ","End":"08:45.050","Text":"the n. Now we know that 1/2^ n goes to 0,"},{"Start":"08:45.050 ","End":"08:46.940","Text":"and it\u0027s a 1/2, a 1/2, an 1/8,"},{"Start":"08:46.940 ","End":"08:49.280","Text":"a 1/16, a 1/32."},{"Start":"08:49.280 ","End":"08:52.175","Text":"It just gets smaller and smaller."},{"Start":"08:52.175 ","End":"08:55.690","Text":"The other way to see it is you could look at it as 1 over 2 to"},{"Start":"08:55.690 ","End":"08:59.120","Text":"the n and when n goes to infinity,"},{"Start":"08:59.120 ","End":"09:01.190","Text":"2 to the n goes to infinity,"},{"Start":"09:01.190 ","End":"09:03.005","Text":"and this also goes to 0."},{"Start":"09:03.005 ","End":"09:07.940","Text":"So altogether, we have that the limit as n goes to"},{"Start":"09:07.940 ","End":"09:14.060","Text":"infinity of A_n is just 1 because this bit goes to 0."},{"Start":"09:14.060 ","End":"09:18.830","Text":"In other words, this limit here is equal to 1,"},{"Start":"09:18.830 ","End":"09:21.500","Text":"and so it does have a limit."},{"Start":"09:21.500 ","End":"09:24.099","Text":"This also makes sense intuitively,"},{"Start":"09:24.099 ","End":"09:28.790","Text":"if you think about it a 1/2 and a 1/4 and an 1/8 and a 1/16,"},{"Start":"09:28.790 ","End":"09:30.490","Text":"each time you are filling in 1/2"},{"Start":"09:30.490 ","End":"09:33.790","Text":"the remaining distance and you\u0027re getting closer and closer to 1."},{"Start":"09:33.790 ","End":"09:36.820","Text":"So this is one of those few cases where we can use"},{"Start":"09:36.820 ","End":"09:41.860","Text":"the definition to find out whether the series converges or diverges."},{"Start":"09:41.860 ","End":"09:45.365","Text":"In this case it definitely converges to 1."},{"Start":"09:45.365 ","End":"09:48.015","Text":"That\u0027s what we sometimes say, converges to,"},{"Start":"09:48.015 ","End":"09:51.155","Text":"or that the sum of this series is 1."},{"Start":"09:51.155 ","End":"09:53.495","Text":"Now let\u0027s look at a third example."},{"Start":"09:53.495 ","End":"09:58.840","Text":"I\u0027ve chosen the example to give you a representative of 3 different kinds of case."},{"Start":"09:58.840 ","End":"10:03.160","Text":"This case, it diverges to infinity or minus infinity."},{"Start":"10:03.160 ","End":"10:05.455","Text":"This case converges."},{"Start":"10:05.455 ","End":"10:09.290","Text":"The last case I\u0027ll give you is one which has no limit at all."},{"Start":"10:09.290 ","End":"10:13.065","Text":"An example of the kind which has no limit at all,"},{"Start":"10:13.065 ","End":"10:14.990","Text":"I can give you that."},{"Start":"10:14.990 ","End":"10:19.105","Text":"Let\u0027s take the limit from, let\u0027s say,"},{"Start":"10:19.105 ","End":"10:26.280","Text":"1^ infinity of minus 1^n."},{"Start":"10:26.800 ","End":"10:30.860","Text":"Now this will converge or diverge all depending on what"},{"Start":"10:30.860 ","End":"10:35.074","Text":"happens with the limit of the partial sums."},{"Start":"10:35.074 ","End":"10:40.740","Text":"So that\u0027s minus 1 is to the power of 1,"},{"Start":"10:40.740 ","End":"10:43.305","Text":"and then to the power of 2 is plus 1,"},{"Start":"10:43.305 ","End":"10:48.975","Text":"then minus 1, then plus 1, minus 1, etc."},{"Start":"10:48.975 ","End":"10:54.560","Text":"The last one would be minus 1 to the n. Now what"},{"Start":"10:54.560 ","End":"11:00.650","Text":"happens with this is that the answer turns out to depend on whether n is odd or even,"},{"Start":"11:00.650 ","End":"11:04.435","Text":"because you see if I start off with minus 1 and when I add the 1,"},{"Start":"11:04.435 ","End":"11:07.340","Text":"I get altogether a total of 0."},{"Start":"11:07.340 ","End":"11:11.645","Text":"Then I subtract 1, so minus 1 and add 1 and I\u0027m up to 0."},{"Start":"11:11.645 ","End":"11:13.995","Text":"Minus 1, 0."},{"Start":"11:13.995 ","End":"11:16.000","Text":"Each time I take 2,"},{"Start":"11:16.000 ","End":"11:19.475","Text":"it\u0027s the same because each pair cancel each other out."},{"Start":"11:19.475 ","End":"11:23.720","Text":"So it just alternates and oscillates between minus 1 and 0."},{"Start":"11:23.720 ","End":"11:26.750","Text":"Now this is a series that doesn\u0027t have a limit."},{"Start":"11:26.750 ","End":"11:30.755","Text":"I\u0027m claiming that this has no limit"},{"Start":"11:30.755 ","End":"11:35.420","Text":"because a limit is a number that series gets closer and closer to."},{"Start":"11:35.420 ","End":"11:38.210","Text":"So eventually, when n is big enough,"},{"Start":"11:38.210 ","End":"11:40.280","Text":"it gets very, very close."},{"Start":"11:40.280 ","End":"11:44.660","Text":"I can say, I want it within within a 1/100,"},{"Start":"11:44.660 ","End":"11:49.640","Text":"0.01 of my limit or even closer a millionth."},{"Start":"11:49.640 ","End":"11:51.200","Text":"It\u0027s going to get closer and closer,"},{"Start":"11:51.200 ","End":"11:53.390","Text":"very close to 1 single number."},{"Start":"11:53.390 ","End":"11:56.300","Text":"But here it keeps going back and forth from minus 1 to 0."},{"Start":"11:56.300 ","End":"12:00.635","Text":"There is no one single number that the series gets closer and closer to."},{"Start":"12:00.635 ","End":"12:02.560","Text":"So there is no limit."},{"Start":"12:02.560 ","End":"12:05.540","Text":"Therefore the series also diverges."},{"Start":"12:05.540 ","End":"12:08.390","Text":"So now we have 3 examples."},{"Start":"12:08.390 ","End":"12:12.110","Text":"This one, which diverges to infinity,"},{"Start":"12:12.110 ","End":"12:13.880","Text":"though I haven\u0027t proved it."},{"Start":"12:13.880 ","End":"12:20.495","Text":"This one, which we computed does converge and we were able to do it using the definition,"},{"Start":"12:20.495 ","End":"12:22.415","Text":"which is usually not possible."},{"Start":"12:22.415 ","End":"12:27.095","Text":"The last example which we showed using the definition that it has no limit."},{"Start":"12:27.095 ","End":"12:29.780","Text":"I think that\u0027ll do for now for this clip,"},{"Start":"12:29.780 ","End":"12:33.755","Text":"but I still owe you to show you why this series,"},{"Start":"12:33.755 ","End":"12:35.390","Text":"the sum of 1 over n,"},{"Start":"12:35.390 ","End":"12:38.075","Text":"in fact diverges to infinity."},{"Start":"12:38.075 ","End":"12:40.800","Text":"That\u0027s in a future clip."}],"ID":6500},{"Watched":false,"Name":"Infinite Geometric Series","Duration":"9m 59s","ChapterTopicVideoID":6474,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6474.jpeg","UploadDate":"2020-01-16T06:56:52.4730000","DurationForVideoObject":"PT9M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.109","Text":"In this clip, I\u0027m going to talk about infinite geometric series."},{"Start":"00:04.109 ","End":"00:08.025","Text":"We actually have encountered a geometric series before."},{"Start":"00:08.025 ","End":"00:13.560","Text":"I remember that we had the series 1/2 plus"},{"Start":"00:13.560 ","End":"00:20.160","Text":"1/2 squared plus 1/2 cubed plus and so on,"},{"Start":"00:20.160 ","End":"00:25.365","Text":"and this was the sum from 1 to infinity"},{"Start":"00:25.365 ","End":"00:31.380","Text":"of 1/2 to the power of n. We actually computed it sum."},{"Start":"00:31.380 ","End":"00:34.365","Text":"It turned out to converge and it\u0027s sum was equal to 1."},{"Start":"00:34.365 ","End":"00:39.150","Text":"In general, we\u0027ll say that a geometric series is something"},{"Start":"00:39.150 ","End":"00:44.355","Text":"of the form the sum to infinity."},{"Start":"00:44.355 ","End":"00:46.715","Text":"Now, I\u0027ll write n equals 0,"},{"Start":"00:46.715 ","End":"00:48.770","Text":"but I\u0027ll get back to that in a moment,"},{"Start":"00:48.770 ","End":"00:50.975","Text":"of q to the power of n,"},{"Start":"00:50.975 ","End":"00:52.925","Text":"where q is some number."},{"Start":"00:52.925 ","End":"00:54.875","Text":"The 0 is not essential."},{"Start":"00:54.875 ","End":"00:59.000","Text":"I could have any number here."},{"Start":"00:59.000 ","End":"01:01.950","Text":"For example, it could be 1."},{"Start":"01:02.020 ","End":"01:05.705","Text":"This thing is optional,"},{"Start":"01:05.705 ","End":"01:07.265","Text":"and it could be something else."},{"Start":"01:07.265 ","End":"01:15.210","Text":"The other thing is that I would like to eliminate or omit the possibility of q being 0."},{"Start":"01:15.210 ","End":"01:20.345","Text":"I prefer q not equal to 0 because various things don\u0027t come out right if q is 0."},{"Start":"01:20.345 ","End":"01:27.530","Text":"Yet another thing I want to say is that in some places or some books,"},{"Start":"01:27.530 ","End":"01:28.760","Text":"most books in fact,"},{"Start":"01:28.760 ","End":"01:32.345","Text":"actually generalize this to include a constant."},{"Start":"01:32.345 ","End":"01:39.230","Text":"What I mean is that I generalize it to the form a times q to the n. Also,"},{"Start":"01:39.230 ","End":"01:41.930","Text":"n goes from something to, say,"},{"Start":"01:41.930 ","End":"01:45.295","Text":"0 to infinity because"},{"Start":"01:45.295 ","End":"01:49.294","Text":"the school definition of geometric series is they have a common ratio,"},{"Start":"01:49.294 ","End":"01:55.520","Text":"q, and this would also fit the notion that successive terms have a common ratio."},{"Start":"01:55.520 ","End":"01:57.500","Text":"But here we\u0027re just going to talk about"},{"Start":"01:57.500 ","End":"02:01.340","Text":"this q to the n. If you do get a problem with this,"},{"Start":"02:01.340 ","End":"02:04.940","Text":"then no problem because this is equal to"},{"Start":"02:04.940 ","End":"02:09.290","Text":"a times the sum of q to the n. You can take the constant outside,"},{"Start":"02:09.290 ","End":"02:12.080","Text":"so we can always reduce it to this case,"},{"Start":"02:12.080 ","End":"02:14.945","Text":"which is our case with,"},{"Start":"02:14.945 ","End":"02:19.200","Text":"as I said, the option of starting not from 0."},{"Start":"02:19.200 ","End":"02:22.850","Text":"This basically the geometric series."},{"Start":"02:22.850 ","End":"02:27.755","Text":"Let\u0027s continue. I like to give some examples."},{"Start":"02:27.755 ","End":"02:32.735","Text":"The 1st 1 I\u0027ll take is just the same 1 above that we are already are familiar with."},{"Start":"02:32.735 ","End":"02:41.755","Text":"The sum from 1 to infinity of 1/2 to the power of n. Next example,"},{"Start":"02:41.755 ","End":"02:46.905","Text":"the sum from 0 to infinity,"},{"Start":"02:46.905 ","End":"02:49.245","Text":"1/4 to the power of n,"},{"Start":"02:49.245 ","End":"02:53.120","Text":"and the example sum from,"},{"Start":"02:53.120 ","End":"03:00.440","Text":"let\u0027s make it from 7 to infinity minus 1/5 to the power of n. I"},{"Start":"03:00.440 ","End":"03:08.300","Text":"could take the sum from 0 to infinity of minus 4 to the power of n,"},{"Start":"03:08.300 ","End":"03:10.220","Text":"and the last example,"},{"Start":"03:10.220 ","End":"03:13.610","Text":"let\u0027s say 5/4 to the power of n,"},{"Start":"03:13.610 ","End":"03:18.289","Text":"where n goes, I\u0027ll take it from 0 to infinity."},{"Start":"03:18.289 ","End":"03:22.805","Text":"1, 2, 3, 5 examples of geometric series."},{"Start":"03:22.805 ","End":"03:25.460","Text":"When it comes to geometric series,"},{"Start":"03:25.460 ","End":"03:30.659","Text":"there\u0027s a main proposition or theorem or claim,"},{"Start":"03:30.659 ","End":"03:33.405","Text":"so I\u0027ll write down this proposition,"},{"Start":"03:33.405 ","End":"03:35.885","Text":"and that is, the series,"},{"Start":"03:35.885 ","End":"03:42.380","Text":"the geometric series, infinite geometric series I\u0027m talking about, the series converges."},{"Start":"03:42.380 ","End":"03:47.450","Text":"I\u0027ll give you conditions for when it converges and when it diverges."},{"Start":"03:47.450 ","End":"03:55.720","Text":"It converges if and only if q is between minus 1 and 1."},{"Start":"03:55.720 ","End":"04:00.395","Text":"This is precisely the condition for the series to converge."},{"Start":"04:00.395 ","End":"04:03.110","Text":"If I\u0027m between minus 1 and 1,"},{"Start":"04:03.110 ","End":"04:06.065","Text":"it converges, otherwise, it diverges."},{"Start":"04:06.065 ","End":"04:12.770","Text":"Sometimes this is written in the book as absolute value of q is less than 1,"},{"Start":"04:12.770 ","End":"04:15.900","Text":"but you can see that that\u0027s the same thing."},{"Start":"04:16.010 ","End":"04:21.365","Text":"If we look at these and we want to know which ones converge,"},{"Start":"04:21.365 ","End":"04:25.070","Text":"just look for the q here is 1/2 and it is"},{"Start":"04:25.070 ","End":"04:28.720","Text":"between minus 1 and 1 or its absolute size is less than 1."},{"Start":"04:28.720 ","End":"04:30.715","Text":"This 1 converges."},{"Start":"04:30.715 ","End":"04:35.060","Text":"I\u0027m going to put a checkmark next to the ones that converge."},{"Start":"04:35.180 ","End":"04:38.355","Text":"1/4, yeah, that\u0027s between minus 1 and 1,"},{"Start":"04:38.355 ","End":"04:40.440","Text":"so this 1 converges."},{"Start":"04:40.440 ","End":"04:45.030","Text":"Minus a 1/5. Yes, that\u0027s between minus 1 and 1."},{"Start":"04:45.030 ","End":"04:47.165","Text":"It converges."},{"Start":"04:47.165 ","End":"04:49.220","Text":"The next 1, minus 4."},{"Start":"04:49.220 ","End":"04:51.860","Text":"No, it is not between minus 1 and 1,"},{"Start":"04:51.860 ","End":"04:54.665","Text":"so it does not converge, it diverges."},{"Start":"04:54.665 ","End":"04:58.175","Text":"Here also, 5/4 is bigger than 1,"},{"Start":"04:58.175 ","End":"05:00.725","Text":"so it also diverges."},{"Start":"05:00.725 ","End":"05:07.070","Text":"These diverged and these"},{"Start":"05:07.070 ","End":"05:14.260","Text":"3 converge just because of this condition."},{"Start":"05:16.340 ","End":"05:18.795","Text":"There\u0027s more I can say."},{"Start":"05:18.795 ","End":"05:20.795","Text":"If a series converges,"},{"Start":"05:20.795 ","End":"05:22.595","Text":"I can also tell you what its sum."},{"Start":"05:22.595 ","End":"05:26.975","Text":"The sum of the series in the convergent case,"},{"Start":"05:26.975 ","End":"05:36.110","Text":"n equals 0 to infinity of q to the n in the case where q is between minus 1 and 1,"},{"Start":"05:36.110 ","End":"05:41.429","Text":"is equal to 1 over"},{"Start":"05:41.429 ","End":"05:47.165","Text":"1 minus q. I\u0027ll show you an example."},{"Start":"05:47.165 ","End":"05:53.780","Text":"In fact, let\u0027s take the example, this 1."},{"Start":"05:53.780 ","End":"06:02.280","Text":"We know that the sum of N from 0 to infinity of 1/4 to the n,"},{"Start":"06:02.280 ","End":"06:05.130","Text":"my q here is equal to 1/4,"},{"Start":"06:05.130 ","End":"06:10.890","Text":"will equal 1 over 1 minus, now,"},{"Start":"06:10.890 ","End":"06:15.240","Text":"this same q is 1/4,"},{"Start":"06:15.240 ","End":"06:19.320","Text":"and the answer turns out to be 1 1/3."},{"Start":"06:19.320 ","End":"06:22.190","Text":"This makes sense because if you write it out,"},{"Start":"06:22.190 ","End":"06:30.775","Text":"we get 1 plus 1/4 plus 1/4 squared plus 1/4 cubed and so on."},{"Start":"06:30.775 ","End":"06:32.930","Text":"It\u0027s larger than 1 and a 1/4,"},{"Start":"06:32.930 ","End":"06:35.165","Text":"but not that much."},{"Start":"06:35.165 ","End":"06:39.450","Text":"It makes sense. Let\u0027s take another example."},{"Start":"06:39.800 ","End":"06:43.665","Text":"I want to take the other 2 that converge here,"},{"Start":"06:43.665 ","End":"06:44.915","Text":"this 1 on this 1."},{"Start":"06:44.915 ","End":"06:49.235","Text":"But I can\u0027t use this formula as is because here,"},{"Start":"06:49.235 ","End":"06:53.815","Text":"n starts with 0 and these 2 start not at 0."},{"Start":"06:53.815 ","End":"07:00.160","Text":"I\u0027m going to generalize this formula and say that in general,"},{"Start":"07:00.160 ","End":"07:05.200","Text":"the sum n goes let\u0027s say K to infinity,"},{"Start":"07:05.200 ","End":"07:08.905","Text":"where K could be 0 but could be larger than 0."},{"Start":"07:08.905 ","End":"07:12.130","Text":"This time it\u0027s equal to,"},{"Start":"07:12.130 ","End":"07:14.155","Text":"similar to the previous formula,"},{"Start":"07:14.155 ","End":"07:16.630","Text":"has the 1 minus q on the denominator."},{"Start":"07:16.630 ","End":"07:18.190","Text":"We\u0027ve done the numerator."},{"Start":"07:18.190 ","End":"07:23.400","Text":"It\u0027s q to the power of K. If K is 0,"},{"Start":"07:23.400 ","End":"07:26.855","Text":"of course, it just becomes this same formula as this."},{"Start":"07:26.855 ","End":"07:33.635","Text":"This K is here 7 and here it\u0027s 1."},{"Start":"07:33.635 ","End":"07:36.825","Text":"I\u0027ve decided to use a different color."},{"Start":"07:36.825 ","End":"07:41.145","Text":"In this case, the 1/2 to the power of n,"},{"Start":"07:41.145 ","End":"07:48.450","Text":"it comes out to be 1/2 to the n. It will be q,"},{"Start":"07:48.450 ","End":"07:52.650","Text":"which is 1/2 to the power of 1,"},{"Start":"07:52.650 ","End":"07:57.915","Text":"over 1 minus a 1/2."},{"Start":"07:57.915 ","End":"08:01.035","Text":"I\u0027ll just highlight that."},{"Start":"08:01.035 ","End":"08:03.330","Text":"This is going to equal,"},{"Start":"08:03.330 ","End":"08:05.325","Text":"let\u0027s see if we simplify it,"},{"Start":"08:05.325 ","End":"08:09.120","Text":"1/2 over 1/2, which is equal to 1."},{"Start":"08:09.120 ","End":"08:14.330","Text":"That\u0027s what we got in the previous clip where I tried to do the sum of this."},{"Start":"08:14.330 ","End":"08:21.630","Text":"We get a 1/2 plus a 1/4 plus an 1/8 and so on, a 16th."},{"Start":"08:21.630 ","End":"08:23.105","Text":"If you think about it,"},{"Start":"08:23.105 ","End":"08:27.170","Text":"it does go to 1 because we\u0027re at a 1/2 and then we go 1/2 the distance again,"},{"Start":"08:27.170 ","End":"08:33.785","Text":"they\u0027re up to 3/4 and another 7/8, 15, 16ths, 31/32."},{"Start":"08:33.785 ","End":"08:36.220","Text":"It\u0027s getting very close to 1."},{"Start":"08:36.220 ","End":"08:40.785","Text":"That makes sense. I\u0027ll just do the last 1 for completeness."},{"Start":"08:40.785 ","End":"08:47.085","Text":"The sum from n equals 7 to infinity"},{"Start":"08:47.085 ","End":"08:54.090","Text":"of minus 1/5 to the power of n is equal to our denominator,"},{"Start":"08:54.090 ","End":"08:56.130","Text":"it\u0027s always 1 minus q,"},{"Start":"08:56.130 ","End":"08:58.395","Text":"but q is negative,"},{"Start":"08:58.395 ","End":"09:08.720","Text":"so it\u0027s plus 1/5."},{"Start":"09:08.720 ","End":"09:13.225","Text":"Let\u0027s just highlight this K is 7."},{"Start":"09:13.225 ","End":"09:21.835","Text":"On the numerator, it will be minus a 1/5 to the power of 7."},{"Start":"09:21.835 ","End":"09:23.640","Text":"We could compute it."},{"Start":"09:23.640 ","End":"09:25.590","Text":"I\u0027m not going to do that now."},{"Start":"09:25.590 ","End":"09:29.740","Text":"Leave it to you to finish off."},{"Start":"09:30.040 ","End":"09:32.495","Text":"I\u0027m basically done."},{"Start":"09:32.495 ","End":"09:36.800","Text":"But just in case you do encounter the other kind,"},{"Start":"09:36.800 ","End":"09:39.075","Text":"the more general 1 with the a in front of it,"},{"Start":"09:39.075 ","End":"09:42.635","Text":"I\u0027ll give you the formula that the sum of a,"},{"Start":"09:42.635 ","End":"09:53.100","Text":"q to the n let\u0027s say from 0 to infinity happens to equal a over 1 minus q,"},{"Start":"09:53.470 ","End":"09:57.210","Text":"and now I am done."}],"ID":6530},{"Watched":false,"Name":"Exercise 1 part a","Duration":"2m 32s","ChapterTopicVideoID":6485,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6485.jpeg","UploadDate":"2016-07-04T16:25:57.9670000","DurationForVideoObject":"PT2M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.610","Text":"In this exercise, we have an infinite series and we have to say what\u0027s up with it:"},{"Start":"00:05.610 ","End":"00:10.620","Text":"Does it diverge or if it converges, and then give the sum, the value."},{"Start":"00:10.620 ","End":"00:13.990","Text":"Let me just copy it over here."},{"Start":"00:14.960 ","End":"00:19.020","Text":"It looks to me like a geometric series, and now to show you"},{"Start":"00:19.020 ","End":"00:23.080","Text":"the formula that I have for geometric series."},{"Start":"00:23.080 ","End":"00:30.645","Text":"In this case, this thing looks very much like this."},{"Start":"00:30.645 ","End":"00:34.830","Text":"I\u0027ll highlight what I mean. This part here,"},{"Start":"00:34.830 ","End":"00:44.745","Text":"if we let q equal 0.44 and we let k equals 1,"},{"Start":"00:44.745 ","End":"00:48.260","Text":"this 1 here, then this is exactly what we have."},{"Start":"00:48.260 ","End":"00:53.180","Text":"Notice that q is between minus 1 and 1."},{"Start":"00:53.180 ","End":"00:56.540","Text":"This is a check that you just have to mentally note."},{"Start":"00:56.540 ","End":"00:59.015","Text":"In that case, if I have this,"},{"Start":"00:59.015 ","End":"01:02.879","Text":"then what it equals is what\u0027s on the right-hand side."},{"Start":"01:04.820 ","End":"01:07.140","Text":"Let me just continue here,"},{"Start":"01:07.140 ","End":"01:09.830","Text":"is equal to q,"},{"Start":"01:09.830 ","End":"01:14.810","Text":"which is 0.44^k,"},{"Start":"01:14.810 ","End":"01:22.990","Text":"which is 1, over 1 minus 0.44."},{"Start":"01:22.990 ","End":"01:25.040","Text":"This is just, well,"},{"Start":"01:25.040 ","End":"01:26.915","Text":"power of 1 doesn\u0027t mean anything,"},{"Start":"01:26.915 ","End":"01:33.960","Text":"it\u0027s just itself, and 1 minus 0.44 is 0.56."},{"Start":"01:33.960 ","End":"01:36.380","Text":"I could do this straight away on the calculator,"},{"Start":"01:36.380 ","End":"01:40.160","Text":"but I\u0027d like to just do a bit by myself first."},{"Start":"01:40.160 ","End":"01:42.170","Text":"Then we can multiply by a 100,"},{"Start":"01:42.170 ","End":"01:49.520","Text":"44/56, and if I divide top and bottom by 4,"},{"Start":"01:49.520 ","End":"01:51.350","Text":"I can get what?"},{"Start":"01:51.350 ","End":"01:56.410","Text":"11/14."},{"Start":"01:56.600 ","End":"02:02.420","Text":"I could say the answer is 11 over 14 or I can do it on the calculator."},{"Start":"02:02.420 ","End":"02:04.985","Text":"You could have done it from any point on the calculator,"},{"Start":"02:04.985 ","End":"02:08.035","Text":"numerically comes out approximately"},{"Start":"02:08.035 ","End":"02:16.500","Text":"0.7857 something."},{"Start":"02:17.210 ","End":"02:22.910","Text":"I\u0027ll prefer the exact answer of the fraction either way."},{"Start":"02:22.910 ","End":"02:24.860","Text":"In answer to the question,"},{"Start":"02:24.860 ","End":"02:32.640","Text":"the series converges and this is the value it converges to. We\u0027re done."}],"ID":6531},{"Watched":false,"Name":"Exercise 1 part b","Duration":"3m 19s","ChapterTopicVideoID":6486,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6486.jpeg","UploadDate":"2016-07-04T16:29:36.2400000","DurationForVideoObject":"PT3M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, we have an infinite series here."},{"Start":"00:05.370 ","End":"00:07.200","Text":"We have to decide first of all,"},{"Start":"00:07.200 ","End":"00:09.735","Text":"if it converges or diverges."},{"Start":"00:09.735 ","End":"00:13.990","Text":"If it converges, we want to know the value of the sum."},{"Start":"00:14.780 ","End":"00:16.830","Text":"Just by the look of it,"},{"Start":"00:16.830 ","End":"00:20.129","Text":"it looks a bit like a geometric series."},{"Start":"00:20.129 ","End":"00:21.930","Text":"Let me just copy it here."},{"Start":"00:21.930 ","End":"00:25.410","Text":"I\u0027ll show you what the geometric series is."},{"Start":"00:25.410 ","End":"00:28.485","Text":"It\u0027s when we have the sum of something to the power of"},{"Start":"00:28.485 ","End":"00:32.770","Text":"n. Now we don\u0027t quite have that here."},{"Start":"00:33.530 ","End":"00:36.630","Text":"If this was also an n,"},{"Start":"00:36.630 ","End":"00:38.195","Text":"then we could say, yeah,"},{"Start":"00:38.195 ","End":"00:44.280","Text":"4/7 to the power of n. We might have to tweak this a bit to get it to look like this."},{"Start":"00:44.290 ","End":"00:49.115","Text":"What I propose is to rewrite it a bit as to say,"},{"Start":"00:49.115 ","End":"00:55.275","Text":"this is the sum from zero to infinity."},{"Start":"00:55.275 ","End":"00:59.280","Text":"I can say that this is 4 to the n,"},{"Start":"00:59.280 ","End":"01:02.910","Text":"over 7 to the n,"},{"Start":"01:02.910 ","End":"01:06.340","Text":"times 1 over 7."},{"Start":"01:06.340 ","End":"01:08.630","Text":"Because 7 to the n plus 1,"},{"Start":"01:08.630 ","End":"01:11.915","Text":"is 7 to the n times 7 and I just split the fraction up."},{"Start":"01:11.915 ","End":"01:15.140","Text":"A constant can come in front of the series."},{"Start":"01:15.140 ","End":"01:20.380","Text":"This is equal to 1/7."},{"Start":"01:20.380 ","End":"01:27.880","Text":"The sum, again from 0 to infinity of just this bit,"},{"Start":"01:27.880 ","End":"01:33.310","Text":"which I can now rewrite as 4 over 7 to the power of"},{"Start":"01:33.310 ","End":"01:41.305","Text":"n. Now it looks very much like this."},{"Start":"01:41.305 ","End":"01:44.619","Text":"I\u0027ll highlight the part I mean,"},{"Start":"01:44.619 ","End":"01:47.680","Text":"ignoring the 1/7 part which stays,"},{"Start":"01:47.680 ","End":"01:54.115","Text":"but if I take q to be equal to 4/7,"},{"Start":"01:54.115 ","End":"01:59.275","Text":"then I\u0027ve got exactly this as far as the sum goes."},{"Start":"01:59.275 ","End":"02:03.460","Text":"Notice also that 4/7 is between minus 1 and 1."},{"Start":"02:03.460 ","End":"02:06.260","Text":"This is a check that you have to do at"},{"Start":"02:06.260 ","End":"02:10.730","Text":"least mentally just to note otherwise the formula doesn\u0027t work."},{"Start":"02:10.730 ","End":"02:16.490","Text":"Now what we get is 1/7 that just sticks around."},{"Start":"02:16.490 ","End":"02:23.015","Text":"This part, I can replace with this formula by 1 over 1 minus q."},{"Start":"02:23.015 ","End":"02:25.385","Text":"But our q is 4/7."},{"Start":"02:25.385 ","End":"02:26.975","Text":"This is what we get."},{"Start":"02:26.975 ","End":"02:30.485","Text":"Now we just need to simplify it a bit using fractions."},{"Start":"02:30.485 ","End":"02:41.040","Text":"1/7 times 1 minus 4/7 is 3/7."},{"Start":"02:41.040 ","End":"02:45.920","Text":"1 over a fraction is the inverse fraction,"},{"Start":"02:45.920 ","End":"02:50.685","Text":"so it\u0027s 1/7 times 7 over 3."},{"Start":"02:50.685 ","End":"02:56.820","Text":"The 7s\u0027 cancel and the answer is 1/3."},{"Start":"02:56.820 ","End":"02:58.800","Text":"I\u0027ll highlight that."},{"Start":"02:58.800 ","End":"03:02.570","Text":"For those of you who don\u0027t like fractions, but like decimals,"},{"Start":"03:02.570 ","End":"03:06.050","Text":"this is 0.333, etc,"},{"Start":"03:06.050 ","End":"03:08.255","Text":"as many as you care to write."},{"Start":"03:08.255 ","End":"03:11.330","Text":"An answer to the original question,"},{"Start":"03:11.330 ","End":"03:19.380","Text":"the series converges and the value it converges to is 1/3, we\u0027re done."}],"ID":6532},{"Watched":false,"Name":"Exercise 1 part c","Duration":"4m 1s","ChapterTopicVideoID":6487,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6487.jpeg","UploadDate":"2016-07-04T16:33:27.0570000","DurationForVideoObject":"PT4M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"We have here this infinite series and the question\u0027s asking us to"},{"Start":"00:05.970 ","End":"00:11.580","Text":"decide if it converges or diverges and if it converges,"},{"Start":"00:11.580 ","End":"00:14.025","Text":"to what value does it converge?"},{"Start":"00:14.025 ","End":"00:18.840","Text":"It looks very much like the geometric series, but not quite."},{"Start":"00:18.840 ","End":"00:25.410","Text":"Let me just copy the question here and I\u0027ll show you the formula for a geometric series."},{"Start":"00:25.410 ","End":"00:27.644","Text":"I\u0027m looking at this,"},{"Start":"00:27.644 ","End":"00:31.450","Text":"and I\u0027m looking at this bit here."},{"Start":"00:32.420 ","End":"00:37.470","Text":"If I just had a single q to the power of n,"},{"Start":"00:37.470 ","End":"00:39.240","Text":"that would do me,"},{"Start":"00:39.240 ","End":"00:41.640","Text":"but I don\u0027t quite have that."},{"Start":"00:41.640 ","End":"00:43.450","Text":"First of all, I have 3 bits,"},{"Start":"00:43.450 ","End":"00:48.465","Text":"I could combine those if they were all the same exponent n,"},{"Start":"00:48.465 ","End":"00:52.310","Text":"but with a little bit of algebra,"},{"Start":"00:52.310 ","End":"00:54.980","Text":"a little bit of powers,"},{"Start":"00:54.980 ","End":"00:58.500","Text":"formulas, we can get it to that form."},{"Start":"00:58.520 ","End":"01:01.895","Text":"What I suggest, first of all,"},{"Start":"01:01.895 ","End":"01:10.080","Text":"is to write the sum as this."},{"Start":"01:10.080 ","End":"01:12.885","Text":"Everything\u0027s the same except the denominator,"},{"Start":"01:12.885 ","End":"01:17.960","Text":"4 to the power of n plus 2 is 4 to the power of n times 4 to the power of 2,"},{"Start":"01:17.960 ","End":"01:22.760","Text":"so it\u0027s 4 to the power of n times 16,"},{"Start":"01:22.760 ","End":"01:24.529","Text":"which is the 4 squared,"},{"Start":"01:24.529 ","End":"01:28.055","Text":"and the sum is from 1 to infinity."},{"Start":"01:28.055 ","End":"01:30.440","Text":"Now what I can do is, first of all,"},{"Start":"01:30.440 ","End":"01:34.250","Text":"I can take the 1/16 out and I can put"},{"Start":"01:34.250 ","End":"01:38.180","Text":"it in front of the series because it doesn\u0027t contain n. So I get"},{"Start":"01:38.180 ","End":"01:48.600","Text":"1/16 of the sum of minus 1 to the n."},{"Start":"01:48.600 ","End":"01:51.530","Text":"I could put it in the top of the fraction, it doesn\u0027t matter."},{"Start":"01:51.530 ","End":"01:56.780","Text":"5 to the n over 4 to the n and again,"},{"Start":"01:56.780 ","End":"02:00.780","Text":"n goes from 1 to infinity."},{"Start":"02:02.060 ","End":"02:05.517","Text":"By using the rules of exponents"},{"Start":"02:05.517 ","End":"02:09.730","Text":"with products and quotients to the power of,"},{"Start":"02:09.730 ","End":"02:14.150","Text":"this is straightforward to see that I could"},{"Start":"02:14.150 ","End":"02:19.565","Text":"actually combine all these as 1 single thing to the power of n,"},{"Start":"02:19.565 ","End":"02:23.765","Text":"which is minus 1 times 5 over 4,"},{"Start":"02:23.765 ","End":"02:27.800","Text":"in other words, minus 5 over 4."},{"Start":"02:27.800 ","End":"02:37.790","Text":"I\u0027m using the formulas that ab to the power of n is a to the n,"},{"Start":"02:37.790 ","End":"02:43.610","Text":"b to the n, only in reverse and also a over b to the power of n is a"},{"Start":"02:43.610 ","End":"02:46.060","Text":"to the n over b to the power of n."},{"Start":"02:46.060 ","End":"02:49.820","Text":"Even though here I have a case like a to the n,"},{"Start":"02:49.820 ","End":"02:53.720","Text":"b to the n over c to the n and combining these,"},{"Start":"02:53.720 ","End":"02:59.465","Text":"I can say that this is ab over c to the power of n, just to combo."},{"Start":"02:59.465 ","End":"03:06.825","Text":"Anyway, this is what we get and now this series,"},{"Start":"03:06.825 ","End":"03:12.635","Text":"without the constant here, looks exactly like this."},{"Start":"03:12.635 ","End":"03:24.135","Text":"If we let k equals 1 and q to be equal to minus 5 over 4,"},{"Start":"03:24.135 ","End":"03:27.150","Text":"I\u0027ll write it as minus 1.25."},{"Start":"03:27.150 ","End":"03:29.805","Text":"I have a reason for writing it in decimal."},{"Start":"03:29.805 ","End":"03:32.613","Text":"That reason is that it\u0027s easier to see"},{"Start":"03:32.613 ","End":"03:36.950","Text":"that it is not between minus 1 and 1."},{"Start":"03:36.950 ","End":"03:40.010","Text":"It\u0027s here in red because it\u0027s important."},{"Start":"03:40.010 ","End":"03:44.345","Text":"When this happens, the series definitely diverges."},{"Start":"03:44.345 ","End":"03:48.650","Text":"The 1/16 doesn\u0027t make any difference to convergences or divergences."},{"Start":"03:48.650 ","End":"03:57.660","Text":"So what I say is that the series diverges and there is no value,"},{"Start":"03:57.660 ","End":"04:00.490","Text":"and so we are done."}],"ID":6533},{"Watched":false,"Name":"Exercise 1 part d","Duration":"4m 2s","ChapterTopicVideoID":6488,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6488.jpeg","UploadDate":"2016-07-04T16:37:10.6970000","DurationForVideoObject":"PT4M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"We have here an infinite series,"},{"Start":"00:03.570 ","End":"00:07.560","Text":"and we have to decide if it converges or diverges."},{"Start":"00:07.560 ","End":"00:08.970","Text":"If it happens to converge,"},{"Start":"00:08.970 ","End":"00:12.765","Text":"you want to actually give the value of the sum of the series."},{"Start":"00:12.765 ","End":"00:18.465","Text":"Let\u0027s see, it looks like a geometric series but not quite."},{"Start":"00:18.465 ","End":"00:21.105","Text":"Let me first of all copy the exercise here,"},{"Start":"00:21.105 ","End":"00:25.530","Text":"and I\u0027ll show you what geometric series formulas are."},{"Start":"00:25.530 ","End":"00:31.680","Text":"Geometric series essentially the general term is q^n."},{"Start":"00:31.680 ","End":"00:33.315","Text":"We don\u0027t quite have that,"},{"Start":"00:33.315 ","End":"00:36.450","Text":"that\u0027s why we have an extra constant in front,"},{"Start":"00:36.450 ","End":"00:39.465","Text":"and also it\u0027s not n but it\u0027s 2n."},{"Start":"00:39.465 ","End":"00:43.310","Text":"With a little bit of algebraic manipulation and trickery,"},{"Start":"00:43.310 ","End":"00:46.135","Text":"we could get it into this form."},{"Start":"00:46.135 ","End":"00:49.545","Text":"What I suggest is this,"},{"Start":"00:49.545 ","End":"00:53.720","Text":"a constant can come out in front of the summation,"},{"Start":"00:53.720 ","End":"01:02.130","Text":"so I can write minus 4 times the sum from 0 to infinity of this."},{"Start":"01:02.130 ","End":"01:06.300","Text":"But I can also change this using the rules of exponents,"},{"Start":"01:13.360 ","End":"01:19.505","Text":"a^b^c is a^bc, you multiply the exponents."},{"Start":"01:19.505 ","End":"01:21.200","Text":"In our case, this"},{"Start":"01:21.200 ","End":"01:30.260","Text":"would be 3/4^ 2^n,"},{"Start":"01:30.260 ","End":"01:32.740","Text":"because 2 times n is 2n."},{"Start":"01:32.740 ","End":"01:43.665","Text":"Now, this really is like this because we have a 0 here."},{"Start":"01:43.665 ","End":"01:52.995","Text":"If I take q to be equal to 3/4 squared,"},{"Start":"01:52.995 ","End":"01:56.500","Text":"I could write this as 9/16."},{"Start":"01:56.610 ","End":"02:01.090","Text":"Then what we have is we have a constant here,"},{"Start":"02:01.090 ","End":"02:06.000","Text":"but the series is the sum from 0 to infinity of q^n,"},{"Start":"02:06.000 ","End":"02:07.990","Text":"and we\u0027ll be able to use this."},{"Start":"02:07.990 ","End":"02:09.670","Text":"We just have to note,"},{"Start":"02:09.670 ","End":"02:16.595","Text":"even just mentally that 9/16 really does fall between minus 1 and 1."},{"Start":"02:16.595 ","End":"02:19.920","Text":"Obviously, 9/16 is less than 1 and bigger than 0,"},{"Start":"02:19.920 ","End":"02:22.470","Text":"so it certainly satisfies this."},{"Start":"02:22.470 ","End":"02:32.400","Text":"We can use this formula to write this instead as a sum as this."},{"Start":"02:32.400 ","End":"02:35.920","Text":"Let me just get some space here."},{"Start":"02:37.940 ","End":"02:44.745","Text":"What we get is the minus 4 stays there,"},{"Start":"02:44.745 ","End":"02:49.005","Text":"the series becomes 1 over 1 minus q,"},{"Start":"02:49.005 ","End":"02:52.215","Text":"so it\u0027s 1 over 1 minus,"},{"Start":"02:52.215 ","End":"02:55.830","Text":"and q we said was 9/16."},{"Start":"02:55.830 ","End":"03:01.605","Text":"Now, it\u0027s just a little bit of simplifying fractions,"},{"Start":"03:01.605 ","End":"03:06.390","Text":"so we get minus 4 times 1 over,"},{"Start":"03:06.390 ","End":"03:13.335","Text":"this is 7/16 because 1 is 16 over 16."},{"Start":"03:13.335 ","End":"03:20.640","Text":"Then 1 over a fraction it\u0027s the reciprocal, the inverse fraction,"},{"Start":"03:20.640 ","End":"03:26.975","Text":"so it\u0027s minus 4 multiplied by 16 over 7,"},{"Start":"03:26.975 ","End":"03:35.860","Text":"and that comes out to be minus 64 over 7."},{"Start":"03:35.860 ","End":"03:38.440","Text":"That\u0027s the answer."},{"Start":"03:38.440 ","End":"03:41.590","Text":"For those of you who like decimals,"},{"Start":"03:41.590 ","End":"03:46.180","Text":"this is approximately equal to 9.14 something,"},{"Start":"03:46.180 ","End":"03:51.515","Text":"something, but in any rate the series converges."},{"Start":"03:51.515 ","End":"03:53.580","Text":"With the geometric series,"},{"Start":"03:53.580 ","End":"03:57.785","Text":"this is exactly the convergence condition which we met,"},{"Start":"03:57.785 ","End":"04:02.490","Text":"converges to this value. We\u0027re done."}],"ID":6534},{"Watched":false,"Name":"Exercise 1 part e","Duration":"7m 13s","ChapterTopicVideoID":6489,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6489.jpeg","UploadDate":"2016-07-04T16:43:31.7930000","DurationForVideoObject":"PT7M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.670","Text":"We have here an infinite series and we have to find out whether it converges or diverges."},{"Start":"00:08.670 ","End":"00:13.845","Text":"If it converges, we actually have to give the value of the sum of the series."},{"Start":"00:13.845 ","End":"00:17.160","Text":"It looks a bit like a geometric series,"},{"Start":"00:17.160 ","End":"00:20.115","Text":"where it has some elements to it, but not quite."},{"Start":"00:20.115 ","End":"00:23.715","Text":"We have this sum in the middle."},{"Start":"00:23.715 ","End":"00:26.850","Text":"But I have an idea that maybe if we split"},{"Start":"00:26.850 ","End":"00:31.380","Text":"up the series into 2 get this plus somehow to split it up,"},{"Start":"00:31.380 ","End":"00:37.065","Text":"then maybe we\u0027ll get the sum of 2 series and each 1 of them will be a geometric."},{"Start":"00:37.065 ","End":"00:40.480","Text":"Let\u0027s try to do some algebra on this."},{"Start":"00:40.480 ","End":"00:44.270","Text":"Let me first copy the exercise and I\u0027ll also show you"},{"Start":"00:44.270 ","End":"00:47.945","Text":"the formula for the geometric series."},{"Start":"00:47.945 ","End":"00:53.390","Text":"Basically the geometric series has to look like"},{"Start":"00:53.390 ","End":"00:59.120","Text":"the sum of something to the power of n and we don\u0027t quite have that,"},{"Start":"00:59.120 ","End":"01:00.695","Text":"but we will get to it."},{"Start":"01:00.695 ","End":"01:06.785","Text":"First thing I can do is split this into 2 fractions using basic algebra."},{"Start":"01:06.785 ","End":"01:09.905","Text":"If I have a plus b over c,"},{"Start":"01:09.905 ","End":"01:13.370","Text":"I can write that as a over c plus b over c,"},{"Start":"01:13.370 ","End":"01:16.000","Text":"fractions with the same denominator."},{"Start":"01:16.000 ","End":"01:26.310","Text":"This equals the sum of 4 to the n over 7 to"},{"Start":"01:26.310 ","End":"01:30.720","Text":"the n plus negative 5 to"},{"Start":"01:30.720 ","End":"01:37.750","Text":"the n over 7 to the n and of course n goes from 1 to infinity."},{"Start":"01:39.020 ","End":"01:44.080","Text":"The other thing we can do now is use the property"},{"Start":"01:44.080 ","End":"01:53.860","Text":"that if I have a sum inside,"},{"Start":"01:53.860 ","End":"01:59.860","Text":"I can take the infinite sum of each of these."},{"Start":"01:59.860 ","End":"02:03.790","Text":"In other words, if I express it right then I\u0027ll just show you what I mean."},{"Start":"02:03.790 ","End":"02:15.890","Text":"This is equal to the sum from n equals 1 to infinity of 4 to the n over 7 to the n plus"},{"Start":"02:15.890 ","End":"02:22.729","Text":"the sum from n goes from 1 to infinity of minus 5"},{"Start":"02:22.729 ","End":"02:30.755","Text":"to the n over 7 to the n, 2 separate series."},{"Start":"02:30.755 ","End":"02:33.845","Text":"This has some strings attached."},{"Start":"02:33.845 ","End":"02:38.060","Text":"It\u0027s certainly true if both of these converge and"},{"Start":"02:38.060 ","End":"02:42.125","Text":"this will turn out to be the case and retroactively justify it."},{"Start":"02:42.125 ","End":"02:46.520","Text":"I\u0027ll just make a remark that in general,"},{"Start":"02:46.520 ","End":"02:51.910","Text":"convergent plus convergent equals convergent"},{"Start":"02:51.910 ","End":"02:58.715","Text":"and convergent plus divergent is divergent."},{"Start":"02:58.715 ","End":"03:03.035","Text":"But if they both turned out to be divergent,"},{"Start":"03:03.035 ","End":"03:06.875","Text":"then we just don\u0027t know what that might be."},{"Start":"03:06.875 ","End":"03:10.730","Text":"I know it will turn out that both of these convergence and that will"},{"Start":"03:10.730 ","End":"03:16.075","Text":"retroactively justify the splitting. Let\u0027s continue."},{"Start":"03:16.075 ","End":"03:23.030","Text":"What I need now is another formula from algebra with exponents is that a over b to"},{"Start":"03:23.030 ","End":"03:29.860","Text":"the power of n is a to the n over b to the n,"},{"Start":"03:29.860 ","End":"03:32.660","Text":"but I need it in the opposite direction."},{"Start":"03:32.660 ","End":"03:43.275","Text":"In short, what we get here is the sum from 1 to infinity of 4/7 to the power of n plus"},{"Start":"03:43.275 ","End":"03:48.210","Text":"the sum from 1 to infinity of minus"},{"Start":"03:48.210 ","End":"03:57.455","Text":"5/7 to the power of n. Now each of these pieces is a geometric series."},{"Start":"03:57.455 ","End":"04:06.900","Text":"Here we have a piece with k equals 1 and q is 4/7."},{"Start":"04:07.130 ","End":"04:10.865","Text":"Here, if we look at this form,"},{"Start":"04:10.865 ","End":"04:13.445","Text":"we also have same k,"},{"Start":"04:13.445 ","End":"04:15.605","Text":"k is this bit here,"},{"Start":"04:15.605 ","End":"04:21.035","Text":"but this time q is minus 5/7."},{"Start":"04:21.035 ","End":"04:25.850","Text":"Now, the important thing is that in each of the cases,"},{"Start":"04:25.850 ","End":"04:29.995","Text":"q is between minus 1 and 1."},{"Start":"04:29.995 ","End":"04:35.140","Text":"It\u0027s obvious, 4/7 is less than 1 and 5/7 is less than 1,"},{"Start":"04:35.140 ","End":"04:36.560","Text":"which means that when it\u0027s negative,"},{"Start":"04:36.560 ","End":"04:38.680","Text":"it\u0027s bigger than minus 1."},{"Start":"04:38.680 ","End":"04:41.540","Text":"I know that both of these are going to converge and now I\u0027ve"},{"Start":"04:41.540 ","End":"04:44.110","Text":"got my justification for the splitting."},{"Start":"04:44.110 ","End":"04:47.420","Text":"All I need now is the actual computation."},{"Start":"04:47.420 ","End":"04:50.000","Text":"I\u0027m using this formula."},{"Start":"04:50.000 ","End":"04:51.875","Text":"In the first case,"},{"Start":"04:51.875 ","End":"04:54.960","Text":"I\u0027ll continue the equals over here."},{"Start":"04:55.450 ","End":"04:59.940","Text":"Just indicate that I\u0027m continuing over here."},{"Start":"05:00.670 ","End":"05:04.070","Text":"The first 1, k is 1,"},{"Start":"05:04.070 ","End":"05:06.425","Text":"so q to the k is just q."},{"Start":"05:06.425 ","End":"05:15.990","Text":"That\u0027s 4/7 over 1 minus 4/7."},{"Start":"05:16.340 ","End":"05:20.765","Text":"The other bit, same formula."},{"Start":"05:20.765 ","End":"05:23.170","Text":"Q to the power of 1 is"},{"Start":"05:23.170 ","End":"05:33.210","Text":"just minus 5/7 over 1 minus, minus 5/7."},{"Start":"05:33.290 ","End":"05:36.770","Text":"All we\u0027re left with now a bit of work in fractions."},{"Start":"05:36.770 ","End":"05:39.110","Text":"Let\u0027s see what we can do with that."},{"Start":"05:39.110 ","End":"05:43.070","Text":"The first bit is 4/7 over"},{"Start":"05:43.070 ","End":"05:51.570","Text":"7/7 minus 4/7 is 3/7."},{"Start":"05:51.610 ","End":"05:59.930","Text":"The 2nd bit is minus 5/7 over 1 minus,"},{"Start":"05:59.930 ","End":"06:03.030","Text":"minus 5/7 is 1 plus 5/7."},{"Start":"06:03.030 ","End":"06:07.225","Text":"1 plus 5/7 is 12/7."},{"Start":"06:07.225 ","End":"06:09.445","Text":"That\u0027s an easy fraction work."},{"Start":"06:09.445 ","End":"06:16.680","Text":"1 is 7/7 another 5/7, so 12/7."},{"Start":"06:16.680 ","End":"06:19.420","Text":"Everywhere we\u0027ll multiply top and bottom by 7."},{"Start":"06:19.420 ","End":"06:28.730","Text":"What I get is 4 over 3 minus 5 over 12."},{"Start":"06:29.330 ","End":"06:35.860","Text":"More fraction work, common denominator 12."},{"Start":"06:38.690 ","End":"06:45.490","Text":"Let see. If you multiply here top and bottom by 4."},{"Start":"06:45.490 ","End":"06:51.880","Text":"I\u0027ve got 16 over 12 here minus 5 over 12 here."},{"Start":"06:51.880 ","End":"06:55.555","Text":"My final answer is"},{"Start":"06:55.555 ","End":"07:02.915","Text":"11 over 12 and this is the answer."},{"Start":"07:02.915 ","End":"07:06.080","Text":"In other words, the original series"},{"Start":"07:06.080 ","End":"07:13.800","Text":"converges and it converges to this value and we\u0027re done."}],"ID":6535},{"Watched":false,"Name":"Exercise 1 part f","Duration":"7m 29s","ChapterTopicVideoID":6490,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6490.jpeg","UploadDate":"2016-07-04T16:49:41.5070000","DurationForVideoObject":"PT7M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.310 ","End":"00:07.770","Text":"this infinite series, and we have to decide if it converges or diverges,"},{"Start":"00:07.770 ","End":"00:11.415","Text":"and if it converges to give the value of the sum."},{"Start":"00:11.415 ","End":"00:15.840","Text":"There are elements here of a geometric series,"},{"Start":"00:15.840 ","End":"00:17.340","Text":"but not quite so."},{"Start":"00:17.340 ","End":"00:18.900","Text":"I think with a bit of work,"},{"Start":"00:18.900 ","End":"00:20.370","Text":"a bit of algebra,"},{"Start":"00:20.370 ","End":"00:23.385","Text":"we could get it to be a geometric series."},{"Start":"00:23.385 ","End":"00:27.450","Text":"Let me just copy it over here and I\u0027ll remind"},{"Start":"00:27.450 ","End":"00:32.550","Text":"you of the formula for infinite geometric series."},{"Start":"00:32.550 ","End":"00:37.080","Text":"What we\u0027re talking about is this one here,"},{"Start":"00:37.080 ","End":"00:40.090","Text":"I\u0027ll just highlight it."},{"Start":"00:40.270 ","End":"00:44.840","Text":"I\u0027d like to get this to be the sum from"},{"Start":"00:44.840 ","End":"00:50.630","Text":"4 to infinity of something to the power of n. I don\u0027t quite have that,"},{"Start":"00:50.630 ","End":"00:59.930","Text":"so let\u0027s do a bit of manipulation on the general term, and I\u0027ll do it as a side exercise."},{"Start":"00:59.930 ","End":"01:05.585","Text":"Let\u0027s see, if I take just 2^3n plus 4,"},{"Start":"01:05.585 ","End":"01:09.244","Text":"times 3^1 minus 2n,"},{"Start":"01:09.244 ","End":"01:15.945","Text":"and I want it to look something like q^n or a variant of it,"},{"Start":"01:15.945 ","End":"01:17.760","Text":"let\u0027s see what I can do."},{"Start":"01:17.760 ","End":"01:23.360","Text":"One thing I can do is use the rules of exponents,"},{"Start":"01:23.360 ","End":"01:24.980","Text":"I\u0027m sure you know them by now."},{"Start":"01:24.980 ","End":"01:26.705","Text":"If I have 2 to the power of a sum,"},{"Start":"01:26.705 ","End":"01:32.185","Text":"I can say it\u0027s the product 2^3n times 2^4,"},{"Start":"01:32.185 ","End":"01:34.920","Text":"I\u0027m not going to write the formula, you should know it by now."},{"Start":"01:34.920 ","End":"01:41.610","Text":"The next one is 3^1, and because it\u0027s a minus,"},{"Start":"01:41.610 ","End":"01:45.580","Text":"divide it by 3^2n."},{"Start":"01:46.210 ","End":"01:50.705","Text":"Now, what I can do is this,"},{"Start":"01:50.705 ","End":"01:59.280","Text":"I can take the 2^4 times 3 separately first,"},{"Start":"01:59.280 ","End":"02:03.345","Text":"so I have 2^4 times 3^1,"},{"Start":"02:03.345 ","End":"02:05.830","Text":"and then I have"},{"Start":"02:10.310 ","End":"02:17.400","Text":"2^3n over 3^2n."},{"Start":"02:17.400 ","End":"02:20.830","Text":"I\u0027m getting closer, but I still need that something to the power of"},{"Start":"02:20.830 ","End":"02:27.460","Text":"n. So let\u0027s use another rule of exponents and say that this is,"},{"Start":"02:27.460 ","End":"02:30.780","Text":"well like this, I can only multiply, 2^4 is 16,"},{"Start":"02:30.780 ","End":"02:35.505","Text":"16 times 3 is 48,"},{"Start":"02:35.505 ","End":"02:39.100","Text":"so 48 is the number part."},{"Start":"02:39.100 ","End":"02:45.520","Text":"Now, this I\u0027ll write as 2^3 to the power of n, again,"},{"Start":"02:45.520 ","End":"02:51.985","Text":"using a rule of exponents that a power of a power, and the denominator as 3^2"},{"Start":"02:51.985 ","End":"02:55.345","Text":"to the power of n,"},{"Start":"02:55.345 ","End":"02:59.240","Text":"and then, I\u0027ll use yet another rule of exponents that when you have"},{"Start":"02:59.240 ","End":"03:03.980","Text":"a fraction and the same power, numerator and denominator,"},{"Start":"03:03.980 ","End":"03:12.545","Text":"I can say that this is just 48 times 2 cubed over 3 squared,"},{"Start":"03:12.545 ","End":"03:17.000","Text":"all to the power of n. Since these are easy numbers,"},{"Start":"03:17.000 ","End":"03:18.695","Text":"let me just do the calculation,"},{"Start":"03:18.695 ","End":"03:22.175","Text":"so it\u0027s 48 times 2 cubed is 8,"},{"Start":"03:22.175 ","End":"03:24.875","Text":"3 squared is 9."},{"Start":"03:24.875 ","End":"03:29.090","Text":"I have 8/9^n."},{"Start":"03:29.090 ","End":"03:33.475","Text":"Now I\u0027m going to take this and go all the way back here."},{"Start":"03:33.475 ","End":"03:41.225","Text":"I can now say that this equals the sum from n equals 4 to infinity"},{"Start":"03:41.225 ","End":"03:51.810","Text":"of 48 times 8/9^n."},{"Start":"03:51.810 ","End":"03:56.880","Text":"A constant that doesn\u0027t contain n can come out of the summation,"},{"Start":"03:56.880 ","End":"04:01.720","Text":"so I can get 48 sum Sigma,"},{"Start":"04:01.720 ","End":"04:12.190","Text":"n equals 4 to n equals infinity of 8/9^n."},{"Start":"04:12.190 ","End":"04:15.495","Text":"Now look, this bit,"},{"Start":"04:15.495 ","End":"04:17.985","Text":"looks exactly like this."},{"Start":"04:17.985 ","End":"04:25.695","Text":"If I let k equals 4 and q equals 8/9,"},{"Start":"04:25.695 ","End":"04:28.960","Text":"this is exactly what I have, and then I\u0027ll be"},{"Start":"04:28.960 ","End":"04:32.285","Text":"able to say it equals what\u0027s on the right-hand side."},{"Start":"04:32.285 ","End":"04:33.880","Text":"Before I do that,"},{"Start":"04:33.880 ","End":"04:36.745","Text":"this thing is in red here for a reason."},{"Start":"04:36.745 ","End":"04:42.865","Text":"It\u0027s very important to check that our q really is between minus 1 and 1,"},{"Start":"04:42.865 ","End":"04:45.305","Text":"and certainly 8/9 is that."},{"Start":"04:45.305 ","End":"04:46.895","Text":"I mean, it\u0027s close to 1,"},{"Start":"04:46.895 ","End":"04:49.365","Text":"but it\u0027s definitely less than 1."},{"Start":"04:49.365 ","End":"04:55.510","Text":"I can now apply the formula and say that this equals,"},{"Start":"04:55.510 ","End":"04:58.425","Text":"and I\u0027m just going down here, 2."},{"Start":"04:58.425 ","End":"05:02.160","Text":"The 48 stays, because it was there."},{"Start":"05:02.160 ","End":"05:06.960","Text":"This bit from the formula is q^k, which means,"},{"Start":"05:06.960 ","End":"05:14.910","Text":"8/9^4, divided by 1 minus q,"},{"Start":"05:14.910 ","End":"05:19.335","Text":"which is 1 minus 8/9."},{"Start":"05:19.335 ","End":"05:22.460","Text":"Now, this is just a fraction and we just have to do a little bit"},{"Start":"05:22.460 ","End":"05:27.220","Text":"of simplification. Let\u0027s see."},{"Start":"05:27.220 ","End":"05:35.100","Text":"This is equal to 48."},{"Start":"05:35.100 ","End":"05:39.185","Text":"Now, 8/9^4,"},{"Start":"05:39.185 ","End":"05:44.460","Text":"let me just leave that for the moment as 8/9^4,"},{"Start":"05:44.460 ","End":"05:46.380","Text":"and then I have 1"},{"Start":"05:46.380 ","End":"05:55.040","Text":"over 1 minus 8/9,"},{"Start":"05:55.040 ","End":"05:56.180","Text":"I\u0027ll just write this right away,"},{"Start":"05:56.180 ","End":"05:58.355","Text":"you can see that this is 1/9."},{"Start":"05:58.355 ","End":"06:02.570","Text":"Also, I\u0027ll erase this, write 1/9,"},{"Start":"06:02.570 ","End":"06:06.865","Text":"and 1 over 1/9 is just 9,"},{"Start":"06:06.865 ","End":"06:09.600","Text":"and so I can multiply the 9, obviously we have like 48 times 9"},{"Start":"06:09.600 ","End":"06:20.780","Text":"times 8, what I meant was,"},{"Start":"06:20.780 ","End":"06:26.580","Text":"I\u0027m going to expand this 8^4 over 9^4."},{"Start":"06:26.980 ","End":"06:30.090","Text":"I know I\u0027m going to need a calculator at some point,"},{"Start":"06:30.090 ","End":"06:32.960","Text":"I\u0027m just trying to hold off for awhile,"},{"Start":"06:32.960 ","End":"06:35.255","Text":"1 of the 9s cancels,"},{"Start":"06:35.255 ","End":"06:45.510","Text":"so I\u0027ve got 48 times 8^4 over 9 cubed."},{"Start":"06:46.990 ","End":"06:50.330","Text":"I\u0027ll just leave it to the calculator."},{"Start":"06:50.330 ","End":"06:55.070","Text":"Anyway, I did the calculation that side, and you don\u0027t need to see it,"},{"Start":"06:55.070 ","End":"06:56.540","Text":"I\u0027ll just give you the answer."},{"Start":"06:56.540 ","End":"07:08.790","Text":"It\u0027s 269 and 169 over 243,"},{"Start":"07:08.790 ","End":"07:10.150","Text":"and I hope that\u0027s right,"},{"Start":"07:10.150 ","End":"07:12.035","Text":"but in any event,"},{"Start":"07:12.035 ","End":"07:14.550","Text":"it\u0027s right up to here."},{"Start":"07:15.320 ","End":"07:21.815","Text":"The series converges, because the question asked us to say,"},{"Start":"07:21.815 ","End":"07:29.130","Text":"converges to the sum for the value, at which is this, and we are done."}],"ID":6536},{"Watched":false,"Name":"Exercise 1 part g","Duration":"4m 22s","ChapterTopicVideoID":6491,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6491.jpeg","UploadDate":"2017-03-20T12:39:23.8230000","DurationForVideoObject":"PT4M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"We have here an infinite series and we have to"},{"Start":"00:03.150 ","End":"00:06.570","Text":"decide if this series converges or diverges."},{"Start":"00:06.570 ","End":"00:08.400","Text":"In the convergence case,"},{"Start":"00:08.400 ","End":"00:13.480","Text":"we actually have to say what is the value of this infinite sum."},{"Start":"00:14.000 ","End":"00:17.925","Text":"Let\u0027s, first of all, copy it here."},{"Start":"00:17.925 ","End":"00:19.770","Text":"It has all the elements of"},{"Start":"00:19.770 ","End":"00:22.920","Text":"a geometric series although we\u0027ll have to do some work on it first"},{"Start":"00:22.920 ","End":"00:28.300","Text":"and I\u0027ll just bring the formulas for geometric series."},{"Start":"00:28.580 ","End":"00:32.780","Text":"What I would like to do would be to manipulate this a bit and to"},{"Start":"00:32.780 ","End":"00:37.075","Text":"be able to use this formula here."},{"Start":"00:37.075 ","End":"00:39.610","Text":"K would be 3."},{"Start":"00:39.610 ","End":"00:41.045","Text":"But the question is,"},{"Start":"00:41.045 ","End":"00:43.595","Text":"how do I get it to be q to the power of n?"},{"Start":"00:43.595 ","End":"00:47.180","Text":"Well, a little bit of algebraic manipulation."},{"Start":"00:47.180 ","End":"00:50.960","Text":"What I\u0027ll do is just work on the general element"},{"Start":"00:50.960 ","End":"00:55.310","Text":"at the side and see if we can simplify it a bit."},{"Start":"00:55.310 ","End":"01:01.025","Text":"I have minus 5 to the power of 3 minus n"},{"Start":"01:01.025 ","End":"01:08.225","Text":"over 8 to the power of 2 minus n. We\u0027re going to use rules of exponents,"},{"Start":"01:08.225 ","End":"01:10.340","Text":"which I\u0027m sure you\u0027re familiar with already."},{"Start":"01:10.340 ","End":"01:18.980","Text":"For example, here, I can break it up into minus 5 to the power of 3 and minus"},{"Start":"01:18.980 ","End":"01:28.955","Text":"5 to the power of minus n divided by 8 to the power of 2,"},{"Start":"01:28.955 ","End":"01:34.655","Text":"8 to the power of minus n. Now, in general,"},{"Start":"01:34.655 ","End":"01:37.790","Text":"when I have a to the power of minus n,"},{"Start":"01:37.790 ","End":"01:40.910","Text":"it\u0027s 1 over a to the n. What I want to"},{"Start":"01:40.910 ","End":"01:45.440","Text":"do is bring the negative exponents to the opposite side."},{"Start":"01:45.440 ","End":"01:49.130","Text":"What I\u0027ll get is, well, this, I can compute,"},{"Start":"01:49.130 ","End":"01:53.850","Text":"the minus 5 to the power of 3 is minus a 125."},{"Start":"01:55.280 ","End":"01:59.355","Text":"The 8 squared is going to be 64."},{"Start":"01:59.355 ","End":"02:03.510","Text":"The 8 to the minus n is going to be 1/8 to the n,"},{"Start":"02:03.510 ","End":"02:07.945","Text":"which means that the 8 to the n comes into the numerator, just like here,"},{"Start":"02:07.945 ","End":"02:09.530","Text":"it becomes plus n,"},{"Start":"02:09.530 ","End":"02:13.130","Text":"and this thing goes into the denominator with a plus"},{"Start":"02:13.130 ","End":"02:18.170","Text":"n. We\u0027re getting already closer to this form."},{"Start":"02:18.170 ","End":"02:21.155","Text":"Because now what I can do is, okay,"},{"Start":"02:21.155 ","End":"02:24.410","Text":"the minus 125/64 stays."},{"Start":"02:24.410 ","End":"02:32.745","Text":"But I\u0027m using the a to the n over b to the n is a/ b to the power of n formula."},{"Start":"02:32.745 ","End":"02:36.980","Text":"Here what I\u0027ll get is 8 over minus 5."},{"Start":"02:36.980 ","End":"02:38.780","Text":"Well, just put the minus at the side,"},{"Start":"02:38.780 ","End":"02:45.650","Text":"minus 8/5 to the power of n. Finally,"},{"Start":"02:45.650 ","End":"02:52.385","Text":"I\u0027ve got it in a form that I can handle, like this."},{"Start":"02:52.385 ","End":"02:56.940","Text":"What we get is back here,"},{"Start":"02:57.140 ","End":"03:04.480","Text":"the sum from 3 to infinity of minus"},{"Start":"03:05.300 ","End":"03:14.970","Text":"125/64 times minus 8/5 to the power of n. A constant can come out in front."},{"Start":"03:14.970 ","End":"03:22.350","Text":"I\u0027ve got minus 125/64 times the sum from 3 to"},{"Start":"03:22.350 ","End":"03:29.895","Text":"infinity of minus 8/5 to the power of n. Now,"},{"Start":"03:29.895 ","End":"03:33.735","Text":"this really looks like this,"},{"Start":"03:33.735 ","End":"03:35.520","Text":"forgetting the constant in front,"},{"Start":"03:35.520 ","End":"03:41.360","Text":"the constant multiplier is not going to change whether the series diverges or converges."},{"Start":"03:41.360 ","End":"03:43.580","Text":"Essentially what I have is this,"},{"Start":"03:43.580 ","End":"03:45.950","Text":"with k equals 3,"},{"Start":"03:45.950 ","End":"03:47.405","Text":"which is less interesting."},{"Start":"03:47.405 ","End":"03:52.780","Text":"What\u0027s more interesting that q is equal to minus 8/5."},{"Start":"03:52.780 ","End":"03:57.320","Text":"If you prefer that in decimal, that\u0027s minus 1.6."},{"Start":"03:57.320 ","End":"04:00.545","Text":"Whichever way helps you to see more clearly,"},{"Start":"04:00.545 ","End":"04:04.340","Text":"the fact that it is not between minus 1 and 1,"},{"Start":"04:04.340 ","End":"04:08.029","Text":"this is definitely less than minus 1,"},{"Start":"04:08.029 ","End":"04:09.664","Text":"so it\u0027s not in range."},{"Start":"04:09.664 ","End":"04:11.885","Text":"Whenever q is not in range,"},{"Start":"04:11.885 ","End":"04:18.695","Text":"then there\u0027s no point continuing because the series diverges."},{"Start":"04:18.695 ","End":"04:21.450","Text":"That\u0027s all there is to it."}],"ID":6537},{"Watched":false,"Name":"Exercise 1 part h","Duration":"3m 40s","ChapterTopicVideoID":6492,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6492.jpeg","UploadDate":"2016-07-04T16:55:38.0470000","DurationForVideoObject":"PT3M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"In this exercise, we have an infinite series"},{"Start":"00:03.645 ","End":"00:08.685","Text":"and we have to decide if it converges or diverges."},{"Start":"00:08.685 ","End":"00:14.805","Text":"If it converges then we have to give its value, the sum of the infinite series."},{"Start":"00:14.805 ","End":"00:19.035","Text":"I\u0027ll just copy it down here."},{"Start":"00:19.035 ","End":"00:24.495","Text":"It has all the elements of a geometric series, but not quite."},{"Start":"00:24.495 ","End":"00:28.290","Text":"Let me just write the formulas for geometric series."},{"Start":"00:28.290 ","End":"00:31.350","Text":"There\u0027s a formula when it starts with 0."},{"Start":"00:31.350 ","End":"00:33.510","Text":"In our case it starts with 2."},{"Start":"00:33.510 ","End":"00:40.210","Text":"What I would like to do is to get it somehow to look like this or a variation of this."},{"Start":"00:40.210 ","End":"00:42.130","Text":"I\u0027ll just highlight it."},{"Start":"00:42.130 ","End":"00:43.820","Text":"If I can get it to look like this,"},{"Start":"00:43.820 ","End":"00:46.535","Text":"then I can say what it\u0027s equal."},{"Start":"00:46.535 ","End":"00:50.395","Text":"Let\u0027s see if we can do some manipulation."},{"Start":"00:50.395 ","End":"00:54.350","Text":"Let\u0027s see, I hope you know your laws of exponents."},{"Start":"00:54.350 ","End":"00:56.270","Text":"This equals the sum,"},{"Start":"00:56.270 ","End":"01:00.200","Text":"and I won\u0027t write every time from 2 to infinity."},{"Start":"01:00.200 ","End":"01:07.440","Text":"This bit I can write as 2 to the power of 3n times 2 to the power of 4."},{"Start":"01:07.440 ","End":"01:14.450","Text":"This part, I can write as 5 to the power of 1 divided by 5 to the power of n,"},{"Start":"01:14.450 ","End":"01:16.795","Text":"a subtraction becomes a division."},{"Start":"01:16.795 ","End":"01:19.295","Text":"This is now equal to the sum."},{"Start":"01:19.295 ","End":"01:25.980","Text":"Like I said, I won\u0027t write every time it\u0027s always from 2 to infinity."},{"Start":"01:26.830 ","End":"01:32.805","Text":"What I\u0027ll do is combine the 2 to the 4th with the 5,"},{"Start":"01:32.805 ","End":"01:34.530","Text":"that\u0027s 16 times 5,"},{"Start":"01:34.530 ","End":"01:39.925","Text":"that\u0027s 80, and then I\u0027ve got 2 to the power of 3,"},{"Start":"01:39.925 ","End":"01:45.845","Text":"n from here, divide it by 5 to the power of n from there."},{"Start":"01:45.845 ","End":"01:48.590","Text":"Now a constant, when I say constant,"},{"Start":"01:48.590 ","End":"01:56.255","Text":"I mean it doesn\u0027t contain n. Write it any way to emphasize that there\u0027s an n here,"},{"Start":"01:56.255 ","End":"01:58.555","Text":"then that constant can come out in front."},{"Start":"01:58.555 ","End":"02:04.115","Text":"I have 80 times the sum of this thing here."},{"Start":"02:04.115 ","End":"02:07.070","Text":"Now, it\u0027s still not quite q to the n,"},{"Start":"02:07.070 ","End":"02:11.710","Text":"but I can do some more algebra and say that this is"},{"Start":"02:11.710 ","End":"02:19.960","Text":"2 cubed to the power of n over 5 to the power of n,"},{"Start":"02:19.960 ","End":"02:27.035","Text":"and then I can use another rule that something to the n over something to the n,"},{"Start":"02:27.035 ","End":"02:32.100","Text":"the same exponent that I can just put this fraction,"},{"Start":"02:32.450 ","End":"02:42.965","Text":"2_3 over 5 to the power of n. Just to be very specific, it\u0027s 80."},{"Start":"02:42.965 ","End":"02:46.640","Text":"Sum n goes from 2 to infinity."},{"Start":"02:46.640 ","End":"02:56.495","Text":"2 cubed over 5 is 8 over 5 to the power of n. Now,"},{"Start":"02:56.495 ","End":"03:03.034","Text":"what I have here is the situation here with k equal 2,"},{"Start":"03:03.034 ","End":"03:07.340","Text":"but q is equal to 8 over"},{"Start":"03:07.340 ","End":"03:13.070","Text":"5 and the 80 just stays there it doesn\u0027t affect to whether it converges or diverges."},{"Start":"03:13.070 ","End":"03:20.330","Text":"However, what\u0027s written in red is exactly the condition on convergence or divergence."},{"Start":"03:20.330 ","End":"03:23.600","Text":"This is the condition on convergence and 8 over 5,"},{"Start":"03:23.600 ","End":"03:24.770","Text":"or if you see it better,"},{"Start":"03:24.770 ","End":"03:29.520","Text":"I\u0027ll write it as 1.6 is definitely out of range."},{"Start":"03:29.540 ","End":"03:33.590","Text":"There\u0027s nothing to continue and we pronounce the seat that"},{"Start":"03:33.590 ","End":"03:40.020","Text":"the series diverges and we are done."}],"ID":6538},{"Watched":false,"Name":"Exercise 1 part i","Duration":"4m 36s","ChapterTopicVideoID":6493,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6493.jpeg","UploadDate":"2016-07-04T17:00:14.5470000","DurationForVideoObject":"PT4M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:05.970","Text":"In this exercise, we are given a series here, an infinite series,"},{"Start":"00:05.970 ","End":"00:10.140","Text":"and if decide if it converges or diverges and in the convergent case,"},{"Start":"00:10.140 ","End":"00:14.280","Text":"we actually have to give the value of the sum of the series."},{"Start":"00:14.280 ","End":"00:17.790","Text":"It\u0027s infinite because that\u0027s what the dot-dot-dot means."},{"Start":"00:17.790 ","End":"00:20.160","Text":"But in this case,"},{"Start":"00:20.160 ","End":"00:24.000","Text":"we don\u0027t have it in Sigma notation."},{"Start":"00:24.000 ","End":"00:26.145","Text":"We don\u0027t have the general term."},{"Start":"00:26.145 ","End":"00:29.009","Text":"I just copied it over here for convenience."},{"Start":"00:29.009 ","End":"00:34.080","Text":"What I\u0027d like to do is see if I can express this in terms of k,"},{"Start":"00:34.080 ","End":"00:37.920","Text":"a number which goes from something to infinity."},{"Start":"00:37.920 ","End":"00:44.589","Text":"If I looked at just 2, 4, 6,"},{"Start":"00:44.589 ","End":"00:46.475","Text":"8, 10 and so on,"},{"Start":"00:46.475 ","End":"00:52.955","Text":"I could easily see that this is twice 1, 2, 3, 4."},{"Start":"00:52.955 ","End":"00:59.415","Text":"Let\u0027s say we have k as an index which goes 1,"},{"Start":"00:59.415 ","End":"01:02.040","Text":"2, 3, 4, etc."},{"Start":"01:02.040 ","End":"01:04.079","Text":"In other words, from 1 to infinity."},{"Start":"01:04.079 ","End":"01:08.135","Text":"Now, let\u0027s build up this general term bit by bit."},{"Start":"01:08.135 ","End":"01:11.565","Text":"I can say that 2k is 2,"},{"Start":"01:11.565 ","End":"01:16.025","Text":"4, 6, 8, and so on."},{"Start":"01:16.025 ","End":"01:19.715","Text":"Getting closer, how about minus 2k?"},{"Start":"01:19.715 ","End":"01:21.935","Text":"That will be minus 2,"},{"Start":"01:21.935 ","End":"01:25.460","Text":"minus 4, minus 6, minus 8."},{"Start":"01:25.460 ","End":"01:28.025","Text":"Still not the general term here."},{"Start":"01:28.025 ","End":"01:32.525","Text":"But if I now say 2 to the power of minus 2k,"},{"Start":"01:32.525 ","End":"01:35.435","Text":"then bingo, I\u0027ve got 2 to the minus 2,"},{"Start":"01:35.435 ","End":"01:36.980","Text":"2 to the minus 4,"},{"Start":"01:36.980 ","End":"01:40.925","Text":"2 to the minus 6 and I\u0027ve even given you a bonus term,"},{"Start":"01:40.925 ","End":"01:43.700","Text":"2 to the minus 8 and so on."},{"Start":"01:43.700 ","End":"01:49.280","Text":"If I take my general term as 2 to the minus 2k,"},{"Start":"01:49.280 ","End":"01:53.660","Text":"now I can express this in sigma form."},{"Start":"01:53.660 ","End":"01:57.870","Text":"This is the sum and k goes from,"},{"Start":"01:57.870 ","End":"01:59.610","Text":"because it goes 1, 2, 3, 4,"},{"Start":"01:59.610 ","End":"02:02.010","Text":"it means it\u0027s from 1 to infinity,"},{"Start":"02:02.010 ","End":"02:08.640","Text":"it starts at 1 and the general term now is 2 to the minus 2k."},{"Start":"02:08.640 ","End":"02:13.640","Text":"That\u0027s what this, well,"},{"Start":"02:13.640 ","End":"02:15.890","Text":"this equals, it comes from here,"},{"Start":"02:15.890 ","End":"02:20.705","Text":"that\u0027s what the equals and just writing the original series in this form."},{"Start":"02:20.705 ","End":"02:25.204","Text":"At this point I can now,"},{"Start":"02:25.204 ","End":"02:28.340","Text":"I\u0027m going to bring out the formulas for geometric series because it looks"},{"Start":"02:28.340 ","End":"02:33.195","Text":"very geometric and oops,"},{"Start":"02:33.195 ","End":"02:35.755","Text":"yeah, this is what I wanted."},{"Start":"02:35.755 ","End":"02:39.350","Text":"I\u0027d like to get it into the form."},{"Start":"02:39.350 ","End":"02:41.120","Text":"Well, 1 of these 2,"},{"Start":"02:41.120 ","End":"02:43.520","Text":"but since it starts from 1 and not from 0,"},{"Start":"02:43.520 ","End":"02:49.490","Text":"I\u0027m going to go with this with k equals 1 and the question is what is Q?"},{"Start":"02:49.490 ","End":"02:56.880","Text":"Just a little bit more algebra and I can write this as the sum from 1 to"},{"Start":"02:56.880 ","End":"03:03.011","Text":"infinity of 2 to the minus 2 to the power of k."},{"Start":"03:03.011 ","End":"03:06.620","Text":"Using the rules of exponents, if I have a product,"},{"Start":"03:06.620 ","End":"03:08.965","Text":"I can make it a power of a power."},{"Start":"03:08.965 ","End":"03:13.595","Text":"Finally, just to make it really visible,"},{"Start":"03:13.595 ","End":"03:17.239","Text":"2 to the minus 2 is 1 over 2 squared which is a quarter."},{"Start":"03:17.239 ","End":"03:21.920","Text":"I have the sum from k equals 1 to infinity of"},{"Start":"03:21.920 ","End":"03:28.280","Text":"1 quarter to the power of k and now this is really like this."},{"Start":"03:28.280 ","End":"03:33.605","Text":"If I say k equals 1 and Q equals 1/3,"},{"Start":"03:33.605 ","End":"03:36.685","Text":"then this is exactly what I have."},{"Start":"03:36.685 ","End":"03:42.350","Text":"At this point, I can tell divergent or convergent by just looking at what\u0027s in red here."},{"Start":"03:42.350 ","End":"03:46.220","Text":"My Q which is a quarter is between minus 1 and 1."},{"Start":"03:46.220 ","End":"03:49.200","Text":"Already I can say it\u0027s convergent,"},{"Start":"03:49.360 ","End":"03:55.670","Text":"but I still have to say what the sum is and so I just use this formula."},{"Start":"03:55.670 ","End":"04:00.590","Text":"What I get is continuing here and say I\u0027ll"},{"Start":"04:00.590 ","End":"04:04.970","Text":"put the equals down here equals from this formula Q which is"},{"Start":"04:04.970 ","End":"04:08.630","Text":"one quarter to the power of k which is"},{"Start":"04:08.630 ","End":"04:15.420","Text":"1 over 1 minus Q which is a 1/4."},{"Start":"04:15.420 ","End":"04:21.860","Text":"In short, I get a 1/4 over 1 minus 1/4 is 3/4 so everyone knows that."},{"Start":"04:21.860 ","End":"04:30.215","Text":"Multiply top and bottom by 4 and I get 1/3 and this is the answer."},{"Start":"04:30.215 ","End":"04:37.720","Text":"Series is convergent and the value is 1/3, done."}],"ID":6539},{"Watched":false,"Name":"Exercise 2 part a","Duration":"8m 40s","ChapterTopicVideoID":6494,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6494.jpeg","UploadDate":"2016-07-04T17:09:17.5300000","DurationForVideoObject":"PT8M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.510","Text":"In this exercise, we have to compute the sum of an infinite series,"},{"Start":"00:06.510 ","End":"00:10.710","Text":"or at least to decide if it converges or diverges and then"},{"Start":"00:10.710 ","End":"00:16.335","Text":"the convergent case to actually say what the value of this infinite sum is."},{"Start":"00:16.335 ","End":"00:19.635","Text":"Let me just copy it over here,"},{"Start":"00:19.635 ","End":"00:23.160","Text":"and get more space."},{"Start":"00:23.160 ","End":"00:26.160","Text":"There\u0027s a standard trick,"},{"Start":"00:26.160 ","End":"00:28.875","Text":"start to recognize what trick to use where."},{"Start":"00:28.875 ","End":"00:31.890","Text":"What we\u0027re gonna do here is use a technique,"},{"Start":"00:31.890 ","End":"00:37.180","Text":"you might recall it, I hope you recall it, of partial fractions."},{"Start":"00:37.180 ","End":"00:40.550","Text":"Let\u0027s leave aside this infinite sum for the moment,"},{"Start":"00:40.550 ","End":"00:49.695","Text":"and let\u0027s just work on the general term which is 1 plus 1 times n plus 2,"},{"Start":"00:49.695 ","End":"00:53.420","Text":"and what we want to do according to partial fractions,"},{"Start":"00:53.420 ","End":"01:03.065","Text":"is to write it as some number over n plus 1 plus another number over n plus 2."},{"Start":"01:03.065 ","End":"01:04.820","Text":"To find what constants,"},{"Start":"01:04.820 ","End":"01:08.125","Text":"A and B will make this true,"},{"Start":"01:08.125 ","End":"01:10.510","Text":"not just in equality and identity,"},{"Start":"01:10.510 ","End":"01:12.259","Text":"make these the same expression."},{"Start":"01:12.259 ","End":"01:16.470","Text":"There\u0027s a standard technique,"},{"Start":"01:17.120 ","End":"01:24.755","Text":"we multiply both sides by the common denominator which is n plus 1 n plus 2,"},{"Start":"01:24.755 ","End":"01:27.910","Text":"which means that in each place,"},{"Start":"01:27.910 ","End":"01:31.820","Text":"we\u0027ll have to multiply by the remaining factors."},{"Start":"01:31.820 ","End":"01:33.665","Text":"Here I multiply by 1,"},{"Start":"01:33.665 ","End":"01:36.680","Text":"here I multiply by n plus 2,"},{"Start":"01:36.680 ","End":"01:39.560","Text":"and here I multiply by n plus 1,"},{"Start":"01:39.560 ","End":"01:43.755","Text":"and that gives us that 1 is equal to"},{"Start":"01:43.755 ","End":"01:51.735","Text":"A times n plus 2 plus B times n plus 1."},{"Start":"01:51.735 ","End":"01:55.590","Text":"This is not an equation in n,"},{"Start":"01:55.590 ","End":"01:58.130","Text":"it\u0027s got to be true for all n. We\u0027re looking for"},{"Start":"01:58.130 ","End":"02:00.890","Text":"A and B to make this much more than equation,"},{"Start":"02:00.890 ","End":"02:04.145","Text":"it\u0027s an identity sometimes written with 3 lines."},{"Start":"02:04.145 ","End":"02:06.280","Text":"If it\u0027s true for all n,"},{"Start":"02:06.280 ","End":"02:10.399","Text":"then we can substitute whatever n we want, and for convenience,"},{"Start":"02:10.399 ","End":"02:16.165","Text":"I suggest to try n equals minus 1 that will make this 0,"},{"Start":"02:16.165 ","End":"02:19.095","Text":"and then later n equals minus 2."},{"Start":"02:19.095 ","End":"02:23.470","Text":"Let\u0027s do that, if I let equal n minus 1,"},{"Start":"02:24.560 ","End":"02:31.005","Text":"then I get that 1 equals A times minus 1"},{"Start":"02:31.005 ","End":"02:38.490","Text":"plus 2 plus B times minus 1 plus 1 is 0."},{"Start":"02:38.490 ","End":"02:41.235","Text":"In other words, this is 0,"},{"Start":"02:41.235 ","End":"02:43.500","Text":"A times 1 is 1,"},{"Start":"02:43.500 ","End":"02:47.980","Text":"so we\u0027ve got that A equals 1."},{"Start":"02:49.730 ","End":"02:53.380","Text":"Let\u0027s try n equals minus 2,"},{"Start":"02:53.380 ","End":"02:55.055","Text":"and see what we get."},{"Start":"02:55.055 ","End":"02:58.250","Text":"If n is minus 2 in here,"},{"Start":"02:58.250 ","End":"03:08.610","Text":"I get that 1 equals A times minus 2 plus 2 is 0 plus B minus 2 plus 1."},{"Start":"03:08.610 ","End":"03:11.005","Text":"This is negative 1,"},{"Start":"03:11.005 ","End":"03:14.505","Text":"so if 1 is negative minus B,"},{"Start":"03:14.505 ","End":"03:19.030","Text":"then B is minus 1."},{"Start":"03:19.030 ","End":"03:26.340","Text":"What that gives us ultimately is an identity that"},{"Start":"03:26.340 ","End":"03:33.520","Text":"1 over n plus 1 n plus 2 from here,"},{"Start":"03:33.520 ","End":"03:42.900","Text":"is actually equal algebraically for all n to 1 plus 1,"},{"Start":"03:42.900 ","End":"03:44.895","Text":"that\u0027s A equals 1,"},{"Start":"03:44.895 ","End":"03:46.220","Text":"and B is minus 1."},{"Start":"03:46.220 ","End":"03:53.060","Text":"So I\u0027ll just put a minus 1 plus 2."},{"Start":"03:53.060 ","End":"03:57.240","Text":"Now, this turns out is helpful."},{"Start":"03:57.240 ","End":"04:00.395","Text":"If we substitute in the original series,"},{"Start":"04:00.395 ","End":"04:03.335","Text":"the right-hand and so the left-hand side,"},{"Start":"04:03.335 ","End":"04:04.940","Text":"something good will happen,"},{"Start":"04:04.940 ","End":"04:07.050","Text":"and I\u0027ll show you."},{"Start":"04:07.340 ","End":"04:13.834","Text":"First of all, let me write this as equal to the sum,"},{"Start":"04:13.834 ","End":"04:21.525","Text":"also from 1 to infinity of, instead of this,"},{"Start":"04:21.525 ","End":"04:25.740","Text":"I write this: 1 plus"},{"Start":"04:25.740 ","End":"04:34.095","Text":"1 minus 1 plus 2 in brackets."},{"Start":"04:34.095 ","End":"04:40.835","Text":"Now, I can start writing out some terms."},{"Start":"04:40.835 ","End":"04:43.585","Text":"If n equals 1,"},{"Start":"04:43.585 ","End":"04:49.140","Text":"I get 1/2 minus 1/3,"},{"Start":"04:49.970 ","End":"04:54.884","Text":"that\u0027s the n equals 1 and then if n equals 2,"},{"Start":"04:54.884 ","End":"05:00.445","Text":"I get 1/3 minus 1/4,"},{"Start":"05:00.445 ","End":"05:03.645","Text":"and then if n equals 3,"},{"Start":"05:03.645 ","End":"05:11.840","Text":"I get 1/4 minus 1/5 plus,"},{"Start":"05:11.840 ","End":"05:14.550","Text":"and so on to infinity."},{"Start":"05:14.660 ","End":"05:16.950","Text":"I see I\u0027m going to need more space."},{"Start":"05:16.950 ","End":"05:19.035","Text":"Let me delete this stuff."},{"Start":"05:19.035 ","End":"05:25.930","Text":"The way we\u0027re going to tackle the infinite sum is not to take an infinite sum,"},{"Start":"05:25.930 ","End":"05:29.434","Text":"but to take a partial sum,"},{"Start":"05:29.434 ","End":"05:33.355","Text":"which is take n from 1 to something,"},{"Start":"05:33.355 ","End":"05:35.940","Text":"I need another letter, say big N,"},{"Start":"05:35.940 ","End":"05:40.405","Text":"so let\u0027s call this the partial sum corresponding to"},{"Start":"05:40.405 ","End":"05:46.090","Text":"letter N. That\u0027s going to be not the infinite sum,"},{"Start":"05:46.090 ","End":"05:52.375","Text":"but the same thing: 1 plus 1 minus 1 plus 2,"},{"Start":"05:52.375 ","End":"05:55.510","Text":"but to stop after we reach n,"},{"Start":"05:55.510 ","End":"05:58.205","Text":"which is going to be the same thing as this,"},{"Start":"05:58.205 ","End":"06:00.870","Text":"and just do the copy-paste from here,"},{"Start":"06:00.870 ","End":"06:05.120","Text":"but instead of going on to infinity,"},{"Start":"06:05.120 ","End":"06:09.380","Text":"we stop after we reach capital N,"},{"Start":"06:09.380 ","End":"06:14.040","Text":"which is going to be 1/N plus 1,"},{"Start":"06:14.040 ","End":"06:16.215","Text":"this is the general term,"},{"Start":"06:16.215 ","End":"06:20.995","Text":"minus 1/N plus 2."},{"Start":"06:20.995 ","End":"06:25.550","Text":"When we have the partial sum just up to N,"},{"Start":"06:25.550 ","End":"06:28.955","Text":"we afterwards will take the limit and let N go to infinity."},{"Start":"06:28.955 ","End":"06:36.795","Text":"Let\u0027s see what the partial sum is up to N. This is actually a,"},{"Start":"06:36.795 ","End":"06:39.195","Text":"it\u0027s called a telescopic series,"},{"Start":"06:39.195 ","End":"06:42.020","Text":"not exactly sure why but anyway,"},{"Start":"06:42.020 ","End":"06:45.185","Text":"it has the property that things cancel in pairs."},{"Start":"06:45.185 ","End":"06:49.410","Text":"See, this minus 1/3 cancels with this plus 1/3,"},{"Start":"06:49.410 ","End":"06:53.505","Text":"and then this minus 1/4 cancels with this plus 1/4,"},{"Start":"06:53.505 ","End":"06:59.610","Text":"and each time the right-hand term"},{"Start":"06:59.610 ","End":"07:04.260","Text":"cancels with the left-hand of the next 1,"},{"Start":"07:04.260 ","End":"07:09.335","Text":"and in short, all the middle terms cancel and what we\u0027re left with"},{"Start":"07:09.335 ","End":"07:18.690","Text":"is just 1/2 minus 1/N plus 2."},{"Start":"07:18.690 ","End":"07:29.405","Text":"Now, I want to take the limit when n goes to infinity of the partial sums up to N,"},{"Start":"07:29.405 ","End":"07:31.864","Text":"let N go to infinity,"},{"Start":"07:31.864 ","End":"07:36.485","Text":"and if this exists, finite limit,"},{"Start":"07:36.485 ","End":"07:38.780","Text":"then that\u0027s called the sum of the series,"},{"Start":"07:38.780 ","End":"07:40.520","Text":"and if this has no limit,"},{"Start":"07:40.520 ","End":"07:42.200","Text":"or it\u0027s an infinite limit,"},{"Start":"07:42.200 ","End":"07:45.500","Text":"then the original series diverges,"},{"Start":"07:45.500 ","End":"07:47.450","Text":"so let\u0027s see what is this limit."},{"Start":"07:47.450 ","End":"07:52.520","Text":"This limit is the limit as N goes to"},{"Start":"07:52.520 ","End":"07:58.755","Text":"infinity of 1/2 minus 1/N plus 2,"},{"Start":"07:58.755 ","End":"08:01.559","Text":"and this is easy, this half is a constant,"},{"Start":"08:01.559 ","End":"08:08.185","Text":"and when N goes to infinity and plus 2 goes to infinity and 1 over infinity is 0,"},{"Start":"08:08.185 ","End":"08:13.960","Text":"symbolically, I can write this as 1/2 minus 1 over infinity just symbolically,"},{"Start":"08:13.960 ","End":"08:18.065","Text":"and that\u0027s 0, so this is equal to 1/2 minus 0,"},{"Start":"08:18.065 ","End":"08:23.760","Text":"which is 1/2, which certainly exists and is finite,"},{"Start":"08:23.870 ","End":"08:28.125","Text":"and this is the answer, and we say,"},{"Start":"08:28.125 ","End":"08:33.405","Text":"yes the original series converges,"},{"Start":"08:33.405 ","End":"08:40.000","Text":"and the value is 1/2, and we\u0027re done."}],"ID":6540},{"Watched":false,"Name":"Exercise 2 part b","Duration":"11m 49s","ChapterTopicVideoID":6495,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6495.jpeg","UploadDate":"2016-07-04T17:21:27.3430000","DurationForVideoObject":"PT11M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"In this exercise, we are given an infinite series."},{"Start":"00:03.855 ","End":"00:06.390","Text":"It\u0027s infinite because of the dot, dot, dot."},{"Start":"00:06.390 ","End":"00:08.955","Text":"It\u0027s called an ellipsis,"},{"Start":"00:08.955 ","End":"00:11.190","Text":"and it means it goes on to infinity."},{"Start":"00:11.190 ","End":"00:14.070","Text":"It could converge or diverge."},{"Start":"00:14.070 ","End":"00:18.045","Text":"If it converges, we also have to give the value."},{"Start":"00:18.045 ","End":"00:21.450","Text":"Just for convenience, I\u0027m going to copy it over"},{"Start":"00:21.450 ","End":"00:26.350","Text":"here and I\u0027ll just get myself some more space here."},{"Start":"00:26.350 ","End":"00:31.040","Text":"What we\u0027d like to do is write it with Sigma notation."},{"Start":"00:31.040 ","End":"00:36.540","Text":"You don\u0027t have to, but it will help if we write it as the sum of something."},{"Start":"00:36.620 ","End":"00:42.080","Text":"Otherwise, I want to write this as the sum as n goes from"},{"Start":"00:42.080 ","End":"00:46.729","Text":"something to infinity of something."},{"Start":"00:46.729 ","End":"00:50.125","Text":"Now, I want to see what is the general pattern."},{"Start":"00:50.125 ","End":"00:56.370","Text":"I think you can easily see that the first bit, 1, 2, 3."},{"Start":"00:56.450 ","End":"01:00.405","Text":"Let\u0027s take n from 1 to infinity."},{"Start":"01:00.405 ","End":"01:04.020","Text":"We\u0027ll put 1 over the first bit, 1, 2,"},{"Start":"01:04.020 ","End":"01:08.195","Text":"3 which is just the value of n. What about the next bit?"},{"Start":"01:08.195 ","End":"01:11.770","Text":"I have 3, 4, 5."},{"Start":"01:14.390 ","End":"01:17.295","Text":"This one is just 2 more than this,"},{"Start":"01:17.295 ","End":"01:18.720","Text":"this is 2 more than this,"},{"Start":"01:18.720 ","End":"01:20.265","Text":"this is 2 more than this,"},{"Start":"01:20.265 ","End":"01:24.275","Text":"so it looks like the pattern is n, n plus 2."},{"Start":"01:24.275 ","End":"01:28.160","Text":"It certainly works if we plug in n equals 1, 2, or 3."},{"Start":"01:28.160 ","End":"01:29.570","Text":"We\u0027ll get 1 times 3,"},{"Start":"01:29.570 ","End":"01:31.040","Text":"2 times 4, 3 times 5."},{"Start":"01:31.040 ","End":"01:35.320","Text":"This looks like what we want to find."},{"Start":"01:35.570 ","End":"01:39.140","Text":"The way we\u0027re going to do this is we\u0027re going to use"},{"Start":"01:39.140 ","End":"01:44.185","Text":"partial fractions and we\u0027ll get what is called a telescoping series."},{"Start":"01:44.185 ","End":"01:46.910","Text":"I hope you remember your partial fractions,"},{"Start":"01:46.910 ","End":"01:49.655","Text":"I\u0027ll do that bit at the side."},{"Start":"01:49.655 ","End":"01:52.955","Text":"The idea is to take this general term,"},{"Start":"01:52.955 ","End":"01:54.860","Text":"forget about the sum for the moment,"},{"Start":"01:54.860 ","End":"02:00.479","Text":"to take 1 over n, n plus 2,"},{"Start":"02:00.479 ","End":"02:04.909","Text":"and to write it algebraically, something equivalent,"},{"Start":"02:04.909 ","End":"02:14.495","Text":"but more convenient as something over n plus something else over n plus 2."},{"Start":"02:14.495 ","End":"02:18.250","Text":"This is called the partial fraction decomposition."},{"Start":"02:18.250 ","End":"02:23.250","Text":"What we do is we put everything over a common denominator,"},{"Start":"02:23.250 ","End":"02:27.385","Text":"or rather, multiply both sides by n, n plus 2."},{"Start":"02:27.385 ","End":"02:29.930","Text":"It\u0027s fairly straightforward to see that if we do that,"},{"Start":"02:29.930 ","End":"02:33.420","Text":"we\u0027ll get 1 is equal to, now,"},{"Start":"02:33.420 ","End":"02:35.460","Text":"A over n times n, n plus 2,"},{"Start":"02:35.460 ","End":"02:40.035","Text":"the n\u0027s cancel, so it\u0027s just A, n plus 2."},{"Start":"02:40.035 ","End":"02:43.200","Text":"In the second case, if I multiply by n, n plus 2,"},{"Start":"02:43.200 ","End":"02:47.620","Text":"I get b times n because the n plus 2 cancels."},{"Start":"02:47.620 ","End":"02:50.055","Text":"Now, this is not an equation in n,"},{"Start":"02:50.055 ","End":"02:51.200","Text":"this is an identity."},{"Start":"02:51.200 ","End":"02:54.470","Text":"It\u0027s true for all n. We have to find A and B."},{"Start":"02:54.470 ","End":"02:56.120","Text":"If it\u0027s true for all n,"},{"Start":"02:56.120 ","End":"02:59.450","Text":"we can substitute convenient values."},{"Start":"02:59.450 ","End":"03:04.375","Text":"For example, if I substitute n equals 0,"},{"Start":"03:04.375 ","End":"03:06.420","Text":"that will make this term 0,"},{"Start":"03:06.420 ","End":"03:15.480","Text":"then we\u0027ll get 1 equals A times, let\u0027s see,"},{"Start":"03:15.480 ","End":"03:24.000","Text":"0 plus 2 is 2 and B times 0, which is nothing,"},{"Start":"03:24.000 ","End":"03:28.020","Text":"so 2A equals 1,"},{"Start":"03:28.020 ","End":"03:33.720","Text":"we get that A equals 1/2."},{"Start":"03:33.720 ","End":"03:35.760","Text":"Intermediate result."},{"Start":"03:35.760 ","End":"03:40.045","Text":"Now, let\u0027s let n equal negative 2."},{"Start":"03:40.045 ","End":"03:42.605","Text":"It could be anything also if you like,"},{"Start":"03:42.605 ","End":"03:45.540","Text":"but negative 2 will make this 0, so it\u0027s easier."},{"Start":"03:45.540 ","End":"03:48.210","Text":"So Let n equals negative 2."},{"Start":"03:48.210 ","End":"03:57.829","Text":"Substitute and we get 1 equals A times 0 plus B times negative 2."},{"Start":"03:57.829 ","End":"04:00.405","Text":"That means minus 2B is 1,"},{"Start":"04:00.405 ","End":"04:03.510","Text":"so B is minus 1/2."},{"Start":"04:03.510 ","End":"04:05.430","Text":"That\u0027s the other thing."},{"Start":"04:05.430 ","End":"04:08.330","Text":"Now we\u0027ve got the algebraic identity,"},{"Start":"04:08.330 ","End":"04:10.700","Text":"that 1 over n,"},{"Start":"04:10.700 ","End":"04:15.965","Text":"n plus 2 equals 1/2 over"},{"Start":"04:15.965 ","End":"04:25.545","Text":"n minus 1/2 over n plus 2."},{"Start":"04:25.545 ","End":"04:31.045","Text":"What I\u0027m going to do with this is substitute this in here"},{"Start":"04:31.045 ","End":"04:37.490","Text":"and I\u0027ll get something that\u0027s more convenient to deal with."},{"Start":"04:37.740 ","End":"04:41.380","Text":"This equals, and I\u0027ll continue on the next line,"},{"Start":"04:41.380 ","End":"04:46.335","Text":"the sum n goes from 1 to infinity."},{"Start":"04:46.335 ","End":"04:49.335","Text":"Instead of this, this."},{"Start":"04:49.335 ","End":"04:56.010","Text":"But I can take the 1/2 outside and write it as 1/2"},{"Start":"04:56.010 ","End":"05:05.305","Text":"of 1 over n minus 1 over n plus 2."},{"Start":"05:05.305 ","End":"05:09.350","Text":"I can do still better and take the 1/2 right out in front."},{"Start":"05:09.350 ","End":"05:12.095","Text":"Let\u0027s do that and we get 1/2,"},{"Start":"05:12.095 ","End":"05:22.050","Text":"the sum 1 to infinity of 1 over n minus 1 over n plus 2."},{"Start":"05:22.610 ","End":"05:27.200","Text":"The way we handle this infinite sum"},{"Start":"05:27.200 ","End":"05:33.260","Text":"sometimes is instead of going up to infinity to stop at a certain point,"},{"Start":"05:33.260 ","End":"05:38.555","Text":"say just take the limit up to N,"},{"Start":"05:38.555 ","End":"05:40.730","Text":"any other letter, what I\u0027m going to do,"},{"Start":"05:40.730 ","End":"05:46.130","Text":"is I\u0027m going to take the sum from n equals 1 up"},{"Start":"05:46.130 ","End":"05:52.660","Text":"to N of the same thing."},{"Start":"05:52.660 ","End":"05:55.010","Text":"Now it\u0027s not an infinite series,"},{"Start":"05:55.010 ","End":"05:56.945","Text":"it\u0027s a finite series."},{"Start":"05:56.945 ","End":"06:00.709","Text":"But after I\u0027ve done this,"},{"Start":"06:00.709 ","End":"06:09.305","Text":"then I\u0027m going to take the limit as n goes to infinity."},{"Start":"06:09.305 ","End":"06:12.220","Text":"Well, there\u0027s the matter of the 1/2."},{"Start":"06:12.220 ","End":"06:15.545","Text":"What I\u0027ve done is just rewritten this bit."},{"Start":"06:15.545 ","End":"06:21.140","Text":"At the end, I\u0027m going to remember if I find that this converges to a limit,"},{"Start":"06:21.140 ","End":"06:25.620","Text":"then plug this in here and multiply by 1/2."},{"Start":"06:26.750 ","End":"06:31.810","Text":"What I\u0027m going to do is just write out a few terms and see what we"},{"Start":"06:31.810 ","End":"06:38.355","Text":"get to write it more explicitly."},{"Start":"06:38.355 ","End":"06:45.844","Text":"When n equals 1, I\u0027ve got 1 over 1 minus 1 over 3."},{"Start":"06:45.844 ","End":"06:50.670","Text":"When n equals 2,"},{"Start":"06:50.670 ","End":"07:00.755","Text":"I get 1 over 2 minus 1 over 4."},{"Start":"07:00.755 ","End":"07:04.150","Text":"When n equals 3,"},{"Start":"07:04.150 ","End":"07:14.340","Text":"I get 1 over 3 minus 1 over 5."},{"Start":"07:14.340 ","End":"07:17.809","Text":"Let me write one more."},{"Start":"07:17.809 ","End":"07:19.790","Text":"Let\u0027s take N equals 4 also."},{"Start":"07:19.790 ","End":"07:26.965","Text":"The next one, I will get 1 over 4 minus 1 over,"},{"Start":"07:26.965 ","End":"07:30.580","Text":"4 plus 2 is 6."},{"Start":"07:33.440 ","End":"07:37.830","Text":"I need some more space. I\u0027m going to erase this."},{"Start":"07:38.590 ","End":"07:45.985","Text":"Then dot, dot, dot up to the last one will be"},{"Start":"07:45.985 ","End":"07:54.975","Text":"1 over n minus 1 over n plus 2."},{"Start":"07:54.975 ","End":"07:58.510","Text":"Now, something happens here."},{"Start":"07:58.510 ","End":"08:02.975","Text":"Things start canceling, but in an unusual pattern."},{"Start":"08:02.975 ","End":"08:07.020","Text":"The minus 1/3 cancels with the 1/3."},{"Start":"08:07.460 ","End":"08:14.835","Text":"Then we have minus 1/4 and plus 1/4."},{"Start":"08:14.835 ","End":"08:22.510","Text":"Eventually, we\u0027ll get minus 1/5 will cancel with plus 1/5."},{"Start":"08:22.610 ","End":"08:24.890","Text":"It seems to me like,"},{"Start":"08:24.890 ","End":"08:26.510","Text":"at any given point,"},{"Start":"08:26.510 ","End":"08:29.180","Text":"all we\u0027re left with is the first 2 pluses,"},{"Start":"08:29.180 ","End":"08:31.085","Text":"this 1 and this 1/2,"},{"Start":"08:31.085 ","End":"08:33.005","Text":"and the last 2 minuses."},{"Start":"08:33.005 ","End":"08:36.265","Text":"You know what, I\u0027m going to write the next to last term also."},{"Start":"08:36.265 ","End":"08:40.660","Text":"I\u0027ll just move this along here."},{"Start":"08:40.660 ","End":"08:44.300","Text":"Now I\u0027ve got room for another term."},{"Start":"08:44.300 ","End":"08:55.565","Text":"The one before last is 1 over n minus 1 takeaway 1 over n plus 1."},{"Start":"08:55.565 ","End":"08:58.145","Text":"Just enough to see."},{"Start":"08:58.145 ","End":"09:00.890","Text":"I must admit this is a bit tricky."},{"Start":"09:00.890 ","End":"09:02.655","Text":"The minus 1/5, obviously,"},{"Start":"09:02.655 ","End":"09:04.190","Text":"cancels with the next one,"},{"Start":"09:04.190 ","End":"09:06.645","Text":"which is going to be 1/5, and so on."},{"Start":"09:06.645 ","End":"09:11.315","Text":"The minus 1/6 will also cancel 2 terms on."},{"Start":"09:11.315 ","End":"09:14.405","Text":"Given that everything cancels,"},{"Start":"09:14.405 ","End":"09:23.010","Text":"this 1 over n is going to cancel with the previous term."},{"Start":"09:26.620 ","End":"09:32.705","Text":"The things that remain are the first 2 pluses and the last 2 minuses."},{"Start":"09:32.705 ","End":"09:35.435","Text":"That\u0027s what we get."},{"Start":"09:35.435 ","End":"09:39.215","Text":"If you think about it, you see this is so."},{"Start":"09:39.215 ","End":"09:45.110","Text":"I can write this as 1 over 1 plus 1 over 2"},{"Start":"09:45.110 ","End":"09:52.650","Text":"minus 1 over n plus 1 minus 1 over n plus 2."},{"Start":"09:56.290 ","End":"10:01.190","Text":"Let\u0027s see where I can write."},{"Start":"10:01.190 ","End":"10:05.580","Text":"I can just scroll down a bit to keep this stuff."},{"Start":"10:06.250 ","End":"10:13.835","Text":"This partial sum, which is also sometimes called S for sum up to n,"},{"Start":"10:13.835 ","End":"10:16.800","Text":"is equal to, well, this expression,"},{"Start":"10:16.800 ","End":"10:18.795","Text":"it\u0027s 1 and a 1/2"},{"Start":"10:18.795 ","End":"10:28.995","Text":"minus 1 over n plus 1 plus 1 over n plus 2."},{"Start":"10:28.995 ","End":"10:35.840","Text":"The infinite sum is the limit as n goes to infinity if it exists."},{"Start":"10:35.840 ","End":"10:38.420","Text":"If not, then the series diverges."},{"Start":"10:38.420 ","End":"10:44.025","Text":"Let\u0027s see what happens when we take n to infinity of S_n."},{"Start":"10:44.025 ","End":"10:50.250","Text":"It\u0027s equal to, these 2 denominators go to infinity,"},{"Start":"10:50.250 ","End":"10:52.760","Text":"and so this whole thing is 0,"},{"Start":"10:52.760 ","End":"10:54.290","Text":"but let me just write it symbolically."},{"Start":"10:54.290 ","End":"10:59.960","Text":"It\u0027s 1 and a 1/2 minus 1 over infinity plus 1 over infinity,"},{"Start":"10:59.960 ","End":"11:04.890","Text":"which is 1 and a 1/2 minus 0 plus 0,"},{"Start":"11:04.890 ","End":"11:08.050","Text":"which is 1 and a 1/2."},{"Start":"11:08.150 ","End":"11:10.910","Text":"I almost forgot."},{"Start":"11:10.910 ","End":"11:15.785","Text":"Remember that we were just dealing with the part that I marked here in green."},{"Start":"11:15.785 ","End":"11:18.155","Text":"There\u0027s a 1/2 still here."},{"Start":"11:18.155 ","End":"11:27.390","Text":"This thing is now equal to 1/2 times this 1 and a 1/2 from here,"},{"Start":"11:27.390 ","End":"11:30.840","Text":"and this is equal to 3/4."},{"Start":"11:30.840 ","End":"11:35.385","Text":"1/2 of 1 and a 1/2 is 3/4, and so yes,"},{"Start":"11:35.385 ","End":"11:40.820","Text":"the infinite series converges and"},{"Start":"11:40.820 ","End":"11:46.520","Text":"it converges to the value of 3/4."},{"Start":"11:46.520 ","End":"11:49.110","Text":"We are done."}],"ID":6541},{"Watched":false,"Name":"Exercise 2 part c","Duration":"10m 7s","ChapterTopicVideoID":6496,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6496.jpeg","UploadDate":"2016-07-04T17:30:43.5430000","DurationForVideoObject":"PT10M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this exercise, we have to compute the sum of"},{"Start":"00:03.150 ","End":"00:07.695","Text":"an infinite series from 1 to infinity of this expression."},{"Start":"00:07.695 ","End":"00:11.010","Text":"If it converges, if it diverges."},{"Start":"00:11.010 ","End":"00:13.515","Text":"It doesn\u0027t have a sum, we just say it diverges,"},{"Start":"00:13.515 ","End":"00:15.825","Text":"we have to decide which."},{"Start":"00:15.825 ","End":"00:20.565","Text":"I\u0027m going to just copy it over here a moment so we get more space."},{"Start":"00:20.565 ","End":"00:22.185","Text":"What we\u0027re going to do is,"},{"Start":"00:22.185 ","End":"00:25.170","Text":"you\u0027ve seen this technique before in other exercises."},{"Start":"00:25.170 ","End":"00:29.490","Text":"We\u0027re going to use a technique of partial fraction decomposition."},{"Start":"00:29.490 ","End":"00:31.785","Text":"What I\u0027m going to do is at the side,"},{"Start":"00:31.785 ","End":"00:36.210","Text":"I\u0027m going to take this general term without the Sigma,"},{"Start":"00:36.210 ","End":"00:42.345","Text":"1 over 16n^2 plus 8n minus 3."},{"Start":"00:42.345 ","End":"00:46.160","Text":"I want to write this in a more convenient form as a partial fraction."},{"Start":"00:46.160 ","End":"00:54.295","Text":"Now, the first step in partial fractions is to factorize the denominator."},{"Start":"00:54.295 ","End":"00:57.750","Text":"I\u0027m going to write it as 1 over."},{"Start":"00:57.750 ","End":"01:03.110","Text":"To save time, I\u0027m going to just factorize"},{"Start":"01:03.110 ","End":"01:05.450","Text":"it for you because otherwise the exercise will get"},{"Start":"01:05.450 ","End":"01:08.215","Text":"too long and you know how to factorize."},{"Start":"01:08.215 ","End":"01:13.025","Text":"It turns out that this is 4n plus"},{"Start":"01:13.025 ","End":"01:19.835","Text":"3 times 4n minus 1."},{"Start":"01:19.835 ","End":"01:23.330","Text":"Actually, I prefer to write it the other way around."},{"Start":"01:24.560 ","End":"01:27.650","Text":"According to partial fractions,"},{"Start":"01:27.650 ","End":"01:32.300","Text":"we should be able to find constants A and B such that this is A over"},{"Start":"01:32.300 ","End":"01:41.935","Text":"4n minus 1 plus B over 4n plus 3."},{"Start":"01:41.935 ","End":"01:46.880","Text":"The usual way is to first of all get a common denominator,"},{"Start":"01:46.880 ","End":"01:51.155","Text":"multiply both sides by 4n minus 1, 4n plus 3,"},{"Start":"01:51.155 ","End":"01:57.900","Text":"and then we get that 1 equals A times the missing factor,"},{"Start":"01:57.900 ","End":"02:03.014","Text":"which is 4n plus 3 plus B,"},{"Start":"02:03.014 ","End":"02:07.350","Text":"and the missing factor here is 4n minus 1."},{"Start":"02:07.350 ","End":"02:15.740","Text":"A and B are constants such that this is equal for all n. It\u0027s not actually inequality,"},{"Start":"02:15.740 ","End":"02:18.350","Text":"it\u0027s really an identity, it\u0027s for all n,"},{"Start":"02:18.350 ","End":"02:22.765","Text":"which means that we can substitute any value of n that we want."},{"Start":"02:22.765 ","End":"02:30.540","Text":"What I\u0027m going to do is substitute values that make 1 or other of these terms 0."},{"Start":"02:31.830 ","End":"02:34.805","Text":"Let\u0027s see what would make this 0?"},{"Start":"02:34.805 ","End":"02:36.230","Text":"Minus 3 over 4."},{"Start":"02:36.230 ","End":"02:38.810","Text":"If n is minus 3 over 4,"},{"Start":"02:38.810 ","End":"02:45.830","Text":"then I get that 1 is equal to A times 0."},{"Start":"02:45.830 ","End":"02:48.320","Text":"I mean, that\u0027s why I chose minus 3 over 4,"},{"Start":"02:48.320 ","End":"02:52.460","Text":"that would make this 0, plus B. Let\u0027s see."},{"Start":"02:52.460 ","End":"03:01.590","Text":"4n is minus 3"},{"Start":"03:01.630 ","End":"03:08.200","Text":"and then minus 1 would be minus 4."},{"Start":"03:08.200 ","End":"03:11.850","Text":"Let\u0027s see, minus 4B is 1."},{"Start":"03:11.850 ","End":"03:19.680","Text":"We get from here that B is equal to minus 1/4."},{"Start":"03:19.680 ","End":"03:22.310","Text":"Now, let\u0027s do the same trick for the other factor."},{"Start":"03:22.310 ","End":"03:26.720","Text":"If I let n equal 1/4,"},{"Start":"03:26.720 ","End":"03:28.850","Text":"that\u0027s what\u0027s going to make this 0."},{"Start":"03:28.850 ","End":"03:34.100","Text":"Then I\u0027ll get that 1 is equal to A times."},{"Start":"03:34.100 ","End":"03:36.005","Text":"Now let\u0027s see, 4n plus 3."},{"Start":"03:36.005 ","End":"03:39.769","Text":"4n is 1 plus 3 is 4,"},{"Start":"03:39.769 ","End":"03:45.580","Text":"8 times 4 plus B times 0."},{"Start":"03:46.310 ","End":"03:52.950","Text":"Therefore, 4A is 1 and A equals 1/4."},{"Start":"03:52.950 ","End":"03:58.800","Text":"I\u0027ve now found A and B here and here."},{"Start":"03:58.800 ","End":"04:08.450","Text":"Going back here, I can rewrite this thing. You know what?"},{"Start":"04:08.450 ","End":"04:10.685","Text":"I\u0027ll just write it up here."},{"Start":"04:10.685 ","End":"04:19.560","Text":"This thing is therefore 1/4 over 4n minus 1 by substituting A."},{"Start":"04:19.560 ","End":"04:29.800","Text":"Then B is minus 1/4 so I\u0027ll put a minus and 1/4 over 4n plus 3."},{"Start":"04:30.920 ","End":"04:36.110","Text":"This form of the same thing is going to"},{"Start":"04:36.110 ","End":"04:41.060","Text":"be much easier to work with when we do the series and I\u0027ll show you."},{"Start":"04:41.060 ","End":"04:43.865","Text":"Let\u0027s get back to here."},{"Start":"04:43.865 ","End":"04:50.750","Text":"Now,, I can take the 1/4 as a common factor and more than that,"},{"Start":"04:50.750 ","End":"04:53.360","Text":"I can take it right outside the summation sign."},{"Start":"04:53.360 ","End":"04:55.535","Text":"We\u0027ve done this trick before."},{"Start":"04:55.535 ","End":"04:57.740","Text":"I\u0027ve done 2 steps in 1."},{"Start":"04:57.740 ","End":"05:00.440","Text":"I\u0027m taking the 1/4 outside completely."},{"Start":"05:00.440 ","End":"05:08.350","Text":"Then I have the sum as n goes from 1 to infinity of,"},{"Start":"05:08.750 ","End":"05:11.160","Text":"without the 1/4 now,"},{"Start":"05:11.160 ","End":"05:15.225","Text":"just 1 over 4n minus 1."},{"Start":"05:15.225 ","End":"05:24.900","Text":"Take away 1 over 4n plus 3."},{"Start":"05:24.900 ","End":"05:30.670","Text":"Now, the way to tackle this thing is to take the sum not from 1 to infinity,"},{"Start":"05:30.670 ","End":"05:32.395","Text":"but like in the tutorial,"},{"Start":"05:32.395 ","End":"05:38.335","Text":"from 1 to some finite number and then we let that finite number go to infinity."},{"Start":"05:38.335 ","End":"05:45.165","Text":"Let\u0027s leave the 1/4 outside for the moment."},{"Start":"05:45.165 ","End":"05:46.590","Text":"We\u0027ll get back to the 1/4,"},{"Start":"05:46.590 ","End":"05:48.915","Text":"I don\u0027t want to keep dragging it along."},{"Start":"05:48.915 ","End":"05:50.325","Text":"Let\u0027s take this thing."},{"Start":"05:50.325 ","End":"05:57.895","Text":"Now, this is an infinite series and if I take the partial sum just from 1 to,"},{"Start":"05:57.895 ","End":"05:59.695","Text":"let\u0027s say, capital N,"},{"Start":"05:59.695 ","End":"06:02.530","Text":"I\u0027ll get a partial sum for capital N,"},{"Start":"06:02.530 ","End":"06:05.730","Text":"use the letter S often for partial sums,"},{"Start":"06:05.730 ","End":"06:10.240","Text":"which means the sum from 1, not to infinity,"},{"Start":"06:10.240 ","End":"06:11.665","Text":"but just to N,"},{"Start":"06:11.665 ","End":"06:15.105","Text":"a whole number of the same thing."},{"Start":"06:15.105 ","End":"06:18.390","Text":"1 over 4n minus 1,"},{"Start":"06:18.390 ","End":"06:21.855","Text":"minus 1 over 4n plus 3."},{"Start":"06:21.855 ","End":"06:26.129","Text":"I\u0027d like to compute this and at the end when I\u0027ve computed S_N,"},{"Start":"06:26.129 ","End":"06:31.295","Text":"later we\u0027ll let n go to infinity, but that\u0027s later."},{"Start":"06:31.295 ","End":"06:37.340","Text":"This is the aim. Now, what we can do here is just start substituting."},{"Start":"06:37.340 ","End":"06:41.220","Text":"Let\u0027s assume n is not just 1 or 2,"},{"Start":"06:41.220 ","End":"06:44.145","Text":"but further along, then we can say,"},{"Start":"06:44.145 ","End":"06:46.520","Text":"if we put n equals 1,"},{"Start":"06:46.520 ","End":"06:51.240","Text":"then we get 1 over, let\u0027s see,"},{"Start":"06:51.240 ","End":"06:54.045","Text":"4 minus 1 is 3,"},{"Start":"06:54.045 ","End":"06:58.395","Text":"minus and then 4 plus 3 is 7,"},{"Start":"06:58.395 ","End":"07:01.630","Text":"minus 1 over 7."},{"Start":"07:02.750 ","End":"07:06.435","Text":"Then let\u0027s put n equal 2."},{"Start":"07:06.435 ","End":"07:13.245","Text":"If n is 2, 4 times 2 minus 1 is 7."},{"Start":"07:13.245 ","End":"07:19.860","Text":"We get 1/7 minus 4 times 2"},{"Start":"07:19.860 ","End":"07:28.155","Text":"plus 3 is 11."},{"Start":"07:28.155 ","End":"07:31.620","Text":"If I let n equals 3,"},{"Start":"07:31.620 ","End":"07:39.945","Text":"then we\u0027ll get 1 over 11 minus 1 over 15."},{"Start":"07:39.945 ","End":"07:44.360","Text":"I think you get the idea of the pattern here."},{"Start":"07:44.360 ","End":"07:46.880","Text":"Let me just squeeze in 1."},{"Start":"07:46.880 ","End":"07:53.100","Text":"The last term would be when I have little n equal big N."},{"Start":"07:53.100 ","End":"08:02.920","Text":"I\u0027ll get 1 over 4 capital N minus 1."},{"Start":"08:03.010 ","End":"08:06.395","Text":"I just push this to the side for a minute."},{"Start":"08:06.395 ","End":"08:14.150","Text":"Minus 1 over 4N plus 3."},{"Start":"08:14.150 ","End":"08:17.660","Text":"Now, look, this is what we call a telescoping"},{"Start":"08:17.660 ","End":"08:22.620","Text":"series and things tend to cancel out in pairs."},{"Start":"08:22.620 ","End":"08:24.870","Text":"Look, minus 1/7 plus 1/7,"},{"Start":"08:24.870 ","End":"08:27.390","Text":"minus 1/11 plus 1/11,"},{"Start":"08:27.390 ","End":"08:29.070","Text":"minus 1/15 plus, well,"},{"Start":"08:29.070 ","End":"08:31.970","Text":"the next one\u0027s going to be plus 1/15 and so on."},{"Start":"08:31.970 ","End":"08:36.780","Text":"Looks like each time we\u0027ll cancel the last here with the first here,"},{"Start":"08:36.780 ","End":"08:38.890","Text":"so what we\u0027re going to be left with at"},{"Start":"08:38.890 ","End":"08:44.550","Text":"the end is the very first 1/3 and the very last to this."},{"Start":"08:46.130 ","End":"08:54.450","Text":"We can say that S_N is 1 over 3 minus 1"},{"Start":"08:54.450 ","End":"09:02.100","Text":"over 4N plus 3."},{"Start":"09:02.100 ","End":"09:04.640","Text":"To get this infinite sum,"},{"Start":"09:04.640 ","End":"09:07.410","Text":"we can take the limit."},{"Start":"09:07.610 ","End":"09:09.795","Text":"Just in case I forget it,"},{"Start":"09:09.795 ","End":"09:12.410","Text":"let me bring the 1/4 back into the picture."},{"Start":"09:12.410 ","End":"09:14.720","Text":"I tend to forget things like that."},{"Start":"09:14.720 ","End":"09:20.030","Text":"1/4, we want the limit as"},{"Start":"09:20.030 ","End":"09:26.330","Text":"N goes to infinity of S_N is equal to 1/4."},{"Start":"09:26.330 ","End":"09:31.820","Text":"The limit when N goes to infinity of this is, 1/3 is a constant."},{"Start":"09:33.100 ","End":"09:36.380","Text":"Here when N goes to infinity,"},{"Start":"09:36.380 ","End":"09:38.060","Text":"so there\u0027s 4N plus 3,"},{"Start":"09:38.060 ","End":"09:39.740","Text":"this whole thing goes to 0."},{"Start":"09:39.740 ","End":"09:43.580","Text":"I\u0027ll just write it symbolically as 1 over infinity so you see what I\u0027m doing."},{"Start":"09:43.580 ","End":"09:48.390","Text":"This goes to infinity and then 1 over infinity is 0."},{"Start":"09:48.390 ","End":"09:53.445","Text":"It\u0027s 1/4 times 1/3 and so the answer is 1/12."},{"Start":"09:53.445 ","End":"10:04.010","Text":"What we would say is that the series converges and the value it converges to is 1/12,"},{"Start":"10:04.010 ","End":"10:07.800","Text":"and we are done."}],"ID":6542},{"Watched":false,"Name":"Exercise 2 part d","Duration":"6m 55s","ChapterTopicVideoID":6497,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6497.jpeg","UploadDate":"2016-07-04T17:36:58.2470000","DurationForVideoObject":"PT6M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we have an infinite series."},{"Start":"00:04.080 ","End":"00:07.050","Text":"I know it\u0027s infinite, that\u0027s what the dot, dot, dot means,"},{"Start":"00:07.050 ","End":"00:12.390","Text":"goes on forever. We have to see if this series converges or diverges."},{"Start":"00:12.390 ","End":"00:14.040","Text":"Of course, in the case it converges,"},{"Start":"00:14.040 ","End":"00:17.055","Text":"we want to actually know the value of the sum of the series."},{"Start":"00:17.055 ","End":"00:22.995","Text":"I just prefer to copy it, and work down here, and get a bit more space."},{"Start":"00:22.995 ","End":"00:29.310","Text":"What I\u0027d like to do first is write it in sigma notation, capital sigma."},{"Start":"00:29.310 ","End":"00:36.790","Text":"I think the pattern is clear enough, so I can say that this is equal to the sum."},{"Start":"00:36.790 ","End":"00:39.260","Text":"I mean, you can see there\u0027s 1, 2, 3,"},{"Start":"00:39.260 ","End":"00:47.210","Text":"4, so that\u0027s going to be our n, and n is going to go from 1 to infinity."},{"Start":"00:47.210 ","End":"00:50.590","Text":"What we\u0027re going to have is the natural logarithm,"},{"Start":"00:50.590 ","End":"00:56.100","Text":"1 plus 1 over, that\u0027s the common bit, and this 1, 2, 3,"},{"Start":"00:56.100 ","End":"01:03.160","Text":"4 is the n-bit, so this is what we have in sigma notation."},{"Start":"01:03.550 ","End":"01:07.715","Text":"Let\u0027s just simplify this a bit."},{"Start":"01:07.715 ","End":"01:11.175","Text":"At the side, I can say,"},{"Start":"01:11.175 ","End":"01:14.490","Text":"I\u0027m talking about the general term, forget the sigma for the moment,"},{"Start":"01:14.490 ","End":"01:18.425","Text":"that the natural log of 1 plus 1 over n,"},{"Start":"01:18.425 ","End":"01:22.125","Text":"I can write it as the natural log of,"},{"Start":"01:22.125 ","End":"01:23.655","Text":"put a common denominator,"},{"Start":"01:23.655 ","End":"01:27.200","Text":"it\u0027s n plus 1 over n. Then using"},{"Start":"01:27.200 ","End":"01:31.730","Text":"the property of the logarithm for the quotient, this is equal"},{"Start":"01:31.730 ","End":"01:35.825","Text":"to the natural log of n plus 1"},{"Start":"01:35.825 ","End":"01:42.090","Text":"minus the natural log of n. This is going to be more useful."},{"Start":"01:42.890 ","End":"01:47.450","Text":"What we have is the infinite sum of n"},{"Start":"01:47.450 ","End":"01:51.680","Text":"goes from 1 to infinity. But instead of this, I\u0027ll write this,"},{"Start":"01:51.680 ","End":"02:02.880","Text":"natural log of n plus 1 minus natural log of n. I\u0027m going to put this in a brackets."},{"Start":"02:03.490 ","End":"02:11.000","Text":"Okay. As usual, the way to do an infinite sum is to take"},{"Start":"02:11.000 ","End":"02:14.480","Text":"a finite sum to make this upper bound, not infinity, but"},{"Start":"02:14.480 ","End":"02:20.375","Text":"some large number, and then let it go to infinity with the limit process. Let\u0027s see,"},{"Start":"02:20.375 ","End":"02:27.385","Text":"let\u0027s take the partial sum, and some other letter beside n, I like to use big N,"},{"Start":"02:27.385 ","End":"02:33.945","Text":"is the partial sum, not infinite, but just from 1 to N of"},{"Start":"02:33.945 ","End":"02:44.405","Text":"the same thing of natural log of n plus 1 minus natural log of n, and in brackets again."},{"Start":"02:44.405 ","End":"02:47.550","Text":"Let\u0027s see what we get."},{"Start":"02:47.830 ","End":"02:51.875","Text":"I\u0027m going to actually just write out some terms."},{"Start":"02:51.875 ","End":"02:58.650","Text":"Let\u0027s assume N is fairly big,"},{"Start":"02:58.700 ","End":"03:02.875","Text":"so what we get is if n is 1,"},{"Start":"03:02.875 ","End":"03:09.875","Text":"we get natural log of 2 minus natural log of 1."},{"Start":"03:09.875 ","End":"03:12.395","Text":"Then the next term, when n is 2,"},{"Start":"03:12.395 ","End":"03:18.770","Text":"is natural log of 3 minus natural log of 2, and then"},{"Start":"03:18.770 ","End":"03:26.310","Text":"we get natural log of 4 minus natural log of 3."},{"Start":"03:29.270 ","End":"03:33.210","Text":"I could write the last term, but what I want to do,"},{"Start":"03:33.210 ","End":"03:35.610","Text":"I can see now already, yeah,"},{"Start":"03:35.610 ","End":"03:40.730","Text":"that\u0027s the last term here, with the N equals n. I\u0027m already looking ahead, and"},{"Start":"03:40.730 ","End":"03:46.715","Text":"seeing that this cancels with this, and this cancels with this,"},{"Start":"03:46.715 ","End":"03:49.145","Text":"but it will look better if they were closer."},{"Start":"03:49.145 ","End":"03:51.245","Text":"I\u0027m going to reverse the order of the brackets."},{"Start":"03:51.245 ","End":"03:54.155","Text":"Let me just write it differently."},{"Start":"03:54.155 ","End":"03:57.625","Text":"It\u0027s like I reversed or minus,"},{"Start":"03:57.625 ","End":"04:00.250","Text":"okay, I\u0027ll show you what I\u0027m doing."},{"Start":"04:00.970 ","End":"04:09.880","Text":"It\u0027s just more convenient if I write minus natural log n plus natural log of n plus 1,"},{"Start":"04:09.880 ","End":"04:13.700","Text":"reversing the order of the brackets, so that here I"},{"Start":"04:13.700 ","End":"04:18.080","Text":"have minus natural log 1 plus natural log 2."},{"Start":"04:18.080 ","End":"04:25.250","Text":"It just look neater to cancel in pairs that are close to each other and then minus log 2,"},{"Start":"04:25.250 ","End":"04:27.710","Text":"I said log, I mean natural log."},{"Start":"04:27.710 ","End":"04:35.690","Text":"Yeah, plus log 3, and then minus log 3"},{"Start":"04:35.690 ","End":"04:43.225","Text":"plus log of 4, and finally the last term from here,"},{"Start":"04:43.225 ","End":"04:52.470","Text":"minus natural log of big N plus natural log of big N plus 1."},{"Start":"04:52.470 ","End":"04:58.680","Text":"So that\u0027s the sum from 1 to n. Now,"},{"Start":"04:58.680 ","End":"05:01.725","Text":"this simplifies telescopic series."},{"Start":"05:01.725 ","End":"05:05.110","Text":"Canceling is now easy, because this cancels with this,"},{"Start":"05:05.110 ","End":"05:07.480","Text":"they\u0027re next to each other, and this cancels with"},{"Start":"05:07.480 ","End":"05:10.755","Text":"this, and this is going to cancel with the 1 next to it,"},{"Start":"05:10.755 ","End":"05:12.935","Text":"this is going to cancel with the 1 before it,"},{"Start":"05:12.935 ","End":"05:15.635","Text":"we\u0027re just going to be left with the first and last."},{"Start":"05:15.635 ","End":"05:22.265","Text":"So what we get is minus natural log of 1 plus"},{"Start":"05:22.265 ","End":"05:30.835","Text":"natural log of N plus 1, and natural log of 1 is 0,"},{"Start":"05:30.835 ","End":"05:35.435","Text":"so it\u0027s just natural log of N plus 1."},{"Start":"05:35.435 ","End":"05:41.525","Text":"Okay, that\u0027s S_N, but what we want, our infinite series, is the limit."},{"Start":"05:41.525 ","End":"05:47.700","Text":"We want the limit as N goes to infinity"},{"Start":"05:47.700 ","End":"05:56.250","Text":"of S_n, and if this has a limit, and it\u0027s finite,"},{"Start":"05:56.250 ","End":"05:59.900","Text":"then that\u0027s the limit of the infinite series."},{"Start":"05:59.900 ","End":"06:05.254","Text":"If this diverges, then our series diverges,"},{"Start":"06:05.254 ","End":"06:06.800","Text":"meaning, if this doesn\u0027t have a limit."},{"Start":"06:06.800 ","End":"06:08.690","Text":"Let\u0027s see what happens."},{"Start":"06:08.690 ","End":"06:13.730","Text":"This is equal to the limit, as N goes to infinity"},{"Start":"06:13.730 ","End":"06:20.240","Text":"of natural log of N plus 1,"},{"Start":"06:20.240 ","End":"06:27.500","Text":"which symbolically I write as a natural log of infinity, or infinity plus 1 is infinity,"},{"Start":"06:27.500 ","End":"06:32.250","Text":"and we know that when a number goes to infinity, so does its natural logarithm,"},{"Start":"06:32.250 ","End":"06:35.595","Text":"this is infinity, but it\u0027s not a number."},{"Start":"06:35.595 ","End":"06:39.240","Text":"This means that our series,"},{"Start":"06:39.240 ","End":"06:48.500","Text":"this 1 here, diverges, and there is no answer."},{"Start":"06:48.500 ","End":"06:50.915","Text":"I mean, there\u0027s no value,"},{"Start":"06:50.915 ","End":"06:54.540","Text":"it just diverges and we\u0027re done."}],"ID":6543},{"Watched":false,"Name":"Exercise 3","Duration":"5m 52s","ChapterTopicVideoID":27353,"CourseChapterTopicPlaylistID":52090,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/27353.jpeg","UploadDate":"2022-02-01T05:18:22.1470000","DurationForVideoObject":"PT5M52S","Description":null,"VideoComments":[],"Subtitles":[],"ID":28540}],"Thumbnail":null,"ID":52090},{"Name":"The Harmonic Series and The P-Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Harmonic Series and the P-Series","Duration":"11m 1s","ChapterTopicVideoID":10062,"CourseChapterTopicPlaylistID":52091,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10062.jpeg","UploadDate":"2020-01-16T06:57:03.8930000","DurationForVideoObject":"PT11M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this clip, I\u0027ll be talking about the harmonic series and"},{"Start":"00:04.050 ","End":"00:08.220","Text":"also one of its generalizations called the p-series."},{"Start":"00:08.220 ","End":"00:11.850","Text":"We\u0027ve actually seen the harmonic series before."},{"Start":"00:11.850 ","End":"00:18.360","Text":"We\u0027ve seen it as the sum n goes from 1 to infinity of 1 over n. If you spell it out,"},{"Start":"00:18.360 ","End":"00:20.790","Text":"it\u0027s 1 plus a 1/2,"},{"Start":"00:20.790 ","End":"00:23.670","Text":"plus a 1/3, plus a 1/4,"},{"Start":"00:23.670 ","End":"00:26.865","Text":"plus a 1/5 plus, and so on."},{"Start":"00:26.865 ","End":"00:31.740","Text":"I think I may have even mentioned before that counter to intuition,"},{"Start":"00:31.740 ","End":"00:36.255","Text":"this actually diverges that this sum is in fact infinity."},{"Start":"00:36.255 ","End":"00:39.370","Text":"I\u0027ll give a proof of this later on in the clip."},{"Start":"00:39.370 ","End":"00:41.450","Text":"I want to mention now the p-series,"},{"Start":"00:41.450 ","End":"00:43.595","Text":"which is a generalization."},{"Start":"00:43.595 ","End":"00:47.630","Text":"It\u0027s the sum also from 1 to infinity."},{"Start":"00:47.630 ","End":"00:50.090","Text":"But instead of 1 over n,"},{"Start":"00:50.090 ","End":"00:57.470","Text":"we have 1 over n to the power of some positive number p. For example,"},{"Start":"00:57.470 ","End":"01:01.210","Text":"if I take p equals 2,"},{"Start":"01:01.210 ","End":"01:05.940","Text":"then what I get is 1 plus 1 over 2 squared,"},{"Start":"01:05.940 ","End":"01:07.830","Text":"plus 1 over 3 squared,"},{"Start":"01:07.830 ","End":"01:09.810","Text":"plus 1 over 4 squared,"},{"Start":"01:09.810 ","End":"01:12.845","Text":"plus 1 over 5 squared, and so on."},{"Start":"01:12.845 ","End":"01:16.385","Text":"In fact, this does converge as you\u0027ll see in a moment."},{"Start":"01:16.385 ","End":"01:17.735","Text":"It\u0027s a famous series."},{"Start":"01:17.735 ","End":"01:23.140","Text":"The sum is actually Pi squared over 6, but that\u0027s irrelevant."},{"Start":"01:23.140 ","End":"01:34.560","Text":"The rule about the p-series is that if p is bigger than 1, then it converges."},{"Start":"01:34.640 ","End":"01:40.935","Text":"If p is less than or equal to 1, it diverges."},{"Start":"01:40.935 ","End":"01:45.395","Text":"We\u0027ve seen 1 example of p equals 1 exactly."},{"Start":"01:45.395 ","End":"01:52.160","Text":"If p equals 1, then we just get the regular harmonic series which diverges."},{"Start":"01:52.160 ","End":"01:54.365","Text":"I want to mention a couple of things."},{"Start":"01:54.365 ","End":"02:00.355","Text":"This condition that p being bigger than 0 is not always."},{"Start":"02:00.355 ","End":"02:03.890","Text":"I\u0027m going to just erase it and not limit p."},{"Start":"02:03.890 ","End":"02:08.290","Text":"The other thing is that this p-series has another name."},{"Start":"02:08.290 ","End":"02:11.425","Text":"It\u0027s sometimes called the generalized harmonic series,"},{"Start":"02:11.425 ","End":"02:14.920","Text":"but I\u0027ve seen other uses for that name generalized,"},{"Start":"02:14.920 ","End":"02:17.230","Text":"so I\u0027ll just call it the p-series."},{"Start":"02:17.230 ","End":"02:19.870","Text":"Still, this thing holds true."},{"Start":"02:19.870 ","End":"02:22.270","Text":"If p bigger than 1, it converges."},{"Start":"02:22.270 ","End":"02:23.890","Text":"If p is less than or equal to 1,"},{"Start":"02:23.890 ","End":"02:27.790","Text":"even if it\u0027s negative, then it diverges."},{"Start":"02:27.790 ","End":"02:34.610","Text":"Let me give some examples of convergent infinite series."},{"Start":"02:35.430 ","End":"02:43.080","Text":"The sum of 1 over n squared."},{"Start":"02:43.080 ","End":"02:46.765","Text":"I\u0027ll write the examples and I\u0027ll show you why they\u0027re convergent."},{"Start":"02:46.765 ","End":"02:50.785","Text":"The sum from 1 to infinity of 1 over n,"},{"Start":"02:50.785 ","End":"02:52.165","Text":"square root of n,"},{"Start":"02:52.165 ","End":"03:00.315","Text":"sum from 1 to infinity of 1 over the 5th root of n to the 7th."},{"Start":"03:00.315 ","End":"03:07.310","Text":"Finally, the sum from 1 to infinity of 1 over n^4th."},{"Start":"03:07.310 ","End":"03:10.645","Text":"The reason that these series are all convergent,"},{"Start":"03:10.645 ","End":"03:12.430","Text":"well, this one we\u0027ve seen before."},{"Start":"03:12.430 ","End":"03:16.750","Text":"Basically here, p is 2 and 2 is bigger than 1."},{"Start":"03:16.750 ","End":"03:19.090","Text":"In this one, it\u0027s n times square root of n,"},{"Start":"03:19.090 ","End":"03:21.595","Text":"which is n^1 and 1/2."},{"Start":"03:21.595 ","End":"03:25.795","Text":"We can take p here as 1 and 1/2 or 3 over 2,"},{"Start":"03:25.795 ","End":"03:27.550","Text":"and this is bigger than 1."},{"Start":"03:27.550 ","End":"03:30.370","Text":"This also is convergent."},{"Start":"03:30.370 ","End":"03:35.055","Text":"Here, p is 7 over 5,"},{"Start":"03:35.055 ","End":"03:37.355","Text":"again bigger than 1, so convergent,"},{"Start":"03:37.355 ","End":"03:41.705","Text":"and here p equals 4 bigger than 1, so convergent."},{"Start":"03:41.705 ","End":"03:45.570","Text":"Now, let\u0027s take some divergent examples."},{"Start":"03:45.820 ","End":"03:48.590","Text":"I just wrote them all out."},{"Start":"03:48.590 ","End":"03:52.835","Text":"Divergent. Why are these divergent here?"},{"Start":"03:52.835 ","End":"03:57.130","Text":"Because square root of n is n^1.5."},{"Start":"03:57.130 ","End":"03:59.295","Text":"It\u0027s p equals 1.5,"},{"Start":"03:59.295 ","End":"04:01.260","Text":"which is less than 1,"},{"Start":"04:01.260 ","End":"04:03.060","Text":"less than or equal to."},{"Start":"04:03.060 ","End":"04:07.305","Text":"Here, we have p equals 4/10ths,"},{"Start":"04:07.305 ","End":"04:09.265","Text":"again less than 1."},{"Start":"04:09.265 ","End":"04:13.955","Text":"This thing here can be written as 1 over n to the minus 1."},{"Start":"04:13.955 ","End":"04:16.810","Text":"That means that p is minus 1,"},{"Start":"04:16.810 ","End":"04:18.760","Text":"which is less than 1,"},{"Start":"04:18.760 ","End":"04:23.480","Text":"and we\u0027re allowing negative values of p. Here also,"},{"Start":"04:23.480 ","End":"04:24.980","Text":"p is minus 2."},{"Start":"04:24.980 ","End":"04:29.050","Text":"All of these values of p are less than 1."},{"Start":"04:29.050 ","End":"04:33.635","Text":"Finally, I want to reiterate the famous 1,"},{"Start":"04:33.635 ","End":"04:37.879","Text":"which is the sum from 1 to infinity,"},{"Start":"04:37.879 ","End":"04:39.410","Text":"the one we talked about before,"},{"Start":"04:39.410 ","End":"04:46.500","Text":"the original harmonic series infamous, well, box it."},{"Start":"04:46.850 ","End":"04:52.805","Text":"Here, it\u0027s also a p-series if you take p equals 1."},{"Start":"04:52.805 ","End":"04:56.360","Text":"We already mentioned that it\u0027s divergent,"},{"Start":"04:56.360 ","End":"05:02.150","Text":"but it also fits this theorem that says that when it\u0027s less than or equal to,"},{"Start":"05:02.150 ","End":"05:04.880","Text":"so even equals 1, it\u0027s divergent."},{"Start":"05:04.880 ","End":"05:09.920","Text":"In fact, what I\u0027m going to do next is give you a proof of why this famous series,"},{"Start":"05:09.920 ","End":"05:12.200","Text":"the 1 plus a 1/2, plus a 1/3, plus a 1/4,"},{"Start":"05:12.200 ","End":"05:15.424","Text":"and so on, why it diverges to infinity."},{"Start":"05:15.424 ","End":"05:18.560","Text":"Let me begin with the proof now. Here we are."},{"Start":"05:18.560 ","End":"05:25.145","Text":"Clean page, we\u0027re going to show that the sum of the harmonic series is infinity."},{"Start":"05:25.145 ","End":"05:27.785","Text":"The harmonic series is divergent."},{"Start":"05:27.785 ","End":"05:30.875","Text":"I\u0027m going to prove this using integrals."},{"Start":"05:30.875 ","End":"05:34.730","Text":"How is this related to integrals? Well, I\u0027ll show you."},{"Start":"05:34.730 ","End":"05:37.629","Text":"I made a rough sketch,"},{"Start":"05:37.629 ","End":"05:41.650","Text":"the graph of y equals 1 over x."},{"Start":"05:41.650 ","End":"05:44.630","Text":"Excuse my sketching skills."},{"Start":"05:44.630 ","End":"05:48.560","Text":"I\u0027ve marked in the points 1, 2, 3, 4,"},{"Start":"05:48.560 ","End":"05:53.555","Text":"and 5, and also the points on the graph above."},{"Start":"05:53.555 ","End":"06:02.720","Text":"What I\u0027d like to do is drawing some rectangles like this."},{"Start":"06:02.720 ","End":"06:06.410","Text":"The above lines are horizontal through here,"},{"Start":"06:06.410 ","End":"06:09.200","Text":"and then from here, and through here."},{"Start":"06:09.200 ","End":"06:13.030","Text":"From here, here, here,"},{"Start":"06:13.030 ","End":"06:15.565","Text":"here, and so on."},{"Start":"06:15.565 ","End":"06:19.940","Text":"I\u0027m assuming that I\u0027m going to continue this to infinity."},{"Start":"06:19.940 ","End":"06:23.600","Text":"Now, what is the height of this rectangle?"},{"Start":"06:23.600 ","End":"06:28.085","Text":"Well, I look at the left and point is 1 and I go up to the graph,"},{"Start":"06:28.085 ","End":"06:29.630","Text":"and it\u0027s 1 over 1."},{"Start":"06:29.630 ","End":"06:32.350","Text":"Here, y equals 1."},{"Start":"06:32.350 ","End":"06:35.730","Text":"On the next rectangle between 2 and 3,"},{"Start":"06:35.730 ","End":"06:40.965","Text":"the height is 1 over 2 so the y here is 1.2,"},{"Start":"06:40.965 ","End":"06:43.515","Text":"and here it\u0027s going to be 1/3,"},{"Start":"06:43.515 ","End":"06:47.085","Text":"and here it\u0027s going to be 1/4, and so on."},{"Start":"06:47.085 ","End":"06:52.010","Text":"The area of the rectangles and I\u0027ll shade them."},{"Start":"06:52.010 ","End":"06:56.690","Text":"Shade this one, shade this one,"},{"Start":"06:56.690 ","End":"07:01.785","Text":"this one, this one, and so on."},{"Start":"07:01.785 ","End":"07:04.770","Text":"The yellow is the shaded area."},{"Start":"07:04.770 ","End":"07:07.350","Text":"The total shaded area,"},{"Start":"07:07.350 ","End":"07:10.185","Text":"it\u0027s going to be the area of this rectangle."},{"Start":"07:10.185 ","End":"07:12.180","Text":"Each base is 1,"},{"Start":"07:12.180 ","End":"07:15.220","Text":"so 1 times 1 is 1."},{"Start":"07:15.560 ","End":"07:21.440","Text":"The area of this rectangle is 1 times a 1/2, which is a 1/2."},{"Start":"07:21.440 ","End":"07:23.945","Text":"The area of this rectangle is 1/3."},{"Start":"07:23.945 ","End":"07:27.410","Text":"The area of this rectangle is 1/4."},{"Start":"07:27.410 ","End":"07:30.860","Text":"What we have is that the area of"},{"Start":"07:30.860 ","End":"07:36.660","Text":"the shaded rectangles to infinity is exactly what we\u0027re looking for."},{"Start":"07:37.040 ","End":"07:39.275","Text":"Now, that\u0027s on the one hand."},{"Start":"07:39.275 ","End":"07:40.460","Text":"On the other hand,"},{"Start":"07:40.460 ","End":"07:44.990","Text":"I\u0027d like to look at the different shaded region."},{"Start":"07:44.990 ","End":"07:55.010","Text":"That shaded region will be what\u0027s under the curve between the x-axis and the curve,"},{"Start":"07:55.010 ","End":"07:58.730","Text":"and from x equals 1 and to infinity."},{"Start":"07:58.730 ","End":"08:01.040","Text":"I\u0027ll get something different."},{"Start":"08:01.040 ","End":"08:08.360","Text":"I\u0027ll get just up to this line here."},{"Start":"08:08.360 ","End":"08:09.800","Text":"I think you get the idea."},{"Start":"08:09.800 ","End":"08:14.375","Text":"The yellow is the rectangles and that is this sum."},{"Start":"08:14.375 ","End":"08:20.120","Text":"The magenta color is for what\u0027s under the curve"},{"Start":"08:20.120 ","End":"08:26.485","Text":"and that is expressed by an integral from 1 to infinity."},{"Start":"08:26.485 ","End":"08:28.485","Text":"Definite integral, I mean,"},{"Start":"08:28.485 ","End":"08:31.575","Text":"of 1 over x, dx."},{"Start":"08:31.575 ","End":"08:34.250","Text":"Now, we have 2 things to measure."},{"Start":"08:34.250 ","End":"08:37.265","Text":"The yellow, which is this,"},{"Start":"08:37.265 ","End":"08:42.065","Text":"the original harmonic series and the purple magenta,"},{"Start":"08:42.065 ","End":"08:43.490","Text":"which is this integral."},{"Start":"08:43.490 ","End":"08:45.860","Text":"Well, let\u0027s compute this integral,"},{"Start":"08:45.860 ","End":"08:51.665","Text":"the integral from 1 to infinity of 1 over x, dx."},{"Start":"08:51.665 ","End":"08:54.695","Text":"It\u0027s an improper integral because of the infinity."},{"Start":"08:54.695 ","End":"08:56.150","Text":"What we do is,"},{"Start":"08:56.150 ","End":"09:06.050","Text":"we take the limit as b goes to infinity of the integral from 1 to b of 1 over x, dx."},{"Start":"09:06.050 ","End":"09:08.045","Text":"Let\u0027s see what this is."},{"Start":"09:08.045 ","End":"09:12.890","Text":"The integral of 1 over x is simply the natural log of x."},{"Start":"09:12.890 ","End":"09:19.225","Text":"It\u0027s the natural log of x and we take it between b and 1,"},{"Start":"09:19.225 ","End":"09:20.630","Text":"which means substitute this,"},{"Start":"09:20.630 ","End":"09:22.250","Text":"substitute this, and subtract."},{"Start":"09:22.250 ","End":"09:28.580","Text":"This is natural log of b minus natural log of 1."},{"Start":"09:28.580 ","End":"09:32.870","Text":"I forgot to say limit as b goes to infinity."},{"Start":"09:32.870 ","End":"09:42.360","Text":"Now, this thing is just the natural log of b because natural log of 1 is 0."},{"Start":"09:42.360 ","End":"09:44.835","Text":"This thing is 0."},{"Start":"09:44.835 ","End":"09:47.120","Text":"As b goes to infinity,"},{"Start":"09:47.120 ","End":"09:50.120","Text":"the natural log of b also goes to infinity."},{"Start":"09:50.120 ","End":"09:53.015","Text":"This is equal to infinity."},{"Start":"09:53.015 ","End":"09:56.585","Text":"Essentially what we have is an inequality."},{"Start":"09:56.585 ","End":"10:03.560","Text":"The area of the yellow is at least as big as the area of the magenta."},{"Start":"10:03.560 ","End":"10:06.835","Text":"The area of the yellow is this, which is this."},{"Start":"10:06.835 ","End":"10:14.655","Text":"We have the sum n equals 1 to infinity of 1 over n,"},{"Start":"10:14.655 ","End":"10:23.780","Text":"is bigger than the integral from 1 to infinity of 1 over x,"},{"Start":"10:23.780 ","End":"10:28.590","Text":"dx because obviously, it\u0027s got more area."},{"Start":"10:28.590 ","End":"10:32.820","Text":"This we just computed is equal to infinity."},{"Start":"10:33.160 ","End":"10:39.425","Text":"If the area of the purple is infinity and the yellow includes the purple,"},{"Start":"10:39.425 ","End":"10:43.780","Text":"then that means that this must also be infinity."},{"Start":"10:43.780 ","End":"10:51.755","Text":"Now, we can really say that this goes as divergence and it goes to infinity."},{"Start":"10:51.755 ","End":"10:57.245","Text":"This thing here finally figured out is infinity."},{"Start":"10:57.245 ","End":"11:02.220","Text":"That is the proof that we were missing before. That\u0027s all."}],"ID":9957},{"Watched":false,"Name":"Exercise 3","Duration":"5m 55s","ChapterTopicVideoID":10063,"CourseChapterTopicPlaylistID":52091,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10063.jpeg","UploadDate":"2017-08-19T11:21:54.2030000","DurationForVideoObject":"PT5M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"This exercise is 6 in 1,"},{"Start":"00:05.580 ","End":"00:08.610","Text":"and in each case, we have an infinite series."},{"Start":"00:08.610 ","End":"00:13.455","Text":"We just have to decide if the series converges or diverges."},{"Start":"00:13.455 ","End":"00:17.925","Text":"We don\u0027t actually have to find the value, the sum, if it converges,"},{"Start":"00:17.925 ","End":"00:21.580","Text":"just to say yes or no to converge."},{"Start":"00:21.890 ","End":"00:27.075","Text":"We do have this formula because each of these is"},{"Start":"00:27.075 ","End":"00:32.380","Text":"a variation of a p-series; what it\u0027s called."},{"Start":"00:33.890 ","End":"00:37.485","Text":"The theorem is that this p-series,"},{"Start":"00:37.485 ","End":"00:39.835","Text":"the sum of 1^p,"},{"Start":"00:39.835 ","End":"00:43.116","Text":"converges or diverges depending on the value of p."},{"Start":"00:43.116 ","End":"00:45.075","Text":"For greater than 1,"},{"Start":"00:45.075 ","End":"00:49.105","Text":"converges, less than or equal to 1, diverges."},{"Start":"00:49.105 ","End":"00:51.395","Text":"Let\u0027s get started."},{"Start":"00:51.395 ","End":"00:58.070","Text":"Scroll down a bit and skip the theorem in sight."},{"Start":"00:58.070 ","End":"01:02.390","Text":"In number 1, we don\u0027t quite have a p-series,"},{"Start":"01:02.390 ","End":"01:04.445","Text":"but if we take a constant,"},{"Start":"01:04.445 ","End":"01:10.025","Text":"the 3/5 outside the summation sign, the Sigma,"},{"Start":"01:10.025 ","End":"01:15.960","Text":"we\u0027ve got 3/5 of the sum from 1 to infinity of"},{"Start":"01:15.960 ","End":"01:22.580","Text":"1, and a constant makes no difference to convergence or divergence."},{"Start":"01:22.580 ","End":"01:24.950","Text":"This is a p-series, well,"},{"Start":"01:24.950 ","End":"01:26.420","Text":"to make it really clear,"},{"Start":"01:26.420 ","End":"01:27.740","Text":"I\u0027ll put a 1 here,"},{"Start":"01:27.740 ","End":"01:36.220","Text":"n is n^1, so we have a p-series with p equals 1."},{"Start":"01:36.220 ","End":"01:41.370","Text":"I needn\u0027t tell you that 1 is less than or equal to 1,"},{"Start":"01:41.370 ","End":"01:43.490","Text":"and so we\u0027re in this case,"},{"Start":"01:43.490 ","End":"01:46.500","Text":"and so it diverges."},{"Start":"01:46.820 ","End":"01:58.115","Text":"This part here is actually the classical harmonic series when p is 1. Next 1."},{"Start":"01:58.115 ","End":"02:03.080","Text":"In this case, we don\u0027t have to take any constants out,"},{"Start":"02:03.080 ","End":"02:07.240","Text":"but we have to rewrite the square root as an exponent."},{"Start":"02:07.240 ","End":"02:12.770","Text":"Of course you remember that the square root of n is n^1/2."},{"Start":"02:12.770 ","End":"02:18.170","Text":"Once again, we have a p-series and this time, p is equal to 1/2,"},{"Start":"02:18.170 ","End":"02:22.200","Text":"and 1/2 is less than or equal to 1."},{"Start":"02:22.550 ","End":"02:24.965","Text":"So in this case,"},{"Start":"02:24.965 ","End":"02:30.540","Text":"we also have that the series diverges."},{"Start":"02:30.610 ","End":"02:35.215","Text":"Then in the next 1,"},{"Start":"02:35.215 ","End":"02:38.490","Text":"it\u0027s just classically a p-series."},{"Start":"02:38.490 ","End":"02:40.930","Text":"We see it right away."},{"Start":"02:41.780 ","End":"02:46.275","Text":"Let\u0027s see, we can see this, we can see this."},{"Start":"02:46.275 ","End":"02:50.385","Text":"It just looks like this with p equals 4."},{"Start":"02:50.385 ","End":"02:58.570","Text":"4 is certainly bigger than 1, and so the answer is converges."},{"Start":"03:00.050 ","End":"03:03.700","Text":"After 3, we\u0027ll do number 4."},{"Start":"03:03.700 ","End":"03:06.905","Text":"This also looks very much like this,"},{"Start":"03:06.905 ","End":"03:14.600","Text":"with p being equal to e. I don\u0027t need to know the exact value of e,"},{"Start":"03:14.600 ","End":"03:16.400","Text":"it\u0027s 2 point something,"},{"Start":"03:16.400 ","End":"03:22.510","Text":"in any event, this is certainly bigger than 1."},{"Start":"03:22.510 ","End":"03:26.165","Text":"It\u0027s 2.718 whatever, but it\u0027s bigger than 1."},{"Start":"03:26.165 ","End":"03:30.750","Text":"Bigger than 1 means converges."},{"Start":"03:31.400 ","End":"03:39.390","Text":"Then on to number 5, constants make no difference,"},{"Start":"03:39.390 ","End":"03:40.730","Text":"so I\u0027ll do 2 things."},{"Start":"03:40.730 ","End":"03:44.090","Text":"First of all, take the 10 outside the brackets,"},{"Start":"03:44.090 ","End":"03:50.840","Text":"and then I\u0027ll get the sum from 1 to"},{"Start":"03:50.840 ","End":"03:57.350","Text":"infinity of 1 over the cube root of n^4."},{"Start":"03:57.350 ","End":"03:59.995","Text":"Now, the cube root of n^4,"},{"Start":"03:59.995 ","End":"04:02.210","Text":"if you remember fractional exponents,"},{"Start":"04:02.210 ","End":"04:05.880","Text":"it\u0027s n^4/3."},{"Start":"04:06.080 ","End":"04:11.705","Text":"This bit is a p-series and the constants make no difference to convergence."},{"Start":"04:11.705 ","End":"04:15.755","Text":"It\u0027s a p-series with p equals 4/3."},{"Start":"04:15.755 ","End":"04:20.320","Text":"Obviously, 4/3 is greater than 1,"},{"Start":"04:20.320 ","End":"04:24.780","Text":"and so we\u0027re in the case where it converges."},{"Start":"04:27.230 ","End":"04:30.640","Text":"In the last 1,"},{"Start":"04:35.180 ","End":"04:39.639","Text":"what we have is the sum."},{"Start":"04:41.120 ","End":"04:45.030","Text":"Notice that it\u0027s not from 1 to infinity,"},{"Start":"04:45.030 ","End":"04:47.805","Text":"it\u0027s from 10 to infinity,"},{"Start":"04:47.805 ","End":"04:55.955","Text":"but a finite number of terms doesn\u0027t affect any series convergence or divergence."},{"Start":"04:55.955 ","End":"05:01.465","Text":"If you add or remove a finite number of terms, it won\u0027t change."},{"Start":"05:01.465 ","End":"05:06.425","Text":"The fact that it\u0027ll change the value it converges to possibly,"},{"Start":"05:06.425 ","End":"05:10.099","Text":"but it won\u0027t change the fact of convergence or divergence,"},{"Start":"05:10.099 ","End":"05:12.160","Text":"so this is irrelevant."},{"Start":"05:12.160 ","End":"05:19.050","Text":"In actual fact in the formula, I could have put from anything to infinity."},{"Start":"05:19.870 ","End":"05:28.020","Text":"The negative means it\u0027s 1^2/3."},{"Start":"05:28.020 ","End":"05:30.060","Text":"We have a p-series again,"},{"Start":"05:30.060 ","End":"05:33.060","Text":"with p equals 2/3."},{"Start":"05:33.060 ","End":"05:36.930","Text":"2/3 is less than or equal to 1,"},{"Start":"05:36.930 ","End":"05:43.900","Text":"so we\u0027re in the case of less or equal to 1 and that means it diverges."},{"Start":"05:44.360 ","End":"05:46.845","Text":"Diverges."},{"Start":"05:46.845 ","End":"05:52.370","Text":"This is the last 1 in the series of exercises."},{"Start":"05:52.370 ","End":"05:54.960","Text":"So we are done."}],"ID":9958}],"Thumbnail":null,"ID":52091},{"Name":"Algebraic Properties of Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 4","Duration":"","ChapterTopicVideoID":10193,"CourseChapterTopicPlaylistID":52092,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"This exercise is made up of 3 separate exercises."},{"Start":"00:04.830 ","End":"00:11.805","Text":"Each one we have an infinite series and we have to decide if it converges or diverges."},{"Start":"00:11.805 ","End":"00:17.205","Text":"We\u0027re going to use the theorem on p series,"},{"Start":"00:17.205 ","End":"00:20.810","Text":"which is here and you should be familiar with it."},{"Start":"00:20.810 ","End":"00:22.610","Text":"I won\u0027t go over it again."},{"Start":"00:22.610 ","End":"00:25.040","Text":"But in all of these,"},{"Start":"00:25.040 ","End":"00:27.680","Text":"do we have an exact p series?"},{"Start":"00:27.680 ","End":"00:29.780","Text":"We\u0027re going to have to do a bit of algebra."},{"Start":"00:29.780 ","End":"00:32.470","Text":"I can see that in most cases it\u0027s the sum of 2 things,"},{"Start":"00:32.470 ","End":"00:36.540","Text":"and if I break it up I\u0027ll get the sum of 2 different p series."},{"Start":"00:36.540 ","End":"00:37.715","Text":"That\u0027s a general idea."},{"Start":"00:37.715 ","End":"00:40.310","Text":"Anyway, I\u0027ll start with number 1,"},{"Start":"00:40.310 ","End":"00:42.390","Text":"scroll down a bit."},{"Start":"00:43.400 ","End":"00:46.830","Text":"Let\u0027s see what we can do here."},{"Start":"00:46.830 ","End":"00:54.005","Text":"Certainly we can break it up into the sum of 2 bits. Let\u0027s see."},{"Start":"00:54.005 ","End":"01:04.000","Text":"The first part will be 10 over n and the second bit is n squared over n cubed."},{"Start":"01:04.000 ","End":"01:08.130","Text":"Sorry, that\u0027s n cubed, apologies."},{"Start":"01:08.130 ","End":"01:12.659","Text":"N squared over n cubed is 1 over n,"},{"Start":"01:12.659 ","End":"01:21.365","Text":"and remember that an infinite sum of a sum,"},{"Start":"01:21.365 ","End":"01:25.560","Text":"I can basically put this Sigma in front of each bit as well."},{"Start":"01:27.440 ","End":"01:35.170","Text":"Certain conditions apply if both of these converge."},{"Start":"01:36.110 ","End":"01:39.690","Text":"I\u0027ll write it first and then I\u0027ll explain."},{"Start":"01:39.690 ","End":"01:43.805","Text":"I want to break it up into 2 separate infinite series,"},{"Start":"01:43.805 ","End":"01:51.850","Text":"10 over n cubed and then also 1 over n. Now,"},{"Start":"01:51.850 ","End":"01:54.355","Text":"if I do this breaking up,"},{"Start":"01:54.355 ","End":"01:58.735","Text":"if one of these converges and the other diverges,"},{"Start":"01:58.735 ","End":"02:01.330","Text":"then the original one diverges."},{"Start":"02:01.330 ","End":"02:05.155","Text":"If they both converge then this one converges."},{"Start":"02:05.155 ","End":"02:09.325","Text":"Thing is if they both diverge then we don\u0027t know."},{"Start":"02:09.325 ","End":"02:14.390","Text":"But that\u0027s not going to be the case as we shall see."},{"Start":"02:15.530 ","End":"02:21.740","Text":"Well, the first one I can just rewrite as take the 10 out,"},{"Start":"02:21.740 ","End":"02:25.900","Text":"constants don\u0027t affect convergence or divergence,"},{"Start":"02:25.900 ","End":"02:28.385","Text":"1 over n cubed,"},{"Start":"02:28.385 ","End":"02:33.555","Text":"and here the sum from 1 to infinity,"},{"Start":"02:33.555 ","End":"02:38.580","Text":"1 over n, but just for emphasis I\u0027ll put 1 over n^1."},{"Start":"02:38.580 ","End":"02:41.050","Text":"I can see I have 2 series."},{"Start":"02:41.050 ","End":"02:49.800","Text":"In one case here I have p equals 3 but in the other case I have p equals 1."},{"Start":"02:51.890 ","End":"02:54.715","Text":"Well, 3 is bigger than 1,"},{"Start":"02:54.715 ","End":"02:57.185","Text":"I don\u0027t need any more explanation on that,"},{"Start":"02:57.185 ","End":"03:01.475","Text":"and 1 is less than or equal to 1."},{"Start":"03:01.475 ","End":"03:04.520","Text":"The first one is"},{"Start":"03:04.520 ","End":"03:13.690","Text":"convergent but the second one diverges."},{"Start":"03:13.880 ","End":"03:19.925","Text":"Like I said, if I have a sum where one converges and the other diverges,"},{"Start":"03:19.925 ","End":"03:22.925","Text":"then that sum diverges."},{"Start":"03:22.925 ","End":"03:29.290","Text":"I can say about this one is that it diverges."},{"Start":"03:33.740 ","End":"03:35.805","Text":"We\u0027re going to the next one."},{"Start":"03:35.805 ","End":"03:37.640","Text":"Just going back again."},{"Start":"03:37.640 ","End":"03:41.270","Text":"The only problem you have is if they\u0027re both divergent then you don\u0027t"},{"Start":"03:41.270 ","End":"03:47.165","Text":"know and you have to use some other technique."},{"Start":"03:47.165 ","End":"03:52.630","Text":"In this one again we\u0027ll use the trick of splitting it up into 2."},{"Start":"03:52.630 ","End":"03:55.460","Text":"I\u0027ll see if I can take a bit of a shortcut here,"},{"Start":"03:55.460 ","End":"03:58.190","Text":"we break it up right away under"},{"Start":"03:58.190 ","End":"04:00.470","Text":"the same assumptions that they\u0027re not going to"},{"Start":"04:00.470 ","End":"04:03.530","Text":"be both divergent and then we\u0027re in trouble."},{"Start":"04:03.530 ","End":"04:10.900","Text":"We get from 1 to infinity of 4n over n squared."},{"Start":"04:11.660 ","End":"04:19.710","Text":"I\u0027m going to do that 3 steps in 1,"},{"Start":"04:19.710 ","End":"04:21.839","Text":"take the 4 out front,"},{"Start":"04:21.839 ","End":"04:28.555","Text":"and the n over n squared I\u0027ll write as 1 over n. The second bit,"},{"Start":"04:28.555 ","End":"04:33.354","Text":"I take the 10 out front and we have the sum,"},{"Start":"04:33.354 ","End":"04:35.295","Text":"again from 1 to infinity,"},{"Start":"04:35.295 ","End":"04:38.845","Text":"of 1 over n squared."},{"Start":"04:38.845 ","End":"04:41.690","Text":"Now, for this one,"},{"Start":"04:41.690 ","End":"04:46.175","Text":"we already saw that we have p equals 1,"},{"Start":"04:46.175 ","End":"04:50.210","Text":"and for this one we have p equals 2."},{"Start":"04:50.210 ","End":"04:54.690","Text":"I mean, we\u0027ll write it again, n ^1, n^2."},{"Start":"04:55.910 ","End":"05:00.780","Text":"I\u0027ll write again, 1 is less than or equal to 1 and 2 is bigger than 1."},{"Start":"05:00.780 ","End":"05:06.585","Text":"Here we have a series that diverges."},{"Start":"05:06.585 ","End":"05:11.440","Text":"Here we have a series that converges."},{"Start":"05:11.600 ","End":"05:14.835","Text":"When we add them together,"},{"Start":"05:14.835 ","End":"05:20.610","Text":"a divergent plus a convergent is a divergent series,"},{"Start":"05:20.610 ","End":"05:25.570","Text":"so this one also diverges."},{"Start":"05:26.540 ","End":"05:29.950","Text":"Now the third one,"},{"Start":"05:31.610 ","End":"05:34.310","Text":"I just realized there\u0027s a typo here."},{"Start":"05:34.310 ","End":"05:35.960","Text":"This should be a 1 not a 0."},{"Start":"05:35.960 ","End":"05:39.600","Text":"I\u0027ll just fix that right away. There we are."},{"Start":"05:39.600 ","End":"05:44.360","Text":"We continue just like before we\u0027ll separate it into 2."},{"Start":"05:44.360 ","End":"05:52.585","Text":"We\u0027ll have the sum from 1 to infinity of 4^n."},{"Start":"05:52.585 ","End":"05:56.300","Text":"Allow me to just use rules of exponents to"},{"Start":"05:56.300 ","End":"06:03.125","Text":"write this as 7^n times 7^1,"},{"Start":"06:03.125 ","End":"06:05.135","Text":"that\u0027s for the first bit."},{"Start":"06:05.135 ","End":"06:12.150","Text":"The second bit, we also have the sum from 1 to infinity, and this time,"},{"Start":"06:12.150 ","End":"06:16.580","Text":"the algebra I\u0027ll use is to change a negative exponent to"},{"Start":"06:16.580 ","End":"06:22.410","Text":"a positive exponent in the denominator, n^1.5."},{"Start":"06:27.380 ","End":"06:31.819","Text":"What we have here is actually a mixture or a hybrid."},{"Start":"06:31.819 ","End":"06:41.840","Text":"This part is a p series with p equaling 1.5,"},{"Start":"06:41.840 ","End":"06:45.080","Text":"which is certainly bigger than 1."},{"Start":"06:45.080 ","End":"06:49.325","Text":"This bit converges."},{"Start":"06:49.325 ","End":"06:55.445","Text":"But the other is actually not a p series."},{"Start":"06:55.445 ","End":"07:05.625","Text":"I can write it if I take the 1/7 in front as the sum n goes from 1 to infinity,"},{"Start":"07:05.625 ","End":"07:11.310","Text":"instead of 4^n over 7^n I can use another property of exponents"},{"Start":"07:11.310 ","End":"07:17.715","Text":"for a fraction that it\u0027s 4/7^n."},{"Start":"07:17.715 ","End":"07:22.750","Text":"This is actually a geometric series, this part here."},{"Start":"07:22.910 ","End":"07:30.030","Text":"It\u0027s of the form the sum from n. It happens to be"},{"Start":"07:30.030 ","End":"07:36.650","Text":"1 but this works for any k to infinity of q^n."},{"Start":"07:36.650 ","End":"07:45.990","Text":"Turns out this converges for minus 1 less than q, less than one of those."},{"Start":"07:45.990 ","End":"07:53.940","Text":"If q is between minus 1 and 1 then it converges otherwise,"},{"Start":"07:53.940 ","End":"07:59.030","Text":"q either bigger or equal to 1"},{"Start":"07:59.030 ","End":"08:05.860","Text":"or q less than or equal to minus 1 then it diverges."},{"Start":"08:05.860 ","End":"08:11.905","Text":"But in our case, 4/7 falls between minus 1 and 1."},{"Start":"08:11.905 ","End":"08:17.740","Text":"Just to summarize, so here we have q is 4/7 and it"},{"Start":"08:17.740 ","End":"08:23.130","Text":"happens to be that q is between minus 1 and 1."},{"Start":"08:23.130 ","End":"08:30.765","Text":"The constant makes no difference and so for this one also converges."},{"Start":"08:30.765 ","End":"08:35.735","Text":"When you break a series up into 2 bits and each one converges,"},{"Start":"08:35.735 ","End":"08:39.965","Text":"then the original one converges too."},{"Start":"08:39.965 ","End":"08:47.790","Text":"We got 2 divergent series and 1 convergent. We are done."}],"ID":10521}],"Thumbnail":null,"ID":52092},{"Name":"The Divergence Test","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 5","Duration":"","ChapterTopicVideoID":10194,"CourseChapterTopicPlaylistID":52093,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.285","Text":"This exercise is made up of 6 separate exercises."},{"Start":"00:06.285 ","End":"00:08.560","Text":"Each of them is an infinite series,"},{"Start":"00:08.560 ","End":"00:14.500","Text":"and we have to decide whether it converges or diverges."},{"Start":"00:14.660 ","End":"00:17.040","Text":"This is a bit of a spoiler,"},{"Start":"00:17.040 ","End":"00:21.215","Text":"but I can tell you right now that all 6 of them diverge."},{"Start":"00:21.215 ","End":"00:25.515","Text":"The way we\u0027re going to show this is with the following theorem,"},{"Start":"00:25.515 ","End":"00:32.100","Text":"that if the general term doesn\u0027t go to 0 as n goes to infinity,"},{"Start":"00:32.100 ","End":"00:35.085","Text":"then the series diverges."},{"Start":"00:35.085 ","End":"00:38.190","Text":"In each case, we\u0027ll take a_n or the general term"},{"Start":"00:38.190 ","End":"00:42.519","Text":"and show that as n goes to infinity, it doesn\u0027t go to 0."},{"Start":"00:42.519 ","End":"00:46.205","Text":"I want to point out that when I say the limit is not 0,"},{"Start":"00:46.205 ","End":"00:47.630","Text":"it could mean 2 things."},{"Start":"00:47.630 ","End":"00:50.285","Text":"It could mean it has a limit but just not 0,"},{"Start":"00:50.285 ","End":"00:52.910","Text":"or it could mean that it has no limit at all,"},{"Start":"00:52.910 ","End":"00:55.225","Text":"or plus or minus infinity or whatever."},{"Start":"00:55.225 ","End":"00:58.910","Text":"There\u0027s many ways for it not to equal 0."},{"Start":"00:58.910 ","End":"01:03.560","Text":"Let\u0027s take them 1 by 1,"},{"Start":"01:03.560 ","End":"01:06.860","Text":"and we\u0027ll start with the first 1."},{"Start":"01:06.860 ","End":"01:11.375","Text":"What I have to do if I\u0027m going to show that this is divergent,"},{"Start":"01:11.375 ","End":"01:17.810","Text":"is to take the limit as n goes to infinity of the general term, I mean,"},{"Start":"01:17.810 ","End":"01:28.710","Text":"just throw out the Sigma of 4n plus 5 over 7n plus 8."},{"Start":"01:29.120 ","End":"01:31.335","Text":"What we can do is,"},{"Start":"01:31.335 ","End":"01:33.405","Text":"we all know various tricks,"},{"Start":"01:33.405 ","End":"01:40.760","Text":"easiest thing to do is to divide top and bottom by n and then"},{"Start":"01:40.760 ","End":"01:48.390","Text":"we get that this is the limit of 4 plus 5 over n,"},{"Start":"01:48.390 ","End":"01:56.540","Text":"that\u0027s the numerator divided by n. Denominator divided by n is 7 plus 8 over n,"},{"Start":"01:56.540 ","End":"01:59.050","Text":"n goes to infinity."},{"Start":"01:59.050 ","End":"02:01.320","Text":"That actually equals, well,"},{"Start":"02:01.320 ","End":"02:07.250","Text":"we write this symbolically as 4 plus 5 over infinity is just a symbolic way of"},{"Start":"02:07.250 ","End":"02:15.035","Text":"writing that this thing goes to 0 as n goes to infinity, 8 over infinity."},{"Start":"02:15.035 ","End":"02:19.140","Text":"Which is just, well,"},{"Start":"02:19.140 ","End":"02:21.794","Text":"this is 0 and this is 0,"},{"Start":"02:21.794 ","End":"02:24.080","Text":"so it\u0027s just 4/7."},{"Start":"02:24.080 ","End":"02:28.490","Text":"Of course, 4/7 is not equal to 0."},{"Start":"02:28.490 ","End":"02:31.170","Text":"This has a limit like the a_n,"},{"Start":"02:31.170 ","End":"02:32.790","Text":"but it\u0027s just not 0."},{"Start":"02:32.790 ","End":"02:35.550","Text":"So diverges."},{"Start":"02:35.550 ","End":"02:39.900","Text":"I won\u0027t write the word diverges, they all diverge."},{"Start":"02:39.900 ","End":"02:44.705","Text":"Number 2, 1, there\u0027s no n in it,"},{"Start":"02:44.705 ","End":"02:49.680","Text":"but the general term a_n is 1."},{"Start":"02:49.680 ","End":"02:50.760","Text":"It\u0027s a constant term."},{"Start":"02:50.760 ","End":"02:52.420","Text":"For all n, it\u0027s 1."},{"Start":"02:52.420 ","End":"02:58.470","Text":"What we need is the limit as n goes to infinity of 1."},{"Start":"02:59.000 ","End":"03:02.910","Text":"Well, it doesn\u0027t matter what n is, this thing is 1,"},{"Start":"03:02.910 ","End":"03:06.755","Text":"so this limit of a constant is just that constant."},{"Start":"03:06.755 ","End":"03:10.015","Text":"Again, this is not equal to 0."},{"Start":"03:10.015 ","End":"03:13.235","Text":"So the general term has a limit."},{"Start":"03:13.235 ","End":"03:19.430","Text":"Constant 1 has a limit of 1 which is not 0."},{"Start":"03:19.430 ","End":"03:23.080","Text":"This one\u0027s a little bit trickier."},{"Start":"03:23.080 ","End":"03:27.370","Text":"When n goes to infinity,"},{"Start":"03:27.370 ","End":"03:31.450","Text":"natural log of n also goes to infinity,"},{"Start":"03:31.450 ","End":"03:39.360","Text":"but the cosine tends to oscillate between 1 and minus 1."},{"Start":"03:39.360 ","End":"03:41.260","Text":"When something goes to infinity,"},{"Start":"03:41.260 ","End":"03:43.510","Text":"even though it\u0027s not continuous,"},{"Start":"03:43.510 ","End":"03:45.669","Text":"I\u0027m just taking sample points."},{"Start":"03:45.669 ","End":"03:52.620","Text":"But without using higher mathematics,"},{"Start":"03:52.620 ","End":"03:55.580","Text":"it\u0027s a bit hard to explain, but generally,"},{"Start":"03:55.580 ","End":"04:02.760","Text":"because natural log of n is going to infinity and it\u0027s going through all kinds of values,"},{"Start":"04:03.380 ","End":"04:08.660","Text":"the cosine is not going to settle down."},{"Start":"04:08.660 ","End":"04:12.440","Text":"Only if the natural log of n was always"},{"Start":"04:12.440 ","End":"04:18.300","Text":"such that the cosine of it was 0 from some point on."},{"Start":"04:24.080 ","End":"04:28.710","Text":"It\u0027s a bit of a hand-waving thing, but generally,"},{"Start":"04:28.860 ","End":"04:37.680","Text":"the limit as n goes to infinity of cosine of"},{"Start":"04:37.680 ","End":"04:46.700","Text":"natural log of n is just non-existent, doesn\u0027t exist."},{"Start":"04:51.440 ","End":"04:56.935","Text":"I\u0027d need to get into a lot more computations to really prove it."},{"Start":"04:56.935 ","End":"05:00.460","Text":"So I\u0027ve given you a sort of explanation that"},{"Start":"05:00.460 ","End":"05:04.290","Text":"the cosine keeps oscillating between plus and minus 1,"},{"Start":"05:04.290 ","End":"05:07.240","Text":"and natural log of n doesn\u0027t have an irregular pattern,"},{"Start":"05:07.240 ","End":"05:10.525","Text":"so it\u0027s going to be hitting a lot of pluses and minuses."},{"Start":"05:10.525 ","End":"05:16.190","Text":"It\u0027s the best I\u0027ll do here. It doesn\u0027t exist."},{"Start":"05:16.190 ","End":"05:18.410","Text":"So in particular, it\u0027s not 0,"},{"Start":"05:18.410 ","End":"05:23.540","Text":"so the series diverges."},{"Start":"05:23.880 ","End":"05:33.175","Text":"The limit as n goes to infinity of n also doesn\u0027t exist."},{"Start":"05:33.175 ","End":"05:35.350","Text":"It\u0027s equal to infinity."},{"Start":"05:35.350 ","End":"05:40.955","Text":"It doesn\u0027t exist, but even if you think of infinity as a number, it\u0027s still not 0."},{"Start":"05:40.955 ","End":"05:44.500","Text":"Either way, you can say it\u0027s nonexistent or you can say it\u0027s infinity,"},{"Start":"05:44.500 ","End":"05:47.330","Text":"but it\u0027s still not going to be 0."},{"Start":"05:47.850 ","End":"05:51.805","Text":"Now, this 1."},{"Start":"05:51.805 ","End":"05:59.880","Text":"For number 5, I\u0027m going to assume that you\u0027ve learnt some calculus,"},{"Start":"05:59.880 ","End":"06:03.760","Text":"you\u0027ve learnt L\u0027Hopital\u0027s rule."},{"Start":"06:04.580 ","End":"06:07.765","Text":"It turns out that"},{"Start":"06:07.765 ","End":"06:13.160","Text":"this limit of the general term is going to be infinity, and I\u0027ll show you."},{"Start":"06:13.160 ","End":"06:16.865","Text":"Suppose I took a set of n, the variable x,"},{"Start":"06:16.865 ","End":"06:26.630","Text":"and I want to know what is the limit as x goes to infinity of e^x over x cubed,"},{"Start":"06:26.630 ","End":"06:29.585","Text":"or just a set of n take a variable x."},{"Start":"06:29.585 ","End":"06:33.470","Text":"Then by L\u0027Hopital\u0027s rule,"},{"Start":"06:33.470 ","End":"06:36.719","Text":"I\u0027ll just write his name."},{"Start":"06:37.540 ","End":"06:41.615","Text":"I think that\u0027s how it\u0027s spelled its French name."},{"Start":"06:41.615 ","End":"06:43.970","Text":"L\u0027Hopital\u0027s rule says that,"},{"Start":"06:43.970 ","End":"06:48.845","Text":"if this goes to infinity and this goes to infinity,"},{"Start":"06:48.845 ","End":"06:55.850","Text":"then we can differentiate"},{"Start":"06:55.850 ","End":"07:00.170","Text":"top and bottom and get an equivalent limit."},{"Start":"07:00.170 ","End":"07:03.380","Text":"This is equal to the limit."},{"Start":"07:03.380 ","End":"07:05.810","Text":"If I differentiate this,"},{"Start":"07:05.810 ","End":"07:07.745","Text":"I also get e^x."},{"Start":"07:07.745 ","End":"07:12.454","Text":"Differentiate the bottom, I get 3x squared,"},{"Start":"07:12.454 ","End":"07:15.054","Text":"still x goes to infinity."},{"Start":"07:15.054 ","End":"07:21.960","Text":"Again, use L\u0027Hopital, differentiate top and bottom,"},{"Start":"07:21.960 ","End":"07:26.830","Text":"e^x over 6x still infinity over infinity,"},{"Start":"07:26.830 ","End":"07:33.885","Text":"differentiate again, limit of e^x over 6."},{"Start":"07:33.885 ","End":"07:37.235","Text":"Finally, this goes to infinity and this is a constant,"},{"Start":"07:37.235 ","End":"07:40.620","Text":"so this is infinity."},{"Start":"07:40.960 ","End":"07:49.320","Text":"It\u0027s certainly not 0."},{"Start":"07:49.320 ","End":"07:53.690","Text":"I know I\u0027ve changed from the discrete variable n to a continuous variable,"},{"Start":"07:53.690 ","End":"07:59.535","Text":"but generally, it\u0027s not a watertight proof,"},{"Start":"07:59.535 ","End":"08:04.385","Text":"but it demonstrate that this thing gets indefinitely large."},{"Start":"08:04.385 ","End":"08:09.470","Text":"This is not 0, and so we\u0027ve got something divergent."},{"Start":"08:09.470 ","End":"08:14.920","Text":"In number 6, there\u0027s a trick we can do."},{"Start":"08:15.080 ","End":"08:20.480","Text":"A_n here is n to the power of n,"},{"Start":"08:20.480 ","End":"08:24.440","Text":"which I can write as n times n times n times,"},{"Start":"08:24.440 ","End":"08:29.645","Text":"and so on and so on and so on times n and times."},{"Start":"08:29.645 ","End":"08:38.180","Text":"Whereas n factorial is n times n minus 1 times n minus 2,"},{"Start":"08:38.180 ","End":"08:42.455","Text":"and so on, down to 1."},{"Start":"08:42.455 ","End":"08:50.680","Text":"Now, this fraction is greater than 1 because this is bigger or equal to."},{"Start":"08:50.680 ","End":"08:52.000","Text":"But then this is bigger than this,"},{"Start":"08:52.000 ","End":"08:55.045","Text":"this is bigger than this, the same number of factors,"},{"Start":"08:55.045 ","End":"09:00.310","Text":"but each factor here is bigger or equal to the factor below."},{"Start":"09:00.310 ","End":"09:05.025","Text":"So this a_n is bigger or equal to 1."},{"Start":"09:05.025 ","End":"09:08.750","Text":"So limit can\u0027t be 0."},{"Start":"09:08.750 ","End":"09:13.010","Text":"It either has no limit,"},{"Start":"09:13.010 ","End":"09:15.905","Text":"actually, it does have a limit."},{"Start":"09:15.905 ","End":"09:23.625","Text":"But let\u0027s see, n goes to infinity of a_n."},{"Start":"09:23.625 ","End":"09:25.205","Text":"Well, let\u0027s just say,"},{"Start":"09:25.205 ","End":"09:28.115","Text":"I don\u0027t know whether it has a limit or not."},{"Start":"09:28.115 ","End":"09:35.600","Text":"But it\u0027s certainly not equal to 0 because it constantly bigger or equal to 1,"},{"Start":"09:35.600 ","End":"09:37.685","Text":"it can never get to 0."},{"Start":"09:37.685 ","End":"09:42.470","Text":"This also diverges, and that\u0027s the last 1 of the set,"},{"Start":"09:42.470 ","End":"09:44.940","Text":"and so we are done."}],"ID":10522}],"Thumbnail":null,"ID":52093},{"Name":"The Integral Test","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 6 Part a","Duration":"","ChapterTopicVideoID":10196,"CourseChapterTopicPlaylistID":52094,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.995","Text":"In this exercise, we have to decide if this infinite series converges or diverges."},{"Start":"00:07.995 ","End":"00:13.395","Text":"We\u0027re going to use the integral test; that\u0027s what it\u0027s called."},{"Start":"00:13.395 ","End":"00:16.260","Text":"Basically, it says that if we have a series"},{"Start":"00:16.260 ","End":"00:20.025","Text":"which is positive, decreasing, and continuous,"},{"Start":"00:20.025 ","End":"00:24.390","Text":"then we can replace the series with an integral"},{"Start":"00:24.390 ","End":"00:29.440","Text":"and 1 converges if and only if the other converges."},{"Start":"00:29.440 ","End":"00:33.860","Text":"The value doesn\u0027t equal the value of this,"},{"Start":"00:33.860 ","End":"00:39.935","Text":"but the convergence of 1, if and only if we have convergence of the other."},{"Start":"00:39.935 ","End":"00:45.370","Text":"Having said that, let me scroll down a bit,"},{"Start":"00:45.370 ","End":"00:48.080","Text":"and let\u0027s check if we have the conditions."},{"Start":"00:48.080 ","End":"00:50.000","Text":"First of all, certainly,"},{"Start":"00:50.000 ","End":"00:52.145","Text":"each term is positive,"},{"Start":"00:52.145 ","End":"00:54.125","Text":"n starts from 1,"},{"Start":"00:54.125 ","End":"00:59.310","Text":"n squared positive n cubed plus 1 is positive, so that\u0027s okay."},{"Start":"01:00.160 ","End":"01:05.600","Text":"Now, f(x), this is the general term a_n."},{"Start":"01:05.600 ","End":"01:15.525","Text":"The function we obtain is x squared over x cubed plus 1."},{"Start":"01:15.525 ","End":"01:21.310","Text":"We\u0027re interested in x from 1 onwards."},{"Start":"01:21.530 ","End":"01:27.940","Text":"It certainly continuous because polynomials are continuous and the quotient"},{"Start":"01:27.940 ","End":"01:33.880","Text":"is continuous and we have no problems with the 0 denominator because it\u0027s positive,"},{"Start":"01:33.880 ","End":"01:37.250","Text":"so we still have to check that it\u0027s decreasing."},{"Start":"01:37.250 ","End":"01:40.240","Text":"The way to check that it\u0027s decreasing,"},{"Start":"01:40.240 ","End":"01:44.905","Text":"easiest thing to do is to show that the derivative is negative."},{"Start":"01:44.905 ","End":"01:48.310","Text":"Well, negative at least on the range from 1 to infinity,"},{"Start":"01:48.310 ","End":"01:50.710","Text":"I\u0027ll just say x bigger than 1,"},{"Start":"01:50.710 ","End":"01:54.700","Text":"and so let\u0027s differentiate."},{"Start":"01:54.700 ","End":"01:58.490","Text":"F\u0027(x) is equal to,"},{"Start":"01:58.490 ","End":"02:04.570","Text":"perhaps this is a good time to remind you of the derivative of a quotient,"},{"Start":"02:04.570 ","End":"02:07.945","Text":"the derivative of u/v say"},{"Start":"02:07.945 ","End":"02:17.310","Text":"is u\u0027v minus uv\u0027/v squared,"},{"Start":"02:17.310 ","End":"02:22.390","Text":"so here, we get derivative of the numerator is"},{"Start":"02:22.390 ","End":"02:28.270","Text":"2x times the denominator x cubed plus"},{"Start":"02:28.270 ","End":"02:36.480","Text":"1 minus numerator times the derivative of denominator,"},{"Start":"02:36.480 ","End":"02:39.030","Text":"which is 3x squared,"},{"Start":"02:39.030 ","End":"02:42.960","Text":"and all this over,"},{"Start":"02:42.960 ","End":"02:51.200","Text":"well, something squared, but because I want to show positivity, I can ignore this."},{"Start":"02:51.200 ","End":"02:53.285","Text":"Let\u0027s just look at the numerator."},{"Start":"02:53.285 ","End":"02:57.950","Text":"I\u0027m just looking at the numerator now and that is equal to, let\u0027s see,"},{"Start":"02:57.950 ","End":"03:03.260","Text":"2x^4"},{"Start":"03:03.260 ","End":"03:11.700","Text":"plus 2x minus 3x^4,"},{"Start":"03:16.000 ","End":"03:19.200","Text":"let\u0027s get some more space here."},{"Start":"03:19.200 ","End":"03:22.659","Text":"This is equal to,"},{"Start":"03:23.210 ","End":"03:29.250","Text":"this is 2x minus x^4,"},{"Start":"03:29.250 ","End":"03:33.530","Text":"and if I take x outside the brackets,"},{"Start":"03:33.530 ","End":"03:40.520","Text":"I\u0027ve got x times 2 minus x cubed,"},{"Start":"03:40.520 ","End":"03:45.300","Text":"and x is positive,"},{"Start":"03:45.320 ","End":"03:51.660","Text":"2 minus x cubed is negative."},{"Start":"03:51.660 ","End":"03:56.505","Text":"It\u0027s negative except for x equals 1,"},{"Start":"03:56.505 ","End":"03:58.730","Text":"and in actual fact,"},{"Start":"03:58.730 ","End":"04:03.430","Text":"we don\u0027t have to be decreasing everywhere just from a certain point on"},{"Start":"04:03.430 ","End":"04:09.310","Text":"and I\u0027ll say that this is negative."},{"Start":"04:12.080 ","End":"04:14.830","Text":"When x is bigger than the cube root of 2,"},{"Start":"04:14.830 ","End":"04:19.020","Text":"but certainly from 2 onwards,"},{"Start":"04:19.020 ","End":"04:21.425","Text":"this would certainly be true,"},{"Start":"04:21.425 ","End":"04:25.730","Text":"because then x cubed is 8 and only increases."},{"Start":"04:25.730 ","End":"04:27.830","Text":"From a certain point on,"},{"Start":"04:27.830 ","End":"04:30.785","Text":"it is negative, and that\u0027s all we need."},{"Start":"04:30.785 ","End":"04:34.230","Text":"This is mostly decreasing."},{"Start":"04:36.890 ","End":"04:39.180","Text":"Just for precision sake,"},{"Start":"04:39.180 ","End":"04:44.090","Text":"let me just replace this with the actual condition that is true"},{"Start":"04:44.090 ","End":"04:49.305","Text":"when x is bigger than cube root of 2, just for precision."},{"Start":"04:49.305 ","End":"04:50.720","Text":"It doesn\u0027t matter, as I said,"},{"Start":"04:50.720 ","End":"04:53.750","Text":"to such a decrease from a certain point onwards."},{"Start":"04:53.750 ","End":"04:55.940","Text":"We\u0027ve met all the conditions,"},{"Start":"04:55.940 ","End":"05:00.285","Text":"and now it\u0027s time to compare the series with the integral."},{"Start":"05:00.285 ","End":"05:02.280","Text":"I have to check the integral,"},{"Start":"05:02.280 ","End":"05:05.055","Text":"I\u0027ll scroll back up so you can see what I\u0027m talking about,"},{"Start":"05:05.055 ","End":"05:06.540","Text":"what we want is this,"},{"Start":"05:06.540 ","End":"05:12.285","Text":"the integral from 1 to infinity of f(x), which is this,"},{"Start":"05:12.285 ","End":"05:20.190","Text":"x squared over x cubed plus 1 dx."},{"Start":"05:20.190 ","End":"05:23.260","Text":"Let\u0027s see if this converges."},{"Start":"05:23.260 ","End":"05:26.480","Text":"The reason we\u0027re talking about the convergence of an"},{"Start":"05:26.480 ","End":"05:29.330","Text":"integral is that this is an improper integral,"},{"Start":"05:29.330 ","End":"05:31.265","Text":"it has the infinity here."},{"Start":"05:31.265 ","End":"05:37.010","Text":"To remind you, the meaning of the convergence here means the existence of"},{"Start":"05:37.010 ","End":"05:43.370","Text":"the limit and we replace the infinity by some lateral,"},{"Start":"05:43.370 ","End":"05:49.790","Text":"let\u0027s say b, and we take the limit as b goes to infinity of the same thing,"},{"Start":"05:49.790 ","End":"05:54.364","Text":"x squared over x cubed plus 1 dx,"},{"Start":"05:54.364 ","End":"05:56.060","Text":"and if this limit exists,"},{"Start":"05:56.060 ","End":"05:58.595","Text":"then the integral converges."},{"Start":"05:58.595 ","End":"06:03.660","Text":"Let\u0027s see if we can do this integral."},{"Start":"06:03.860 ","End":"06:07.940","Text":"I\u0027ll do the integral at the side or first of all,"},{"Start":"06:07.940 ","End":"06:11.510","Text":"compute the indefinite integral."},{"Start":"06:11.510 ","End":"06:16.045","Text":"I want to know what is the indefinite integral,"},{"Start":"06:16.045 ","End":"06:19.975","Text":"what we call a primitive or anti-derivative,"},{"Start":"06:19.975 ","End":"06:23.520","Text":"x cubed plus 1 dx,"},{"Start":"06:23.520 ","End":"06:30.245","Text":"and notice that the derivative of the denominator is 3x squared."},{"Start":"06:30.245 ","End":"06:33.265","Text":"It\u0027s not quite what we have in the numerator,"},{"Start":"06:33.265 ","End":"06:40.210","Text":"but if I put a 3 here and a 1/3 here,"},{"Start":"06:40.210 ","End":"06:42.890","Text":"you\u0027ll agree with me that I haven\u0027t changed anything,"},{"Start":"06:42.890 ","End":"06:49.380","Text":"and now I have the derivative of the denominator in the numerator,"},{"Start":"06:49.380 ","End":"06:57.270","Text":"and then the answer to this is just natural log of the denominator."},{"Start":"06:57.640 ","End":"07:02.255","Text":"I\u0027m not going to repeat all of the integral calculus here,"},{"Start":"07:02.255 ","End":"07:06.140","Text":"but there is a pattern that the integral of f\u0027/f"},{"Start":"07:06.140 ","End":"07:10.840","Text":"is a natural log of f; the denominator."},{"Start":"07:10.840 ","End":"07:16.230","Text":"We don\u0027t need the plus c because we\u0027re going to substitute it here."},{"Start":"07:16.900 ","End":"07:22.340","Text":"What we get here now is this thing,"},{"Start":"07:22.340 ","End":"07:31.145","Text":"natural log of x cubed plus 1 evaluated between 1 and b."},{"Start":"07:31.145 ","End":"07:33.805","Text":"I forgot to write limit."},{"Start":"07:33.805 ","End":"07:43.795","Text":"Now we can say that this is the limit as b goes to infinity."},{"Start":"07:43.795 ","End":"07:50.540","Text":"Now, this expression evaluated like this means substitute b,"},{"Start":"07:50.540 ","End":"07:52.655","Text":"substitute 1, and subtract."},{"Start":"07:52.655 ","End":"07:54.320","Text":"If we substitute b,"},{"Start":"07:54.320 ","End":"08:02.460","Text":"we get natural log of b cubed plus 1, and if we substitute 1,"},{"Start":"08:02.460 ","End":"08:06.030","Text":"we just get natural log of 2,"},{"Start":"08:06.030 ","End":"08:10.235","Text":"and then as b goes to infinity,"},{"Start":"08:10.235 ","End":"08:12.410","Text":"b cubed plus 1,"},{"Start":"08:12.410 ","End":"08:15.740","Text":"this part here also goes to infinity,"},{"Start":"08:15.740 ","End":"08:20.690","Text":"and the natural log of infinity is also infinity."},{"Start":"08:20.690 ","End":"08:23.795","Text":"In short, this is infinity,"},{"Start":"08:23.795 ","End":"08:28.485","Text":"or in other words, it diverges."},{"Start":"08:28.485 ","End":"08:31.890","Text":"Integral has to be some actual number."},{"Start":"08:31.890 ","End":"08:35.270","Text":"This diverges, and if this diverges,"},{"Start":"08:35.270 ","End":"08:38.135","Text":"then the original series diverges."},{"Start":"08:38.135 ","End":"08:41.700","Text":"We are done."}],"ID":10523},{"Watched":false,"Name":"Exercise 6 Part b","Duration":"","ChapterTopicVideoID":10197,"CourseChapterTopicPlaylistID":52094,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.175","Text":"In this exercise, we\u0027re given an infinite series from 1 to infinity of this,"},{"Start":"00:08.175 ","End":"00:12.630","Text":"and we have to decide if this series converges or diverges."},{"Start":"00:12.630 ","End":"00:16.400","Text":"We\u0027re going to use the integral test for convergence,"},{"Start":"00:16.400 ","End":"00:19.340","Text":"which is encapsulated in this box."},{"Start":"00:19.340 ","End":"00:25.275","Text":"Generally speaking, it says that if the general term,"},{"Start":"00:25.275 ","End":"00:30.720","Text":"this 1 over square root of n plus 5 is positive for all n,"},{"Start":"00:30.720 ","End":"00:34.500","Text":"and the function f of x,"},{"Start":"00:34.500 ","End":"00:36.735","Text":"replace n by x you got a function,"},{"Start":"00:36.735 ","End":"00:38.960","Text":"if it\u0027s decreasing and continuous,"},{"Start":"00:38.960 ","End":"00:42.230","Text":"then we can replace the problem of"},{"Start":"00:42.230 ","End":"00:46.969","Text":"the series with an integration problem with an improper integral,"},{"Start":"00:46.969 ","End":"00:48.380","Text":"and if this converges,"},{"Start":"00:48.380 ","End":"00:50.840","Text":"this converges and vice versa."},{"Start":"00:50.840 ","End":"00:59.450","Text":"In our case, the f of x comes out to be just the same thing as the a_n,"},{"Start":"00:59.450 ","End":"01:05.610","Text":"but with x, 1 over square root of x plus 5."},{"Start":"01:05.610 ","End":"01:07.670","Text":"According to the theorem,"},{"Start":"01:07.670 ","End":"01:10.760","Text":"we have to look at the improper integral."},{"Start":"01:10.760 ","End":"01:15.770","Text":"It\u0027s improper because it goes to infinity of 1 over"},{"Start":"01:15.770 ","End":"01:25.215","Text":"this dx square root of x plus 5 dx and see if this converges or not."},{"Start":"01:25.215 ","End":"01:34.095","Text":"Converges for an improper integral means that we have a finite limit when,"},{"Start":"01:34.095 ","End":"01:36.085","Text":"let\u0027s use the letter b,"},{"Start":"01:36.085 ","End":"01:39.620","Text":"goes to infinity, replace the infinity by"},{"Start":"01:39.620 ","End":"01:44.090","Text":"the letter b and take b to infinity of the same thing,"},{"Start":"01:44.090 ","End":"01:53.235","Text":"1 over square root of x plus 5 dx."},{"Start":"01:53.235 ","End":"01:56.610","Text":"We have to evaluate this,"},{"Start":"01:56.610 ","End":"02:00.565","Text":"first of all the integral and then the limit."},{"Start":"02:00.565 ","End":"02:04.835","Text":"I\u0027m going to just do the indefinite integral at the side."},{"Start":"02:04.835 ","End":"02:10.790","Text":"The indefinite integral is like the primitive or anti-derivative."},{"Start":"02:10.790 ","End":"02:20.530","Text":"We have 1 over the square root of x plus 5 dx."},{"Start":"02:20.900 ","End":"02:28.110","Text":"What this equals, if we had 1 over the square root of x,"},{"Start":"02:30.820 ","End":"02:35.585","Text":"well, I\u0027ll do it more slowly."},{"Start":"02:35.585 ","End":"02:41.985","Text":"I\u0027m going to remind you that the derivative"},{"Start":"02:41.985 ","End":"02:51.300","Text":"of square root of x is 1 over twice square root of x."},{"Start":"02:51.430 ","End":"02:59.100","Text":"If I had here a 2 in the denominator,"},{"Start":"02:59.100 ","End":"03:04.665","Text":"and if I also put a 2 here, that wouldn\u0027t change."},{"Start":"03:04.665 ","End":"03:09.425","Text":"I can say that this integral is"},{"Start":"03:09.425 ","End":"03:15.800","Text":"the square root of x plus 5 plus a constant,"},{"Start":"03:15.800 ","End":"03:19.655","Text":"which I\u0027m not going to write because we\u0027re going to use a definite integral."},{"Start":"03:19.655 ","End":"03:23.410","Text":"You might say, \"But it\u0027s not x, it\u0027s x plus 5.\""},{"Start":"03:23.410 ","End":"03:30.210","Text":"Yes, that\u0027s true, but the inner derivative of x plus 5 is 1, so that\u0027s okay."},{"Start":"03:30.550 ","End":"03:40.955","Text":"Now I can go back to here and say this is the limit as b goes to infinity."},{"Start":"03:40.955 ","End":"03:44.430","Text":"In fact, I can put the 2 in front here,"},{"Start":"03:44.430 ","End":"03:48.485","Text":"the 2 is not going make any difference as to the convergence or divergence."},{"Start":"03:48.485 ","End":"03:58.020","Text":"We have the square root of x plus 5,"},{"Start":"03:58.020 ","End":"04:04.910","Text":"this thing we want to evaluate between 1 and b,"},{"Start":"04:04.910 ","End":"04:07.895","Text":"and then we\u0027re going to let b go to infinity."},{"Start":"04:07.895 ","End":"04:16.700","Text":"This is twice the limit of the square root of b"},{"Start":"04:16.700 ","End":"04:26.465","Text":"plus 5 minus the square root of 1 plus 5 as b goes to infinity."},{"Start":"04:26.465 ","End":"04:28.010","Text":"This is a constant,"},{"Start":"04:28.010 ","End":"04:35.130","Text":"when b plus 5 goes to infinity,"},{"Start":"04:35.130 ","End":"04:39.435","Text":"we have here the square root of infinity, which is infinity."},{"Start":"04:39.435 ","End":"04:41.975","Text":"Because this goes to infinity,"},{"Start":"04:41.975 ","End":"04:44.075","Text":"this thing comes out to infinity,"},{"Start":"04:44.075 ","End":"04:46.760","Text":"which means it doesn\u0027t have a limit,"},{"Start":"04:46.760 ","End":"04:48.110","Text":"it doesn\u0027t have a finite limit."},{"Start":"04:48.110 ","End":"04:54.410","Text":"The answer is that the integral diverges and so does the original series."},{"Start":"04:54.410 ","End":"04:58.770","Text":"That\u0027s the answer, it diverges. We\u0027re done."}],"ID":10524},{"Watched":false,"Name":"Exercise 6 Part c","Duration":"","ChapterTopicVideoID":10198,"CourseChapterTopicPlaylistID":52094,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have an infinite series."},{"Start":"00:03.270 ","End":"00:05.790","Text":"It\u0027s infinite, because it\u0027s from 1 to infinity"},{"Start":"00:05.790 ","End":"00:10.395","Text":"and we have to decide if it converges or diverges."},{"Start":"00:10.395 ","End":"00:14.730","Text":"The technique we\u0027re going to use here is the integral test,"},{"Start":"00:14.730 ","End":"00:18.300","Text":"which is summarized in the box here,"},{"Start":"00:18.300 ","End":"00:24.780","Text":"that basically allows us to replace a series with an"},{"Start":"00:24.780 ","End":"00:32.830","Text":"integral and instead of a_n,"},{"Start":"00:33.320 ","End":"00:35.580","Text":"as a function of n,"},{"Start":"00:35.580 ","End":"00:41.870","Text":"we take correspondingly a function of x to be the same thing, but with x."},{"Start":"00:41.870 ","End":"00:45.035","Text":"Like xe to the minus x, in our case."},{"Start":"00:45.035 ","End":"00:48.170","Text":"Now, the theorem to remind you says,"},{"Start":"00:48.170 ","End":"00:50.015","Text":"that if we have a series,"},{"Start":"00:50.015 ","End":"00:53.240","Text":"that\u0027s basically 3 things, positive, decreasing,"},{"Start":"00:53.240 ","End":"00:57.470","Text":"and continuous, then we can decide if"},{"Start":"00:57.470 ","End":"01:02.965","Text":"the series converges or not according to a certain improper integral."},{"Start":"01:02.965 ","End":"01:07.910","Text":"What we have to do, is after we\u0027ve checked that it meets the conditions,"},{"Start":"01:07.910 ","End":"01:12.320","Text":"is to replace this series by the improper integral."},{"Start":"01:12.320 ","End":"01:20.520","Text":"The reason it\u0027s improper is because it\u0027s an infinity here of f of x dx,"},{"Start":"01:20.520 ","End":"01:25.325","Text":"and to see if this converges or diverges,"},{"Start":"01:25.325 ","End":"01:29.960","Text":"and the answer to that is the same as the answer to the series."},{"Start":"01:29.960 ","End":"01:32.935","Text":"Let\u0027s see if we meet the conditions."},{"Start":"01:32.935 ","End":"01:39.585","Text":"Now, a_n is ne to the minus n,"},{"Start":"01:39.585 ","End":"01:44.205","Text":"and n of course, is bigger or equal to 1."},{"Start":"01:44.205 ","End":"01:48.410","Text":"Certainly, n is positive and e to the anything is positive,"},{"Start":"01:48.410 ","End":"01:51.990","Text":"so this is surely positive."},{"Start":"01:51.990 ","End":"01:56.020","Text":"Now, f of x is this."},{"Start":"01:56.270 ","End":"02:00.125","Text":"It\u0027s made up of elementary functions,"},{"Start":"02:00.125 ","End":"02:05.990","Text":"so it\u0027s certainly continuous and there\u0027s no problems with the not defined."},{"Start":"02:05.990 ","End":"02:08.725","Text":"It\u0027s defined everywhere, it\u0027s continuous."},{"Start":"02:08.725 ","End":"02:12.230","Text":"The only thing we still have to show,"},{"Start":"02:12.230 ","End":"02:16.025","Text":"is that it\u0027s decreasing before we can use the theorem."},{"Start":"02:16.025 ","End":"02:18.830","Text":"Let\u0027s check that f is decreasing."},{"Start":"02:18.830 ","End":"02:21.940","Text":"What we have, and I\u0027ll do that at the side,"},{"Start":"02:21.940 ","End":"02:28.580","Text":"we have a function f of x is equal to xe to the minus x."},{"Start":"02:28.580 ","End":"02:30.680","Text":"You want to show it\u0027s decreasing,"},{"Start":"02:30.680 ","End":"02:33.770","Text":"and the best way, not the best way, but in this case,"},{"Start":"02:33.770 ","End":"02:44.035","Text":"my suggestion is that we show that the derivative is negative, decreasing."},{"Start":"02:44.035 ","End":"02:46.750","Text":"If the derivative is negative, then it\u0027s decreasing,"},{"Start":"02:46.750 ","End":"02:51.145","Text":"at least on the range from x bigger than 1,"},{"Start":"02:51.145 ","End":"02:52.900","Text":"because we\u0027re going from 1 to infinity,"},{"Start":"02:52.900 ","End":"02:54.865","Text":"which means x bigger than 1."},{"Start":"02:54.865 ","End":"02:58.955","Text":"Let\u0027s see, what\u0027s the derivative."},{"Start":"02:58.955 ","End":"03:02.465","Text":"Do I need to remind you of the product rule?"},{"Start":"03:02.465 ","End":"03:05.880","Text":"Well, just in case, you should know this."},{"Start":"03:05.880 ","End":"03:07.725","Text":"The derivative of a product,"},{"Start":"03:07.725 ","End":"03:10.239","Text":"derivative of the first times the second"},{"Start":"03:10.239 ","End":"03:13.625","Text":"plus the first times the derivative of the second."},{"Start":"03:13.625 ","End":"03:18.630","Text":"In our case, u prime is 1."},{"Start":"03:18.630 ","End":"03:20.220","Text":"I don\u0027t write anything,"},{"Start":"03:20.220 ","End":"03:21.975","Text":"because 1 times something."},{"Start":"03:21.975 ","End":"03:28.770","Text":"Then v is e to the minus x, 1 times this."},{"Start":"03:28.770 ","End":"03:34.890","Text":"Then plus u, that\u0027s this 1 as is,"},{"Start":"03:34.890 ","End":"03:37.035","Text":"and the derivative of v,"},{"Start":"03:37.035 ","End":"03:39.105","Text":"it\u0027s e to the minus x."},{"Start":"03:39.105 ","End":"03:40.940","Text":"Using the chain rule,"},{"Start":"03:40.940 ","End":"03:43.940","Text":"it\u0027s e to the minus x,"},{"Start":"03:43.940 ","End":"03:46.310","Text":"but because it\u0027s minus x,"},{"Start":"03:46.310 ","End":"03:49.400","Text":"we have to also multiply by minus 1."},{"Start":"03:49.400 ","End":"03:57.695","Text":"In short, if I take e to the minus x outside the brackets, I get 1."},{"Start":"03:57.695 ","End":"04:02.640","Text":"Then x with minus 1 is minus x."},{"Start":"04:04.370 ","End":"04:10.180","Text":"This is certainly negative,"},{"Start":"04:11.300 ","End":"04:14.655","Text":"because x is bigger than 1,"},{"Start":"04:14.655 ","End":"04:20.160","Text":"so 1 minus x is negative and this is positive always."},{"Start":"04:20.160 ","End":"04:22.040","Text":"Positive times negative is negative,"},{"Start":"04:22.040 ","End":"04:25.600","Text":"so we have the conditions of the theorem."},{"Start":"04:25.600 ","End":"04:33.180","Text":"Now, we should go ahead and see if this improper integral converges."},{"Start":"04:34.580 ","End":"04:37.740","Text":"Instead of f of x, let me write this,"},{"Start":"04:37.740 ","End":"04:45.094","Text":"I\u0027ll just erase that and write xe to the minus x dx."},{"Start":"04:45.094 ","End":"04:48.290","Text":"Now, what is this thing with the infinity?"},{"Start":"04:48.290 ","End":"04:50.390","Text":"The improper integral is defined to be"},{"Start":"04:50.390 ","End":"04:55.610","Text":"the limit where instead of infinity we take some number,"},{"Start":"04:55.610 ","End":"04:58.130","Text":"call it b, and then we that b,"},{"Start":"04:58.130 ","End":"05:02.995","Text":"go to infinity and we get a limit of the same thing."},{"Start":"05:02.995 ","End":"05:05.400","Text":"If this limit exists,"},{"Start":"05:05.400 ","End":"05:07.850","Text":"in other words, if this thing converges,"},{"Start":"05:07.850 ","End":"05:13.590","Text":"then we\u0027ll know whether the original series converges or not. Let\u0027s see."},{"Start":"05:13.930 ","End":"05:18.180","Text":"This is equal to, again,"},{"Start":"05:18.180 ","End":"05:25.885","Text":"I need a side exercise to compute the integral of xe to the minus x."},{"Start":"05:25.885 ","End":"05:29.025","Text":"Let me do that over here."},{"Start":"05:29.025 ","End":"05:34.200","Text":"The integral of xe to the minus x dx,"},{"Start":"05:34.200 ","End":"05:42.720","Text":"we\u0027ll do using what you call the integration by parts, sorry."},{"Start":"05:42.720 ","End":"05:47.930","Text":"Integration by parts, I\u0027m not going to go over the whole theory again,"},{"Start":"05:47.930 ","End":"05:52.065","Text":"we need a u and a dv."},{"Start":"05:52.065 ","End":"05:54.110","Text":"The 1 that we want to differentiate,"},{"Start":"05:54.110 ","End":"05:57.800","Text":"the polynomial is usually the part that goes with the dv,"},{"Start":"05:57.800 ","End":"06:05.265","Text":"so will take u to be x and we\u0027ll let"},{"Start":"06:05.265 ","End":"06:13.695","Text":"dv equal e to the minus x dx."},{"Start":"06:13.695 ","End":"06:16.370","Text":"Then we need to complete the other 2."},{"Start":"06:16.370 ","End":"06:26.650","Text":"Du is equal to 1 dx and v is the antiderivative of this,"},{"Start":"06:26.650 ","End":"06:30.035","Text":"is minus e to the minus x."},{"Start":"06:30.035 ","End":"06:36.110","Text":"Then we use the formula that the integral of"},{"Start":"06:36.110 ","End":"06:44.480","Text":"udv is uv minus the integral of vdu."},{"Start":"06:44.480 ","End":"06:49.780","Text":"What we get is that this thing is equal to,"},{"Start":"06:49.780 ","End":"06:55.620","Text":"it was u and this was dv, so now I take uv,"},{"Start":"06:55.620 ","End":"07:04.085","Text":"xe to the minus x minus the integral of v,"},{"Start":"07:04.085 ","End":"07:06.950","Text":"which is minus e to"},{"Start":"07:06.950 ","End":"07:16.090","Text":"the minus x times du, which is dx."},{"Start":"07:18.140 ","End":"07:22.185","Text":"Well, minus with the minus cancels,"},{"Start":"07:22.185 ","End":"07:27.470","Text":"so I can just basically think of this as a plus."},{"Start":"07:27.470 ","End":"07:32.909","Text":"I\u0027ve got xe to the minus x,"},{"Start":"07:37.940 ","End":"07:44.130","Text":"and I\u0027ll just replace these 2 minuses with a plus."},{"Start":"07:44.130 ","End":"07:49.740","Text":"Erase this minus and put a plus here. I hope that\u0027s okay."},{"Start":"07:49.740 ","End":"07:52.450","Text":"The integral of e to the minus x,"},{"Start":"07:52.450 ","End":"07:59.475","Text":"the antiderivative is minus e to the minus x."},{"Start":"07:59.475 ","End":"08:02.235","Text":"That\u0027s the indefinite integral."},{"Start":"08:02.235 ","End":"08:04.905","Text":"Now, let\u0027s go back here."},{"Start":"08:04.905 ","End":"08:10.060","Text":"What we have is the limit as b goes to infinity."},{"Start":"08:10.060 ","End":"08:12.715","Text":"We know what the indefinite integral is."},{"Start":"08:12.715 ","End":"08:15.145","Text":"It\u0027s this, so I\u0027ll just put it in brackets."},{"Start":"08:15.145 ","End":"08:24.035","Text":"It\u0027s xe to the minus x minus e to the minus x."},{"Start":"08:24.035 ","End":"08:28.730","Text":"This has got to be taken between 1 and b,"},{"Start":"08:28.730 ","End":"08:31.480","Text":"which means substitute b, substitute a and subtract."},{"Start":"08:31.480 ","End":"08:36.510","Text":"I just noticed a small mistake,"},{"Start":"08:36.510 ","End":"08:43.430","Text":"uv is x times minus e to the minus x. I need a minus here,"},{"Start":"08:43.430 ","End":"08:48.530","Text":"and so a minus here and a minus here."},{"Start":"08:48.530 ","End":"08:51.720","Text":"Sorry about that, but no harm done."},{"Start":"08:52.450 ","End":"09:00.005","Text":"This is equal to the limit as b goes to infinity."},{"Start":"09:00.005 ","End":"09:03.000","Text":"Let\u0027s substitute b."},{"Start":"09:03.000 ","End":"09:11.745","Text":"We\u0027ve got minus be to the minus b,"},{"Start":"09:11.745 ","End":"09:16.810","Text":"minus e to the minus b."},{"Start":"09:19.490 ","End":"09:25.715","Text":"Well, when I\u0027m going to substitute 1 and I\u0027m going to subtract, they\u0027ll become pluses."},{"Start":"09:25.715 ","End":"09:28.040","Text":"The 2 minuses subtracted,"},{"Start":"09:28.040 ","End":"09:32.164","Text":"so it\u0027s plus 1e to the minus 1,"},{"Start":"09:32.164 ","End":"09:35.550","Text":"plus e to the minus 1."},{"Start":"09:36.680 ","End":"09:41.970","Text":"Next step. I should have written this in brackets,"},{"Start":"09:41.970 ","End":"09:44.280","Text":"because the whole thing is a limit."},{"Start":"09:44.280 ","End":"09:50.120","Text":"We get the limit as b goes to infinity."},{"Start":"09:50.120 ","End":"09:52.450","Text":"Now, the constant part is not really interesting."},{"Start":"09:52.450 ","End":"09:54.055","Text":"This is the interesting part."},{"Start":"09:54.055 ","End":"09:59.335","Text":"What we have, is we can take out of the first 2,"},{"Start":"09:59.335 ","End":"10:03.595","Text":"see we can take e to the minus b and you can also take a minus,"},{"Start":"10:03.595 ","End":"10:08.160","Text":"so we have a minus e to"},{"Start":"10:08.160 ","End":"10:14.020","Text":"the minus b times b"},{"Start":"10:14.020 ","End":"10:21.710","Text":"plus 1 plus a constant."},{"Start":"10:22.470 ","End":"10:26.140","Text":"The constant I can actually leave outside the integral."},{"Start":"10:26.140 ","End":"10:32.080","Text":"I can say, the limit of this plus,"},{"Start":"10:32.300 ","End":"10:36.840","Text":"let\u0027s see, this is e to the minus 1 and this is e to the minus 1,"},{"Start":"10:36.840 ","End":"10:39.470","Text":"2e to the minus 1."},{"Start":"10:39.470 ","End":"10:42.260","Text":"This constant is not going to make any difference."},{"Start":"10:42.260 ","End":"10:47.780","Text":"What we have to do is figure out this limit."},{"Start":"10:48.120 ","End":"10:59.100","Text":"If I write it, instead"},{"Start":"10:59.100 ","End":"11:01.960","Text":"of e to the minus b,"},{"Start":"11:02.510 ","End":"11:05.640","Text":"I\u0027ll put the minus in front,"},{"Start":"11:05.640 ","End":"11:12.755","Text":"b plus 1 over e to the power of b,"},{"Start":"11:12.755 ","End":"11:16.565","Text":"where b goes to infinity."},{"Start":"11:16.565 ","End":"11:20.330","Text":"This in itself is not immediately clear,"},{"Start":"11:20.330 ","End":"11:24.980","Text":"but we can actually use L\u0027Hopital\u0027s rule here."},{"Start":"11:24.980 ","End":"11:32.720","Text":"Because the numerator goes to infinity and the denominator goes to infinity,"},{"Start":"11:32.720 ","End":"11:37.980","Text":"we can actually say that this equals using L\u0027Hopital."},{"Start":"11:37.980 ","End":"11:40.820","Text":"I\u0027ll write his name, L\u0027Hopital,"},{"Start":"11:40.820 ","End":"11:46.295","Text":"the French mathematician who said that if we have an infinity over infinity situation,"},{"Start":"11:46.295 ","End":"11:49.130","Text":"we can replace this limit with"},{"Start":"11:49.130 ","End":"11:52.430","Text":"the derivative at the top and the derivative at the bottom,"},{"Start":"11:52.430 ","End":"11:56.065","Text":"so we\u0027ve got derivative of this is 1,"},{"Start":"11:56.065 ","End":"12:00.170","Text":"derivative of e^b is just e^b."},{"Start":"12:00.170 ","End":"12:01.730","Text":"Derivative with respect to b,"},{"Start":"12:01.730 ","End":"12:03.710","Text":"b is just some variable."},{"Start":"12:03.710 ","End":"12:12.180","Text":"Then this is minus 1 over infinity, which is 0."},{"Start":"12:12.200 ","End":"12:18.470","Text":"In other words, the integral converges to a finite value which happens to be 0,"},{"Start":"12:18.470 ","End":"12:21.635","Text":"and so the original series also converges."},{"Start":"12:21.635 ","End":"12:24.330","Text":"We are done."}],"ID":10525},{"Watched":false,"Name":"Exercise 6 Part d","Duration":"","ChapterTopicVideoID":10199,"CourseChapterTopicPlaylistID":52094,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, we\u0027re given an infinite series from 1 to"},{"Start":"00:05.160 ","End":"00:10.740","Text":"infinity and we have to decide if it converges or diverges,"},{"Start":"00:10.740 ","End":"00:14.730","Text":"and the technique we\u0027re going to use is the integral test,"},{"Start":"00:14.730 ","End":"00:17.460","Text":"which is summarized in the box here."},{"Start":"00:17.460 ","End":"00:20.700","Text":"Basically we have to meet 3 conditions,"},{"Start":"00:20.700 ","End":"00:25.065","Text":"the series has to be positive every term,"},{"Start":"00:25.065 ","End":"00:28.515","Text":"and the corresponding function,"},{"Start":"00:28.515 ","End":"00:30.105","Text":"get to that in a moment,"},{"Start":"00:30.105 ","End":"00:33.480","Text":"has to be decreasing and continuous."},{"Start":"00:33.480 ","End":"00:36.630","Text":"The function corresponding to the series,"},{"Start":"00:36.630 ","End":"00:38.415","Text":"I\u0027ll show you what I mean,"},{"Start":"00:38.415 ","End":"00:41.580","Text":"means that the general term,"},{"Start":"00:41.580 ","End":"00:47.069","Text":"a n is n squared e to the minus n cubed,"},{"Start":"00:47.069 ","End":"00:49.220","Text":"and the corresponding function is,"},{"Start":"00:49.220 ","End":"00:58.010","Text":"we replace n by x and we get a function of x which is x squared e to the minus x cubed."},{"Start":"00:58.010 ","End":"01:01.055","Text":"Once again, the terms have to be positive,"},{"Start":"01:01.055 ","End":"01:03.845","Text":"and this function has to be decreasing and continuous."},{"Start":"01:03.845 ","End":"01:10.900","Text":"Let\u0027s see, now, as for the positivity,"},{"Start":"01:10.900 ","End":"01:15.525","Text":"n is positive, so n squared is positive and e to"},{"Start":"01:15.525 ","End":"01:20.360","Text":"anything is positive, positive, we are okay."},{"Start":"01:20.360 ","End":"01:22.175","Text":"As for the function continuous,"},{"Start":"01:22.175 ","End":"01:23.840","Text":"well it\u0027s elementary functions,"},{"Start":"01:23.840 ","End":"01:30.050","Text":"x squared is continuous everywhere and e to the power of is continuous,"},{"Start":"01:30.050 ","End":"01:33.275","Text":"x cubed continuous, everything is continuous."},{"Start":"01:33.275 ","End":"01:37.640","Text":"I\u0027ll just note that this function we only care about when"},{"Start":"01:37.640 ","End":"01:43.380","Text":"x is bigger than 1 because we\u0027re going from 1 to infinity."},{"Start":"01:43.380 ","End":"01:48.110","Text":"The only thing that we still missing is the decreasing,"},{"Start":"01:48.110 ","End":"01:54.140","Text":"and the decreasing, easiest way to do that is to take the derivative,"},{"Start":"01:54.140 ","End":"01:56.730","Text":"and show that it\u0027s negative."},{"Start":"01:57.470 ","End":"02:04.984","Text":"Let\u0027s do that, f prime of x is,"},{"Start":"02:04.984 ","End":"02:10.665","Text":"now I\u0027m just going to remind you of the product rule just in case you\u0027ve forgotten it,"},{"Start":"02:10.665 ","End":"02:14.870","Text":"that when you have a product u times v and take the derivative,"},{"Start":"02:14.870 ","End":"02:17.810","Text":"it\u0027s the first times the derivative of the"},{"Start":"02:17.810 ","End":"02:21.635","Text":"second plus derivative of the first times the second,"},{"Start":"02:21.635 ","End":"02:23.330","Text":"and in our case,"},{"Start":"02:23.330 ","End":"02:25.120","Text":"if this is u and this is v,"},{"Start":"02:25.120 ","End":"02:28.319","Text":"we have x squared as is,"},{"Start":"02:28.319 ","End":"02:30.300","Text":"and then we differentiate this,"},{"Start":"02:30.300 ","End":"02:31.680","Text":"it\u0027s e to the something,"},{"Start":"02:31.680 ","End":"02:34.310","Text":"so we start off with e to the same thing,"},{"Start":"02:34.310 ","End":"02:36.350","Text":"but then we need the inner derivative,"},{"Start":"02:36.350 ","End":"02:40.225","Text":"which is minus 3x squared."},{"Start":"02:40.225 ","End":"02:43.625","Text":"The second bit, the derivative of the first part,"},{"Start":"02:43.625 ","End":"02:48.125","Text":"which is 2x times the second bit as is,"},{"Start":"02:48.125 ","End":"02:50.915","Text":"e to the minus x cubed."},{"Start":"02:50.915 ","End":"02:58.060","Text":"What I can do is I can take the e to the minus x cubed outside the brackets,"},{"Start":"02:58.060 ","End":"03:03.180","Text":"and we\u0027ve got e to the minus x cubed,"},{"Start":"03:03.180 ","End":"03:08.655","Text":"and then what we have left is, let\u0027s see,"},{"Start":"03:08.655 ","End":"03:12.040","Text":"I\u0027d rather take the 2x first from here,"},{"Start":"03:12.380 ","End":"03:18.060","Text":"and from here I get minus 3x to the fourth,"},{"Start":"03:18.060 ","End":"03:22.150","Text":"as a minus 3x squared together with the x squared."},{"Start":"03:24.380 ","End":"03:29.510","Text":"I could even take an extra x outside the brackets,"},{"Start":"03:29.510 ","End":"03:33.155","Text":"so it\u0027s, x e to the minus x cubed,"},{"Start":"03:33.155 ","End":"03:44.190","Text":"and then 2 minus 3x cubed."},{"Start":"03:45.280 ","End":"03:55.305","Text":"x cubed is bigger than 1, and so,"},{"Start":"03:55.305 ","End":"03:59.790","Text":"certainly 3x cubed is bigger than 3,"},{"Start":"03:59.790 ","End":"04:03.585","Text":"so 2 minus this is certainly negative,"},{"Start":"04:03.585 ","End":"04:05.675","Text":"the x is positive,"},{"Start":"04:05.675 ","End":"04:07.850","Text":"e to the anything is positive,"},{"Start":"04:07.850 ","End":"04:10.715","Text":"so we have a plus, plus minus."},{"Start":"04:10.715 ","End":"04:19.920","Text":"The whole thing, f prime of x is negative,"},{"Start":"04:19.920 ","End":"04:25.725","Text":"now we\u0027ve got the decreasing part."},{"Start":"04:25.725 ","End":"04:28.390","Text":"We\u0027ve fulfilled all the conditions,"},{"Start":"04:28.390 ","End":"04:33.835","Text":"what we do now is we replace the series by an integral,"},{"Start":"04:33.835 ","End":"04:36.350","Text":"instead of this infinite series,"},{"Start":"04:36.350 ","End":"04:38.645","Text":"we put the improper integral,"},{"Start":"04:38.645 ","End":"04:42.470","Text":"the integral from 1 to infinity, the infinities,"},{"Start":"04:42.470 ","End":"04:44.650","Text":"what makes it improper,"},{"Start":"04:44.650 ","End":"04:48.830","Text":"of x squared, this function here,"},{"Start":"04:48.830 ","End":"04:53.270","Text":"e to the minus x cubed dx."},{"Start":"04:53.270 ","End":"04:55.130","Text":"When we have an improper integral,"},{"Start":"04:55.130 ","End":"04:58.505","Text":"we can talk about its convergence because really,"},{"Start":"04:58.505 ","End":"05:05.340","Text":"the definition of this is the limit of a finite integral,"},{"Start":"05:05.340 ","End":"05:08.345","Text":"we go from 1 to some large number b,"},{"Start":"05:08.345 ","End":"05:12.590","Text":"and then we let b go to infinity of the same thing,"},{"Start":"05:12.590 ","End":"05:17.940","Text":"x squared e to the minus x cubed dx,"},{"Start":"05:17.940 ","End":"05:20.010","Text":"and if this limit exists,"},{"Start":"05:20.010 ","End":"05:21.675","Text":"it is a finite number,"},{"Start":"05:21.675 ","End":"05:26.000","Text":"then this converges, and then the original series converges."},{"Start":"05:26.000 ","End":"05:30.210","Text":"What we have to do is figure this out, and for this,"},{"Start":"05:30.210 ","End":"05:35.955","Text":"we first need the indefinite integral,"},{"Start":"05:35.955 ","End":"05:39.240","Text":"so again I\u0027m going to do that as a side exercise,"},{"Start":"05:39.240 ","End":"05:50.345","Text":"what I need to compute is the indefinite integral of x squared e to the minus x cubed dx."},{"Start":"05:50.345 ","End":"06:00.860","Text":"Now notice that, if I had here minus 3x squared,"},{"Start":"06:00.860 ","End":"06:04.175","Text":"just like over here,"},{"Start":"06:04.175 ","End":"06:08.690","Text":"we got a minus 3x squared from the derivative of this,"},{"Start":"06:08.690 ","End":"06:11.660","Text":"then we\u0027d have the derivative of this here."},{"Start":"06:11.660 ","End":"06:20.915","Text":"I\u0027m going to write it as minus 3x squared e to the minus x cubed dx,"},{"Start":"06:20.915 ","End":"06:22.910","Text":"but if I just did that, I\u0027ll be cheating."},{"Start":"06:22.910 ","End":"06:23.990","Text":"I\u0027ve changed the results,"},{"Start":"06:23.990 ","End":"06:25.535","Text":"so we have to compensate,"},{"Start":"06:25.535 ","End":"06:27.965","Text":"if I multiply by minus a 1/3,"},{"Start":"06:27.965 ","End":"06:30.545","Text":"then the minus 1/3 cancels with the minus 3,"},{"Start":"06:30.545 ","End":"06:32.790","Text":"and now I\u0027m okay."},{"Start":"06:34.280 ","End":"06:38.790","Text":"This is now equal to minus 1/3,"},{"Start":"06:38.790 ","End":"06:43.310","Text":"and the anti-derivative of this is just e to"},{"Start":"06:43.310 ","End":"06:47.780","Text":"the minus x cubed because I already have the derivative of this here,"},{"Start":"06:47.780 ","End":"06:49.670","Text":"so if I differentiate this,"},{"Start":"06:49.670 ","End":"06:51.740","Text":"I get this times minus 3x squared,"},{"Start":"06:51.740 ","End":"06:53.970","Text":"just like I did here."},{"Start":"06:54.170 ","End":"07:00.000","Text":"This is the primitive or anti-derivative or indefinite integral,"},{"Start":"07:00.000 ","End":"07:03.114","Text":"normally I would add a plus c but I\u0027m not bothering,"},{"Start":"07:03.114 ","End":"07:06.955","Text":"because we\u0027re going to plug it in here to do a definite integral."},{"Start":"07:06.955 ","End":"07:15.569","Text":"What we get here is the limit from the same b goes to infinity, but this time,"},{"Start":"07:15.580 ","End":"07:18.024","Text":"instead of this integral,"},{"Start":"07:18.024 ","End":"07:22.240","Text":"I can write minus 1/3,"},{"Start":"07:23.420 ","End":"07:28.565","Text":"e to the minus x cubed,"},{"Start":"07:28.565 ","End":"07:35.515","Text":"taken between 1 and b."},{"Start":"07:35.515 ","End":"07:41.040","Text":"The minus 1/3 will come out in front,"},{"Start":"07:41.040 ","End":"07:42.930","Text":"so I\u0027ve got minus 1/3,"},{"Start":"07:42.930 ","End":"07:52.390","Text":"that won\u0027t affect convergence or divergence limit as b goes to infinity,"},{"Start":"07:52.580 ","End":"07:57.875","Text":"and what we get here is just the e to the minus x cubed part."},{"Start":"07:57.875 ","End":"08:07.260","Text":"We\u0027ve got e to the minus b cubed minus when I plug 1 in,"},{"Start":"08:07.260 ","End":"08:10.720","Text":"e to the minus 1."},{"Start":"08:12.170 ","End":"08:18.135","Text":"When b goes to infinity, this part,"},{"Start":"08:18.135 ","End":"08:26.055","Text":"the minus b cubed will go to minus infinity."},{"Start":"08:26.055 ","End":"08:29.000","Text":"Basically what we get is, here,"},{"Start":"08:29.000 ","End":"08:33.935","Text":"and we can even actually compute the value of this integral is minus 1/3,"},{"Start":"08:33.935 ","End":"08:36.905","Text":"e to the minus infinity,"},{"Start":"08:36.905 ","End":"08:41.300","Text":"this is just symbolically writing that this way to"},{"Start":"08:41.300 ","End":"08:46.025","Text":"show that it\u0027s e to the power of something that goes to minus infinity,"},{"Start":"08:46.025 ","End":"08:51.515","Text":"now e to the minus infinity is well known to be 0,"},{"Start":"08:51.515 ","End":"08:55.275","Text":"this part here is 0,"},{"Start":"08:55.275 ","End":"08:57.135","Text":"all I get is,"},{"Start":"08:57.135 ","End":"08:58.830","Text":"and a minus and the minus is plus,"},{"Start":"08:58.830 ","End":"09:01.230","Text":"I get 1/3, 1 over e,"},{"Start":"09:01.230 ","End":"09:03.180","Text":"it\u0027s 1 over 3e."},{"Start":"09:03.180 ","End":"09:06.390","Text":"In fact, the value is not important,"},{"Start":"09:06.390 ","End":"09:09.500","Text":"the point is that it exists and it\u0027s a finite number,"},{"Start":"09:09.500 ","End":"09:18.030","Text":"and so this integral in fact converges,"},{"Start":"09:18.030 ","End":"09:20.780","Text":"and if the integral converges,"},{"Start":"09:20.780 ","End":"09:22.685","Text":"and so does the series,"},{"Start":"09:22.685 ","End":"09:25.550","Text":"and so that\u0027s the answer,"},{"Start":"09:25.550 ","End":"09:29.100","Text":"the series converges and we\u0027re done."}],"ID":10526},{"Watched":false,"Name":"Exercise 6 Part e","Duration":"","ChapterTopicVideoID":10200,"CourseChapterTopicPlaylistID":52094,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this exercise we\u0027re given an infinite series,"},{"Start":"00:03.390 ","End":"00:07.245","Text":"n goes from 2 up to infinity of this expression."},{"Start":"00:07.245 ","End":"00:09.060","Text":"This is our a_n,"},{"Start":"00:09.060 ","End":"00:13.695","Text":"and we have to decide if it converges or diverges."},{"Start":"00:13.695 ","End":"00:16.680","Text":"We\u0027re going to use the integral test,"},{"Start":"00:16.680 ","End":"00:19.720","Text":"which is summarized in the box here."},{"Start":"00:26.540 ","End":"00:29.445","Text":"We wrote the a_n."},{"Start":"00:29.445 ","End":"00:31.620","Text":"This is our a_n,"},{"Start":"00:31.620 ","End":"00:40.110","Text":"which is 1 over n log cubed to the n, natural log."},{"Start":"00:40.110 ","End":"00:45.950","Text":"If we can show 3 things that this a_n is a positive series,"},{"Start":"00:45.950 ","End":"00:47.330","Text":"each term is positive,"},{"Start":"00:47.330 ","End":"00:51.155","Text":"and f of x. I have to explain what f of x is."},{"Start":"00:51.155 ","End":"00:54.390","Text":"F is the same as a_n,"},{"Start":"00:54.390 ","End":"00:57.330","Text":"except with the n replaced by x."},{"Start":"00:57.330 ","End":"00:59.835","Text":"It\u0027s 1 over x,"},{"Start":"00:59.835 ","End":"01:01.925","Text":"natural log cubed of x."},{"Start":"01:01.925 ","End":"01:06.685","Text":"We can show that f of x also is decreasing and continuous."},{"Start":"01:06.685 ","End":"01:09.779","Text":"That\u0027s 3 things: series positive,"},{"Start":"01:09.779 ","End":"01:12.215","Text":"function decreasing, function continuous."},{"Start":"01:12.215 ","End":"01:17.750","Text":"Then we can conclude that the convergence or not of"},{"Start":"01:17.750 ","End":"01:24.390","Text":"the series is the same as the convergence or not of this integral."},{"Start":"01:24.890 ","End":"01:29.155","Text":"Let\u0027s see if we can meet the conditions of the theorem."},{"Start":"01:29.155 ","End":"01:32.860","Text":"Certainly, a_n is positive,"},{"Start":"01:32.860 ","End":"01:40.420","Text":"because, well, needs maybe a word of explanation."},{"Start":"01:40.420 ","End":"01:46.240","Text":"N is positive because n goes from 2 onwards."},{"Start":"01:46.240 ","End":"01:49.759","Text":"N is bigger or equal to 2,"},{"Start":"01:50.790 ","End":"01:53.425","Text":"and if n is bigger or equal to 2,"},{"Start":"01:53.425 ","End":"01:55.450","Text":"this is certainly positive,"},{"Start":"01:55.450 ","End":"01:58.990","Text":"a natural log of n is also positive."},{"Start":"01:58.990 ","End":"02:00.750","Text":"When n is 1, it\u0027s 0,"},{"Start":"02:00.750 ","End":"02:02.575","Text":"that\u0027s why we started at 2."},{"Start":"02:02.575 ","End":"02:04.280","Text":"But from 2 onwards,"},{"Start":"02:04.280 ","End":"02:08.090","Text":"the natural log is already positive."},{"Start":"02:08.480 ","End":"02:12.090","Text":"Everything is positive, so a_n is positive."},{"Start":"02:12.090 ","End":"02:14.275","Text":"This is okay."},{"Start":"02:14.275 ","End":"02:16.660","Text":"Now, is the function continuous?"},{"Start":"02:16.660 ","End":"02:18.520","Text":"We\u0027ll do that part first."},{"Start":"02:18.520 ","End":"02:25.210","Text":"The natural log is a continuous function wherever it\u0027s defined,"},{"Start":"02:25.210 ","End":"02:27.030","Text":"which is x bigger than 0,"},{"Start":"02:27.030 ","End":"02:29.845","Text":"and it\u0027s certainly true for x bigger than 2,"},{"Start":"02:29.845 ","End":"02:34.470","Text":"which by the way is the domain bigger or equal to,"},{"Start":"02:34.470 ","End":"02:38.155","Text":"doesn\u0027t really matter, because we\u0027re going from 2 onwards."},{"Start":"02:38.155 ","End":"02:41.065","Text":"In this range certainly natural log is defined,"},{"Start":"02:41.065 ","End":"02:43.824","Text":"and it\u0027s not 0, it\u0027s bigger than 0."},{"Start":"02:43.824 ","End":"02:46.735","Text":"Just the same explanation as here."},{"Start":"02:46.735 ","End":"02:50.265","Text":"This thing is always positive."},{"Start":"02:50.265 ","End":"02:54.935","Text":"It\u0027s continuous because it\u0027s made up of elementary functions,"},{"Start":"02:54.935 ","End":"02:57.425","Text":"and there\u0027s no 0s in the denominator."},{"Start":"02:57.425 ","End":"03:00.890","Text":"The decreasing is the more difficult part."},{"Start":"03:00.890 ","End":"03:03.165","Text":"We do that by"},{"Start":"03:03.165 ","End":"03:13.435","Text":"differentiating f and showing that the derivative is negative."},{"Start":"03:13.435 ","End":"03:16.050","Text":"There is a trick one could use,"},{"Start":"03:16.050 ","End":"03:17.270","Text":"and I\u0027ll mention it,"},{"Start":"03:17.270 ","End":"03:18.710","Text":"but I won\u0027t use it."},{"Start":"03:18.710 ","End":"03:22.430","Text":"If we show that the denominator is increasing,"},{"Start":"03:22.430 ","End":"03:25.545","Text":"then 1 over an increasing something is decreasing,"},{"Start":"03:25.545 ","End":"03:27.600","Text":"but I\u0027ll be conservative,"},{"Start":"03:27.600 ","End":"03:30.890","Text":"I\u0027ll do it the old-fashioned way of just differentiating and"},{"Start":"03:30.890 ","End":"03:35.570","Text":"seeing that we get something negative."},{"Start":"03:35.570 ","End":"03:37.880","Text":"Before we start differentiating,"},{"Start":"03:37.880 ","End":"03:41.540","Text":"I\u0027d like to remind you of some rules that you may have forgotten."},{"Start":"03:41.540 ","End":"03:45.850","Text":"First of all, the derivative."},{"Start":"03:45.850 ","End":"03:49.349","Text":"I can either look at this as a quotient,"},{"Start":"03:49.349 ","End":"03:52.040","Text":"as a fraction, or as something to the minus 1."},{"Start":"03:52.040 ","End":"03:53.540","Text":"Let\u0027s say I look at it as a fraction,"},{"Start":"03:53.540 ","End":"04:01.380","Text":"as u over v. I\u0027ll remind you that u over v derivative"},{"Start":"04:01.380 ","End":"04:11.830","Text":"is the derivative of u times v minus u times the derivative of v over v squared."},{"Start":"04:11.830 ","End":"04:16.940","Text":"The other thing is that we\u0027re going to need is a product rule."},{"Start":"04:16.940 ","End":"04:21.260","Text":"We\u0027ll need u times v derivative,"},{"Start":"04:21.260 ","End":"04:30.035","Text":"is derivative of u times v plus u times derivative of v. I guess exponents now."},{"Start":"04:30.035 ","End":"04:31.895","Text":"You will remember those."},{"Start":"04:31.895 ","End":"04:34.280","Text":"Let\u0027s get started with"},{"Start":"04:34.280 ","End":"04:39.295","Text":"the differentiation with the aim of hoping that we get something negative."},{"Start":"04:39.295 ","End":"04:41.555","Text":"So f prime of x."},{"Start":"04:41.555 ","End":"04:43.670","Text":"First of all, if I use the quotient rule,"},{"Start":"04:43.670 ","End":"04:45.410","Text":"and this is u and this is v,"},{"Start":"04:45.410 ","End":"04:47.750","Text":"u prime is 0,"},{"Start":"04:47.750 ","End":"04:51.160","Text":"so that\u0027s nothing, so I just have a minus."},{"Start":"04:51.160 ","End":"04:53.370","Text":"Then u is 1,"},{"Start":"04:53.370 ","End":"04:55.965","Text":"so I just got v prime."},{"Start":"04:55.965 ","End":"05:00.770","Text":"V prime, I\u0027ll first of all write it as"},{"Start":"05:00.770 ","End":"05:08.430","Text":"x natural log of x cubed prime."},{"Start":"05:08.430 ","End":"05:13.129","Text":"All this, over the denominator squared."},{"Start":"05:13.129 ","End":"05:15.920","Text":"Not even going to write what the denominator is,"},{"Start":"05:15.920 ","End":"05:20.870","Text":"just something squared, because I\u0027m just looking for positivity or negativity,"},{"Start":"05:20.870 ","End":"05:23.580","Text":"so squared will mean that it\u0027s positive."},{"Start":"05:24.010 ","End":"05:29.930","Text":"You know what? Let\u0027s even just continue with the numerator here,"},{"Start":"05:29.930 ","End":"05:34.279","Text":"because we just care about positive or negative."},{"Start":"05:34.279 ","End":"05:40.260","Text":"Pardon me. Let\u0027s continue with the derivative of this."},{"Start":"05:40.760 ","End":"05:52.140","Text":"In fact, let me just go with x natural log cubed of x, the derivative."},{"Start":"05:52.140 ","End":"05:54.710","Text":"If this comes out positive,"},{"Start":"05:54.710 ","End":"05:55.910","Text":"then because of the minus,"},{"Start":"05:55.910 ","End":"05:57.290","Text":"this whole thing will be negative."},{"Start":"05:57.290 ","End":"06:01.110","Text":"If we have minus a positive over something squared,"},{"Start":"06:01.110 ","End":"06:03.155","Text":"so let\u0027s just do this part."},{"Start":"06:03.155 ","End":"06:06.800","Text":"First of all, I\u0027m going to use the product rule."},{"Start":"06:06.800 ","End":"06:08.660","Text":"It\u0027s the derivative of x,"},{"Start":"06:08.660 ","End":"06:15.705","Text":"which is 1 times v natural log cubed of x,"},{"Start":"06:15.705 ","End":"06:21.805","Text":"plus this thing as is and the derivative of that."},{"Start":"06:21.805 ","End":"06:25.385","Text":"I\u0027m going to need the chain rule because it\u0027s something cubed."},{"Start":"06:25.385 ","End":"06:30.350","Text":"Something cubed means it\u0027s 3 times that something squared."},{"Start":"06:30.350 ","End":"06:34.040","Text":"But we also need the inner derivative of natural log x,"},{"Start":"06:34.040 ","End":"06:36.899","Text":"which is 1 over x."},{"Start":"06:38.060 ","End":"06:41.390","Text":"We can cancel this x with this x,"},{"Start":"06:41.390 ","End":"06:43.960","Text":"that will make things easier."},{"Start":"06:43.960 ","End":"06:50.300","Text":"In fact, I can also take natural log squared of x outside the brackets,"},{"Start":"06:50.300 ","End":"06:53.225","Text":"which notice it\u0027s a positive quantity as well."},{"Start":"06:53.225 ","End":"06:55.400","Text":"What we\u0027re left with is natural log"},{"Start":"06:55.400 ","End":"07:05.850","Text":"of x plus 3."},{"Start":"07:05.850 ","End":"07:08.325","Text":"X is, as I said, bigger than 2."},{"Start":"07:08.325 ","End":"07:13.129","Text":"Natural log, well, it\u0027s positive in any case because it\u0027s squared."},{"Start":"07:13.129 ","End":"07:19.400","Text":"This is certainly bigger than 0."},{"Start":"07:19.400 ","End":"07:22.730","Text":"It\u0027s even bigger than natural log of 2."},{"Start":"07:22.730 ","End":"07:25.325","Text":"All together this is a positive thing."},{"Start":"07:25.325 ","End":"07:28.290","Text":"Positive times positive."},{"Start":"07:30.410 ","End":"07:35.240","Text":"All in all we\u0027ve got that this thing is minus,"},{"Start":"07:35.240 ","End":"07:42.240","Text":"I\u0027ll just write it as a positive. That\u0027s not good."},{"Start":"07:42.410 ","End":"07:48.780","Text":"I\u0027ll just write positive over something,"},{"Start":"07:48.780 ","End":"07:51.880","Text":"because I don\u0027t even know what it is, squared."},{"Start":"07:52.220 ","End":"07:57.535","Text":"All together I have something that\u0027s negative."},{"Start":"07:57.535 ","End":"08:02.055","Text":"Derivative negative means the function\u0027s decreasing,"},{"Start":"08:02.055 ","End":"08:08.525","Text":"and that means that I have now done the last of the 3 conditions: series positive,"},{"Start":"08:08.525 ","End":"08:12.925","Text":"function continuous, and now function decreasing."},{"Start":"08:12.925 ","End":"08:15.125","Text":"I\u0027ve met the conditions,"},{"Start":"08:15.125 ","End":"08:17.525","Text":"I can then use the conclusions."},{"Start":"08:17.525 ","End":"08:23.800","Text":"The conclusion basically says that I can replace this series by the integral,"},{"Start":"08:23.800 ","End":"08:25.845","Text":"it\u0027s an improper integral,"},{"Start":"08:25.845 ","End":"08:26.975","Text":"it goes to infinity,"},{"Start":"08:26.975 ","End":"08:28.565","Text":"the same lower bound,"},{"Start":"08:28.565 ","End":"08:37.330","Text":"of the function, which is x natural log cubed of x dx."},{"Start":"08:37.330 ","End":"08:40.140","Text":"If this converges, this converges,"},{"Start":"08:40.140 ","End":"08:43.265","Text":"and if this doesn\u0027t converge, this doesn\u0027t converge."},{"Start":"08:43.265 ","End":"08:46.820","Text":"Now, what do I mean by convergence of an integral?"},{"Start":"08:46.820 ","End":"08:48.815","Text":"Well, it\u0027s an improper integral,"},{"Start":"08:48.815 ","End":"08:52.130","Text":"so this is written as a limit."},{"Start":"08:52.130 ","End":"08:55.325","Text":"Instead of 2 to infinity,"},{"Start":"08:55.325 ","End":"08:57.170","Text":"we go from 2 to some number,"},{"Start":"08:57.170 ","End":"09:02.190","Text":"I\u0027ll call it b, and then let b go to infinity of the same thing."},{"Start":"09:02.190 ","End":"09:09.825","Text":"1 over x natural log cubed of x dx."},{"Start":"09:09.825 ","End":"09:12.320","Text":"If this limit exists and is finite,"},{"Start":"09:12.320 ","End":"09:15.740","Text":"then this is a convergent integral,"},{"Start":"09:15.740 ","End":"09:17.855","Text":"and then the series will be convergent."},{"Start":"09:17.855 ","End":"09:20.149","Text":"How do we do this integral?"},{"Start":"09:20.149 ","End":"09:24.305","Text":"My recommendation is to use a substitution."},{"Start":"09:24.305 ","End":"09:28.340","Text":"I see here that there\u0027s a natural log of x, it\u0027s cubed."},{"Start":"09:28.340 ","End":"09:31.025","Text":"But I also see that there is a derivative of it,"},{"Start":"09:31.025 ","End":"09:34.400","Text":"1 over x here."},{"Start":"09:34.400 ","End":"09:40.430","Text":"What I\u0027m going to do, I\u0027ll first compute the indefinite integral at the side."},{"Start":"09:40.430 ","End":"09:43.310","Text":"As I said, we\u0027re going to use substitution."},{"Start":"09:43.310 ","End":"09:48.415","Text":"I want to compute the integral indefinite of,"},{"Start":"09:48.415 ","End":"09:55.950","Text":"I\u0027ll just write it as dx over x natural log cubed of x."},{"Start":"09:55.950 ","End":"09:59.700","Text":"What I\u0027m going to do is the substitution."},{"Start":"09:59.740 ","End":"10:10.195","Text":"I\u0027m going to substitute t equals natural log of x."},{"Start":"10:10.195 ","End":"10:12.135","Text":"If that\u0027s the case,"},{"Start":"10:12.135 ","End":"10:19.270","Text":"then dt is 1 over x dx."},{"Start":"10:19.910 ","End":"10:25.205","Text":"I noticed that 1 over x dx is exactly this first bit."},{"Start":"10:25.205 ","End":"10:28.000","Text":"This over this is dt,"},{"Start":"10:28.370 ","End":"10:32.070","Text":"natural log of x is t,"},{"Start":"10:32.070 ","End":"10:35.890","Text":"so we have dt over t cubed."},{"Start":"10:36.110 ","End":"10:45.115","Text":"This is equal to the integral of t to the power of minus 3."},{"Start":"10:45.115 ","End":"10:46.960","Text":"This is equal to,"},{"Start":"10:46.960 ","End":"10:49.160","Text":"you raise the power by 1,"},{"Start":"10:49.160 ","End":"10:50.890","Text":"so it\u0027s t to the minus 2,"},{"Start":"10:50.890 ","End":"10:54.295","Text":"and divide by the new power."},{"Start":"10:54.295 ","End":"10:57.030","Text":"This is what we have, plus a constant,"},{"Start":"10:57.030 ","End":"11:01.640","Text":"which we needn\u0027t write because we\u0027re going to use a definite integral."},{"Start":"11:01.640 ","End":"11:07.470","Text":"But I have to get it back from t to x."},{"Start":"11:07.550 ","End":"11:10.020","Text":"With a slight rewrite,"},{"Start":"11:10.020 ","End":"11:13.185","Text":"it\u0027s minus 1/2, that\u0027s the constant bit."},{"Start":"11:13.185 ","End":"11:20.180","Text":"T_minus 2, I can write it as 1 over natural log of x squared,"},{"Start":"11:20.180 ","End":"11:22.000","Text":"squared we write here."},{"Start":"11:22.000 ","End":"11:25.680","Text":"Now, I go back here,"},{"Start":"11:25.680 ","End":"11:30.605","Text":"and I can compute this definite integral by means of the"},{"Start":"11:30.605 ","End":"11:36.790","Text":"indefinite integral just by writing what I have here."},{"Start":"11:36.790 ","End":"11:41.700","Text":"You know what? The minus 1/2 can come right out in front."},{"Start":"11:41.700 ","End":"11:43.470","Text":"I write the minus 1/2."},{"Start":"11:43.470 ","End":"11:47.840","Text":"A constant won\u0027t affect the convergence or divergence."},{"Start":"11:47.840 ","End":"11:52.149","Text":"Then I have 1 over"},{"Start":"11:52.149 ","End":"12:00.210","Text":"natural log squared of x evaluated between 2 and b,"},{"Start":"12:00.210 ","End":"12:01.650","Text":"which means plug in b,"},{"Start":"12:01.650 ","End":"12:03.765","Text":"plug in 2, and subtract."},{"Start":"12:03.765 ","End":"12:05.900","Text":"I\u0027ll forget about the minus 1/2,"},{"Start":"12:05.900 ","End":"12:09.505","Text":"I\u0027ll just continue with this part here."},{"Start":"12:09.505 ","End":"12:13.155","Text":"As I say, this doesn\u0027t affect the convergence."},{"Start":"12:13.155 ","End":"12:17.240","Text":"I\u0027ve got the limit as b goes to infinity of."},{"Start":"12:17.240 ","End":"12:23.070","Text":"When I put in b, I have 1 over natural log squared of b."},{"Start":"12:23.070 ","End":"12:30.240","Text":"When I put in 2, I have 1 over natural log squared of 2."},{"Start":"12:30.260 ","End":"12:34.340","Text":"This is a constant, and this thing,"},{"Start":"12:34.340 ","End":"12:36.320","Text":"when b goes to infinity,"},{"Start":"12:36.320 ","End":"12:40.805","Text":"the natural log of b also goes to infinity."},{"Start":"12:40.805 ","End":"12:51.195","Text":"Basically what I have here is 1 over infinity squared minus 1 over,"},{"Start":"12:51.195 ","End":"12:54.140","Text":"it\u0027s a constant, doesn\u0027t matter what it is."},{"Start":"12:54.140 ","End":"12:57.295","Text":"Anyway, 1 over infinity is 0,"},{"Start":"12:57.295 ","End":"13:01.460","Text":"so all we get is a finite number,"},{"Start":"13:01.460 ","End":"13:10.160","Text":"1 or minus 1 over natural log squared of 2."},{"Start":"13:10.160 ","End":"13:11.990","Text":"That\u0027s just this part."},{"Start":"13:11.990 ","End":"13:14.705","Text":"If we actually wanted to figure out the integral,"},{"Start":"13:14.705 ","End":"13:17.390","Text":"we\u0027d have to multiply also by minus 1/2."},{"Start":"13:17.390 ","End":"13:21.200","Text":"But we don\u0027t care about the value of the integral,"},{"Start":"13:21.200 ","End":"13:24.380","Text":"the theorem just states that if the integral converges,"},{"Start":"13:24.380 ","End":"13:25.760","Text":"and so does the series."},{"Start":"13:25.760 ","End":"13:29.240","Text":"So here we\u0027ve shown that the integral converges,"},{"Start":"13:29.240 ","End":"13:31.585","Text":"because it came out of finite number,"},{"Start":"13:31.585 ","End":"13:36.250","Text":"and so the series does too, and we\u0027re done."}],"ID":10527},{"Watched":false,"Name":"Exercise 6 Part f","Duration":"","ChapterTopicVideoID":10195,"CourseChapterTopicPlaylistID":52094,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we\u0027re given an infinite series,"},{"Start":"00:04.080 ","End":"00:06.105","Text":"it\u0027s from 2 to infinity."},{"Start":"00:06.105 ","End":"00:09.070","Text":"In a moment, we\u0027ll see why the 2."},{"Start":"00:09.110 ","End":"00:11.700","Text":"What we\u0027re going to do is,"},{"Start":"00:11.700 ","End":"00:17.130","Text":"decide whether it\u0027s convergent or divergent using the integral test,"},{"Start":"00:17.130 ","End":"00:19.530","Text":"which is summarized in the box here."},{"Start":"00:19.530 ","End":"00:21.915","Text":"There\u0027s quite a bit here, but basically,"},{"Start":"00:21.915 ","End":"00:28.770","Text":"we have to show 3 things to meet 3 conditions and if we meet these conditions,"},{"Start":"00:28.770 ","End":"00:32.579","Text":"we can replace the series by an improper integral,"},{"Start":"00:32.579 ","End":"00:38.655","Text":"which is hopefully easier to deal with the decision about convergence or not."},{"Start":"00:38.655 ","End":"00:41.980","Text":"I\u0027ll just make some more room here."},{"Start":"00:41.980 ","End":"00:48.260","Text":"This is the same original series where a_n is 1"},{"Start":"00:48.260 ","End":"00:55.530","Text":"over n square root of the natural log of n. However,"},{"Start":"00:55.530 ","End":"00:58.925","Text":"at this point, I can explain why we didn\u0027t start from 1 like usual."},{"Start":"00:58.925 ","End":"01:00.875","Text":"Because if n is 1,"},{"Start":"01:00.875 ","End":"01:03.395","Text":"the natural log of 1 is 0,"},{"Start":"01:03.395 ","End":"01:10.280","Text":"and we get the square root of 0 is 0 on the denominator so we start from 2,"},{"Start":"01:10.280 ","End":"01:12.995","Text":"and then everything makes sense."},{"Start":"01:12.995 ","End":"01:16.220","Text":"It also answers the question about positivity."},{"Start":"01:16.220 ","End":"01:19.515","Text":"If n is 2 or more,"},{"Start":"01:19.515 ","End":"01:22.620","Text":"certainly the n is positive."},{"Start":"01:22.620 ","End":"01:25.560","Text":"The natural log of anything bigger than"},{"Start":"01:25.560 ","End":"01:28.700","Text":"1 is positive and the square root of positive is positive."},{"Start":"01:28.700 ","End":"01:33.000","Text":"In short, the positive part is okay."},{"Start":"01:33.290 ","End":"01:35.775","Text":"Now, what is this f of x?"},{"Start":"01:35.775 ","End":"01:41.540","Text":"This f of x is just the same expression as in a_n,"},{"Start":"01:41.540 ","End":"01:44.210","Text":"but instead of a discrete variable n,"},{"Start":"01:44.210 ","End":"01:46.580","Text":"we have a continuous variable x."},{"Start":"01:46.580 ","End":"01:54.875","Text":"This will replace n by x and get a function x square root of natural log of x."},{"Start":"01:54.875 ","End":"01:57.485","Text":"But if n is,"},{"Start":"01:57.485 ","End":"01:59.135","Text":"goes from 2 to infinity,"},{"Start":"01:59.135 ","End":"02:03.290","Text":"then x also is from 2 to infinity,"},{"Start":"02:03.290 ","End":"02:05.610","Text":"in other words, bigger than 2."},{"Start":"02:06.610 ","End":"02:12.830","Text":"The continuous part, to show that f is continuous,"},{"Start":"02:12.830 ","End":"02:16.640","Text":"that\u0027s also straightforward because it\u0027s made up of all continuous bits."},{"Start":"02:16.640 ","End":"02:20.720","Text":"Natural log is continuous and when x is bigger than 2,"},{"Start":"02:20.720 ","End":"02:22.805","Text":"it\u0027s positive and no problems."},{"Start":"02:22.805 ","End":"02:26.660","Text":"Never 0. Square root is continuous."},{"Start":"02:26.660 ","End":"02:30.320","Text":"One over x is continuous and just combining,"},{"Start":"02:30.320 ","End":"02:32.540","Text":"multiplying, dividing, and so on."},{"Start":"02:32.540 ","End":"02:36.200","Text":"It\u0027s made up of elementary functions, so it\u0027s continuous."},{"Start":"02:36.200 ","End":"02:43.205","Text":"The only bit of work we have to do is showing the decreasing part."},{"Start":"02:43.205 ","End":"02:45.950","Text":"We have to show this in order to be able to"},{"Start":"02:45.950 ","End":"02:49.825","Text":"benefit from the conclusions of the integral test."},{"Start":"02:49.825 ","End":"02:54.395","Text":"Let\u0027s show that this function is decreasing by differentiating."},{"Start":"02:54.395 ","End":"02:57.690","Text":"I\u0027m showing that the derivative is negative."},{"Start":"02:58.520 ","End":"03:05.810","Text":"One way of differentiating is instead of having the 1 over is to write it"},{"Start":"03:05.810 ","End":"03:13.625","Text":"as x square root of natural log of x to the power of minus 1."},{"Start":"03:13.625 ","End":"03:16.865","Text":"Or you could do it with the quotient rule."},{"Start":"03:16.865 ","End":"03:19.085","Text":"Either would work."},{"Start":"03:19.085 ","End":"03:26.540","Text":"Then we get that f prime of x is equal to because of the minus 1,"},{"Start":"03:26.540 ","End":"03:30.570","Text":"we get minus the same thing to the minus"},{"Start":"03:30.570 ","End":"03:36.710","Text":"2 and that thing is x square root of natural log of x."},{"Start":"03:36.710 ","End":"03:39.590","Text":"But we have to multiply by the inner derivative."},{"Start":"03:39.590 ","End":"03:42.910","Text":"Now, the inner derivative is a product."},{"Start":"03:42.910 ","End":"03:49.450","Text":"I\u0027ll remind you that the derivative of a product uv,"},{"Start":"03:49.450 ","End":"03:54.170","Text":"each time you differentiate 1 and multiply by the other,"},{"Start":"03:54.170 ","End":"03:57.005","Text":"first times derivative of the second."},{"Start":"03:57.005 ","End":"04:00.890","Text":"I guess the other thing we should remember is that the derivative of"},{"Start":"04:00.890 ","End":"04:05.210","Text":"the square root of something if I have the square root of u,"},{"Start":"04:05.210 ","End":"04:07.850","Text":"some expression and take the derivative,"},{"Start":"04:07.850 ","End":"04:11.180","Text":"it\u0027s 1 over twice the square root of u."},{"Start":"04:11.180 ","End":"04:14.615","Text":"But because of the chain rule also the inner derivative."},{"Start":"04:14.615 ","End":"04:17.725","Text":"Altogether using these here,"},{"Start":"04:17.725 ","End":"04:22.325","Text":"the missing bit which is the derivative of this part,"},{"Start":"04:22.325 ","End":"04:24.770","Text":"is from the chain rule."},{"Start":"04:24.770 ","End":"04:26.510","Text":"Derivative of the first,"},{"Start":"04:26.510 ","End":"04:32.060","Text":"which is 1 times the second square root of natural log of"},{"Start":"04:32.060 ","End":"04:38.990","Text":"x plus the first as is which is x and the derivative of this."},{"Start":"04:38.990 ","End":"04:40.970","Text":"Now the derivative of the square root,"},{"Start":"04:40.970 ","End":"04:47.165","Text":"we said is 1 over twice the square root of this thing,"},{"Start":"04:47.165 ","End":"04:50.989","Text":"but we also have to multiply by the inner derivative u prime,"},{"Start":"04:50.989 ","End":"04:53.335","Text":"which is 1 over x."},{"Start":"04:53.335 ","End":"05:01.000","Text":"It looks messy, not so bad as it appears."},{"Start":"05:01.490 ","End":"05:06.750","Text":"Let me just keep this as is or you know what?"},{"Start":"05:06.750 ","End":"05:09.470","Text":"I\u0027ll write this in the denominator,"},{"Start":"05:09.470 ","End":"05:11.855","Text":"it might just look a bit better."},{"Start":"05:11.855 ","End":"05:13.745","Text":"I need some more space here."},{"Start":"05:13.745 ","End":"05:23.149","Text":"This is minus 1 over x square root of natural log x squared."},{"Start":"05:23.149 ","End":"05:25.055","Text":"Doesn\u0027t really matter what\u0027s in here."},{"Start":"05:25.055 ","End":"05:27.934","Text":"I\u0027m just noticing that it\u0027s squared, so it\u0027s positive."},{"Start":"05:27.934 ","End":"05:29.690","Text":"This part is negative."},{"Start":"05:29.690 ","End":"05:34.259","Text":"Then hopefully if this part comes up positive will end up negative."},{"Start":"05:39.980 ","End":"05:46.540","Text":"I can cancel the x with the 1 over x."},{"Start":"05:46.700 ","End":"05:53.285","Text":"If I put this over a common denominator,"},{"Start":"05:53.285 ","End":"06:01.805","Text":"so let\u0027s write it as all over twice square root of natural log of x."},{"Start":"06:01.805 ","End":"06:06.710","Text":"The second part is the easy part is plus 1."},{"Start":"06:06.710 ","End":"06:10.310","Text":"This part, I have to multiply top and"},{"Start":"06:10.310 ","End":"06:14.495","Text":"bottom by twice the square root of natural log of x."},{"Start":"06:14.495 ","End":"06:17.930","Text":"Square root times square root is the thing itself."},{"Start":"06:17.930 ","End":"06:20.635","Text":"This is what I get."},{"Start":"06:20.635 ","End":"06:23.690","Text":"At this point, it\u0027s easy to see that this thing is"},{"Start":"06:23.690 ","End":"06:26.540","Text":"negative because look, this is positive."},{"Start":"06:26.540 ","End":"06:29.570","Text":"We already said that natural log is positive,"},{"Start":"06:29.570 ","End":"06:30.650","Text":"its square root is positive,"},{"Start":"06:30.650 ","End":"06:34.440","Text":"2 is positive, something squared is positive."},{"Start":"06:34.450 ","End":"06:37.130","Text":"We have here a negative,"},{"Start":"06:37.130 ","End":"06:39.440","Text":"but this thing is positive,"},{"Start":"06:39.440 ","End":"06:41.240","Text":"plus 1 is still positive."},{"Start":"06:41.240 ","End":"06:43.865","Text":"Everything is positive except for this minus."},{"Start":"06:43.865 ","End":"06:47.120","Text":"Altogether we end up with negative and"},{"Start":"06:47.120 ","End":"06:50.615","Text":"negative derivative means the function is decreasing."},{"Start":"06:50.615 ","End":"06:55.040","Text":"Finally, we\u0027ve met all the conditions and then we"},{"Start":"06:55.040 ","End":"06:59.240","Text":"can draw the conclusion that we can check convergence of the series"},{"Start":"06:59.240 ","End":"07:07.670","Text":"by replacing it with an improper integral that is the integral from 2 to infinity."},{"Start":"07:07.670 ","End":"07:16.900","Text":"The infinity is what makes it improper of the function 1 over x,"},{"Start":"07:16.900 ","End":"07:23.055","Text":"square root of natural log x dx."},{"Start":"07:23.055 ","End":"07:26.375","Text":"Now, what does it mean to take an integral to infinity?"},{"Start":"07:26.375 ","End":"07:28.430","Text":"How does an improper integral work?"},{"Start":"07:28.430 ","End":"07:33.080","Text":"The way it works is that instead of taking infinity,"},{"Start":"07:33.080 ","End":"07:38.195","Text":"we take the integral from 2 but up to some number,"},{"Start":"07:38.195 ","End":"07:42.230","Text":"say b and then take"},{"Start":"07:42.230 ","End":"07:47.840","Text":"the limit of this thing as b goes to infinity and here the same thing as above,"},{"Start":"07:47.840 ","End":"07:55.410","Text":"1 over x square root of natural log x dx."},{"Start":"07:56.630 ","End":"08:00.100","Text":"Now we have an integration to do,"},{"Start":"08:00.100 ","End":"08:03.320","Text":"so let\u0027s also do that as a side exercise."},{"Start":"08:03.320 ","End":"08:08.675","Text":"We need the indefinite integral first, the anti-derivative primitive."},{"Start":"08:08.675 ","End":"08:11.404","Text":"What I want is the integral."},{"Start":"08:11.404 ","End":"08:20.920","Text":"I\u0027ll just write it as dx over x square root of natural log of x."},{"Start":"08:20.920 ","End":"08:26.165","Text":"The technique I\u0027m going to use on this integral is substitution."},{"Start":"08:26.165 ","End":"08:37.460","Text":"Let\u0027s substitute t will equal natural log of x."},{"Start":"08:37.460 ","End":"08:46.640","Text":"The reason this is good is because dt is the derivative of this dx."},{"Start":"08:46.640 ","End":"08:51.245","Text":"In other words, 1 over x dx."},{"Start":"08:51.245 ","End":"08:56.270","Text":"Look, 1 over x dx is what we have here, dx over x."},{"Start":"08:56.270 ","End":"09:02.060","Text":"We end up when we convert to the language of t instead of x,"},{"Start":"09:02.060 ","End":"09:07.230","Text":"then this integral becomes dx over x is dt."},{"Start":"09:07.540 ","End":"09:15.090","Text":"On the denominator, we have the square root of t. Now,"},{"Start":"09:15.090 ","End":"09:17.950","Text":"this is a well-known integral."},{"Start":"09:17.950 ","End":"09:24.884","Text":"Allow me to write a 2 here and a 2 here."},{"Start":"09:24.884 ","End":"09:27.140","Text":"They\u0027ll cancel because this isn\u0027t a denominator,"},{"Start":"09:27.140 ","End":"09:28.505","Text":"this is a numerator."},{"Start":"09:28.505 ","End":"09:30.080","Text":"The reason I did that because"},{"Start":"09:30.080 ","End":"09:33.769","Text":"1 over twice the square root of t is one of those immediate integrals."},{"Start":"09:33.769 ","End":"09:37.685","Text":"It\u0027s just the square root of t plus a constant."},{"Start":"09:37.685 ","End":"09:43.465","Text":"But we don\u0027t need the constant because we\u0027re going to do a definite integral."},{"Start":"09:43.465 ","End":"09:48.780","Text":"However, we do need to get back from t to"},{"Start":"09:48.780 ","End":"09:56.870","Text":"x. t is natural log of x so this is the square root of natural log of x."},{"Start":"09:56.870 ","End":"10:03.905","Text":"At this point, I go back here and say that what I need now is the limit"},{"Start":"10:03.905 ","End":"10:13.205","Text":"from 2 to b of natural log of x under the square root sign."},{"Start":"10:13.205 ","End":"10:15.965","Text":"Sorry, not the integral."},{"Start":"10:15.965 ","End":"10:19.235","Text":"This is the integral, silly me."},{"Start":"10:19.235 ","End":"10:22.190","Text":"Just that it has to be evaluated"},{"Start":"10:22.190 ","End":"10:31.819","Text":"between 2 and b and so this is equal to,"},{"Start":"10:31.819 ","End":"10:34.850","Text":"am I forgetting to write the equals?"},{"Start":"10:34.850 ","End":"10:40.385","Text":"Yeah. This is equal to limit."},{"Start":"10:40.385 ","End":"10:42.620","Text":"That\u0027s still the same."},{"Start":"10:42.620 ","End":"10:46.030","Text":"Now, this evaluation means plugin b,"},{"Start":"10:46.030 ","End":"10:48.990","Text":"plugin 2, and subtract the lower from the upper."},{"Start":"10:48.990 ","End":"10:58.260","Text":"I\u0027ve got the square root of natural log of b minus the square root of natural log of 2."},{"Start":"10:58.260 ","End":"11:00.665","Text":"B goes to infinity."},{"Start":"11:00.665 ","End":"11:02.885","Text":"If b goes to infinity,"},{"Start":"11:02.885 ","End":"11:10.680","Text":"natural log of b goes to infinity and the square root of infinity is still infinity."},{"Start":"11:10.790 ","End":"11:14.270","Text":"Infinity minus the constant is infinity."},{"Start":"11:14.270 ","End":"11:16.760","Text":"In other words, this limit is infinity,"},{"Start":"11:16.760 ","End":"11:19.950","Text":"which means that it diverges."},{"Start":"11:20.660 ","End":"11:24.145","Text":"Since the integral diverges,"},{"Start":"11:24.145 ","End":"11:30.065","Text":"by the theorem, the infinite series also diverges."},{"Start":"11:30.065 ","End":"11:32.970","Text":"That\u0027s the answer. We\u0027re done."}],"ID":10528}],"Thumbnail":null,"ID":52094},{"Name":"The Limit Comparison Test","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 7 Part a","Duration":"7m 46s","ChapterTopicVideoID":10202,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10202.jpeg","UploadDate":"2017-08-22T04:37:37.0870000","DurationForVideoObject":"PT7M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.389","Text":"In this exercise, we have an infinite series."},{"Start":"00:03.389 ","End":"00:06.660","Text":"We can see because the sum goes to infinity"},{"Start":"00:06.660 ","End":"00:14.085","Text":"and we have to determine if it\u0027s convergent or divergent."},{"Start":"00:14.085 ","End":"00:16.200","Text":"There\u0027s more than 1 way to do this,"},{"Start":"00:16.200 ","End":"00:20.580","Text":"but here we\u0027ll do it using what is called the comparison test,"},{"Start":"00:20.580 ","End":"00:22.800","Text":"which is summarized here."},{"Start":"00:22.800 ","End":"00:30.840","Text":"Which basically says that when we\u0027re talking about positive or non-negative series,"},{"Start":"00:30.840 ","End":"00:36.210","Text":"that if we are less than a convergent series,"},{"Start":"00:36.210 ","End":"00:38.055","Text":"then we\u0027re also convergent."},{"Start":"00:38.055 ","End":"00:43.380","Text":"The logical consequence is that if something"},{"Start":"00:43.380 ","End":"00:49.590","Text":"is bigger than a divergent series, then it diverges."},{"Start":"00:49.990 ","End":"00:55.040","Text":"But before we start doing anything formally,"},{"Start":"00:55.040 ","End":"01:00.890","Text":"we have to decide which of the 2 cases we\u0027re aiming for and we use"},{"Start":"01:00.890 ","End":"01:04.370","Text":"some vague techniques just to at least decide whether we\u0027re going to"},{"Start":"01:04.370 ","End":"01:08.570","Text":"try for convergent or divergent and then do it more formally."},{"Start":"01:08.570 ","End":"01:10.940","Text":"Let\u0027s see. Roughly speaking,"},{"Start":"01:10.940 ","End":"01:13.715","Text":"with polynomials and rational functions,"},{"Start":"01:13.715 ","End":"01:18.030","Text":"what really counts are the leading terms."},{"Start":"01:18.030 ","End":"01:20.390","Text":"In general, this series behaves like,"},{"Start":"01:20.390 ","End":"01:22.610","Text":"I\u0027ll write that using a squiggly line,"},{"Start":"01:22.610 ","End":"01:26.600","Text":"behaves like the leading terms,"},{"Start":"01:26.600 ","End":"01:36.275","Text":"which is 5n squared over 14n to the 5th,"},{"Start":"01:36.275 ","End":"01:38.650","Text":"I forgot the sigma,"},{"Start":"01:38.650 ","End":"01:44.570","Text":"and from 1 to infinity and constants don\u0027t really matter."},{"Start":"01:44.570 ","End":"01:52.755","Text":"You can ignore the 5 and the 14 and this behaves like n squared over n to the 5th."},{"Start":"01:52.755 ","End":"01:56.745","Text":"It behaves like 1 over n cubed."},{"Start":"01:56.745 ","End":"01:59.120","Text":"When I say behaves like,"},{"Start":"01:59.120 ","End":"02:02.300","Text":"I mean, if this converges,"},{"Start":"02:02.300 ","End":"02:10.435","Text":"this converges and vice versa and we throw constants and other stuff out."},{"Start":"02:10.435 ","End":"02:12.650","Text":"We\u0027re just trying to, for ourselves,"},{"Start":"02:12.650 ","End":"02:18.125","Text":"decide which are the 2 answers to expect and then we go and do something more formal."},{"Start":"02:18.125 ","End":"02:25.820","Text":"Now, n squared over n to the 5th is n cubed and this is actually convergent."},{"Start":"02:25.820 ","End":"02:29.975","Text":"The reason I know it\u0027s convergent because it\u0027s a p series."},{"Start":"02:29.975 ","End":"02:37.495","Text":"In general, if I have the sum to infinity of 1 over n^p,"},{"Start":"02:37.495 ","End":"02:42.800","Text":"it converges if and only if p is bigger than 1 and in our case,"},{"Start":"02:42.800 ","End":"02:47.785","Text":"p is 3, which is definitely bigger than 1."},{"Start":"02:47.785 ","End":"02:52.625","Text":"We know we\u0027re expecting the answer to be convergent."},{"Start":"02:52.625 ","End":"02:56.765","Text":"What we\u0027re going to do is go for case 1 and we\u0027ll find"},{"Start":"02:56.765 ","End":"03:03.335","Text":"a b_n series which is simpler and more intuitive than the original 1,"},{"Start":"03:03.335 ","End":"03:07.535","Text":"but which is bigger or equal to these terms."},{"Start":"03:07.535 ","End":"03:12.205","Text":"Now, what we do to find such a b_n,"},{"Start":"03:12.205 ","End":"03:16.330","Text":"let me just label this as a_n first of all,"},{"Start":"03:18.320 ","End":"03:22.320","Text":"this is a_n and now I have to produce b_n."},{"Start":"03:22.320 ","End":"03:30.560","Text":"My b_n will be, let\u0027s see."},{"Start":"03:30.560 ","End":"03:34.125","Text":"Let me just, first of all, write the dividing line."},{"Start":"03:34.125 ","End":"03:41.255","Text":"What I\u0027m going to do is increase the numerator and decrease the denominator."},{"Start":"03:41.255 ","End":"03:42.785","Text":"If I do that,"},{"Start":"03:42.785 ","End":"03:45.690","Text":"then it will only get bigger."},{"Start":"03:46.790 ","End":"03:50.410","Text":"I\u0027m going to decrease the denominator by"},{"Start":"03:50.410 ","End":"03:55.570","Text":"just writing 14n to the 5th and dropping this."},{"Start":"03:55.570 ","End":"03:58.270","Text":"Like I said, decreasing the denominator can"},{"Start":"03:58.270 ","End":"04:01.000","Text":"only increase the value and I\u0027m going to increase"},{"Start":"04:01.000 ","End":"04:07.395","Text":"the numerator and I don\u0027t like to have all this mixed stuff,"},{"Start":"04:07.395 ","End":"04:12.680","Text":"so let\u0027s just make them all n squared."},{"Start":"04:12.680 ","End":"04:17.860","Text":"This thing is certainly less than or equal to 5n"},{"Start":"04:17.860 ","End":"04:24.100","Text":"squared plus 4n squared plus 8n squared."},{"Start":"04:24.100 ","End":"04:29.550","Text":"If I call this b_n,"},{"Start":"04:29.550 ","End":"04:38.895","Text":"then certainly a_n is less than or equal to b_n,"},{"Start":"04:38.895 ","End":"04:41.820","Text":"or b_n bigger or equal to a_n."},{"Start":"04:41.820 ","End":"04:43.685","Text":"I also should have mentioned,"},{"Start":"04:43.685 ","End":"04:48.130","Text":"of course, a_n is bigger or equal to 0."},{"Start":"04:48.130 ","End":"04:50.090","Text":"That\u0027s another part of the condition because"},{"Start":"04:50.090 ","End":"04:52.670","Text":"everything here is positive, notice it\u0027s all pluses."},{"Start":"04:52.670 ","End":"04:56.150","Text":"That\u0027s something we had to mentally note."},{"Start":"04:56.150 ","End":"04:58.590","Text":"Noted, verified."},{"Start":"04:58.590 ","End":"05:02.720","Text":"Now we\u0027re ready for the comparison test."},{"Start":"05:02.720 ","End":"05:05.990","Text":"All I have to show now is this part here"},{"Start":"05:05.990 ","End":"05:10.170","Text":"that a_n is less than or equal to b_n."},{"Start":"05:11.090 ","End":"05:19.600","Text":"But, well, we don\u0027t really have to show it because certainly,"},{"Start":"05:19.600 ","End":"05:21.895","Text":"we could just write it in words."},{"Start":"05:21.895 ","End":"05:30.070","Text":"This is because the numerator"},{"Start":"05:32.120 ","End":"05:37.915","Text":"in b_n is bigger than the numerator,"},{"Start":"05:37.915 ","End":"05:42.475","Text":"bigger or equal to numerator in a_n,"},{"Start":"05:42.475 ","End":"05:46.905","Text":"and the denominator in"},{"Start":"05:46.905 ","End":"05:54.560","Text":"b_n is less than or equal to the denominator."},{"Start":"05:57.750 ","End":"06:01.120","Text":"I don\u0027t think you need to go into this any deeper."},{"Start":"06:01.120 ","End":"06:05.620","Text":"If you have positive stuff and you just decrease here and increase here,"},{"Start":"06:05.620 ","End":"06:07.330","Text":"you can only get bigger."},{"Start":"06:07.330 ","End":"06:14.460","Text":"The only part that\u0027s missing is to show that b_n is convergent."},{"Start":"06:14.460 ","End":"06:25.060","Text":"This series, the sum of b_n is equal to the sum of 5 plus 4 plus 8,"},{"Start":"06:25.060 ","End":"06:32.620","Text":"9 plus 8, 17n squared over"},{"Start":"06:32.620 ","End":"06:41.590","Text":"14n to the 5th and this definitely behaves like it converges or diverges."},{"Start":"06:41.590 ","End":"06:43.750","Text":"A constant won\u0027t make any difference."},{"Start":"06:43.750 ","End":"06:47.350","Text":"It\u0027s exactly like 1 over n cubed,"},{"Start":"06:47.350 ","End":"06:50.350","Text":"which is what we originally compared to,"},{"Start":"06:50.350 ","End":"06:52.520","Text":"but this was more informal."},{"Start":"06:52.520 ","End":"06:54.915","Text":"I could write actually,"},{"Start":"06:54.915 ","End":"07:00.660","Text":"if you don\u0027t like this squiggly line, we could say,"},{"Start":"07:00.660 ","End":"07:02.890","Text":"it\u0027s equal to this,"},{"Start":"07:02.890 ","End":"07:06.685","Text":"but then we need the 17 over 14 and then we can say,"},{"Start":"07:06.685 ","End":"07:09.087","Text":"a constant makes no difference and"},{"Start":"07:09.087 ","End":"07:12.380","Text":"this is a convergent series, so it\u0027s a p-series."},{"Start":"07:12.380 ","End":"07:20.570","Text":"Let me just write everywhere, 1 to infinity, 1 to infinity."},{"Start":"07:20.570 ","End":"07:25.190","Text":"It gets a bit tedious, so sometimes, leave these things out."},{"Start":"07:25.190 ","End":"07:30.893","Text":"This converges and therefore because"},{"Start":"07:30.893 ","End":"07:33.710","Text":"of the comparison test, b_n converges,"},{"Start":"07:33.710 ","End":"07:37.565","Text":"a_n is less than or equal to b_n, so a_n converges,"},{"Start":"07:37.565 ","End":"07:45.430","Text":"which implies that a_n also converges and we are done."}],"ID":10529},{"Watched":false,"Name":"Exercise 7 Part b","Duration":"4m 43s","ChapterTopicVideoID":10203,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10203.jpeg","UploadDate":"2017-08-22T04:37:47.7770000","DurationForVideoObject":"PT4M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.120","Text":"In this exercise, we\u0027re given an infinite series it\u0027s from 1 to infinity,"},{"Start":"00:06.120 ","End":"00:09.960","Text":"and we have to decide if it converges or diverges."},{"Start":"00:09.960 ","End":"00:14.580","Text":"There are several tests we could use."},{"Start":"00:14.580 ","End":"00:15.960","Text":"Techniques."},{"Start":"00:15.960 ","End":"00:20.475","Text":"What I\u0027m going to use here is something called the comparison test,"},{"Start":"00:20.475 ","End":"00:22.695","Text":"which is summarized here."},{"Start":"00:22.695 ","End":"00:30.930","Text":"Which basically says that if I\u0027m less than or equal to a convergent series,"},{"Start":"00:30.930 ","End":"00:32.130","Text":"then I converge to."},{"Start":"00:32.130 ","End":"00:36.505","Text":"Or if I\u0027m bigger or equal to a divergent, then I diverge."},{"Start":"00:36.505 ","End":"00:42.820","Text":"I being the series here and it relates to non-negative series."},{"Start":"00:43.020 ","End":"00:47.640","Text":"The idea is initially to guess."},{"Start":"00:47.640 ","End":"00:52.225","Text":"We\u0027ll not guess but make an intelligent estimate of what we expect,"},{"Start":"00:52.225 ","End":"00:55.510","Text":"whether we\u0027re expecting convergent or divergent"},{"Start":"00:55.510 ","End":"00:55.511","Text":"and then we choose the other series accordingly."},{"Start":"00:55.511 ","End":"01:03.760","Text":"Let\u0027s first of all informally see if we can figure"},{"Start":"01:03.760 ","End":"01:08.445","Text":"out if this series is going to converge or diverge."},{"Start":"01:08.445 ","End":"01:12.880","Text":"By the way, note that the word almost here means"},{"Start":"01:12.880 ","End":"01:16.495","Text":"that it could be from a certain and onwards."},{"Start":"01:16.495 ","End":"01:21.400","Text":"First, a finite number of n, is not going to make any difference."},{"Start":"01:21.400 ","End":"01:27.500","Text":"I\u0027m going to assume that n is bigger than 3 and if I look at this,"},{"Start":"01:27.500 ","End":"01:34.590","Text":"I could separate it like putting the line here and saying,"},{"Start":"01:34.590 ","End":"01:43.795","Text":"it\u0027s 2 over n squared from here and the rest of it is certainly less than 1."},{"Start":"01:43.795 ","End":"01:45.555","Text":"Less than or equal to 1,"},{"Start":"01:45.555 ","End":"01:50.700","Text":"because here have all n\u0027s and here I have 3, or 4, or 5 up to n."},{"Start":"01:50.700 ","End":"01:57.365","Text":"Each number here is less than or equal to the corresponding 1 below it."},{"Start":"01:57.365 ","End":"02:06.320","Text":"Actually, before that I wanted to say that"},{"Start":"02:06.320 ","End":"02:09.080","Text":"the series 2 over n squared we know about"},{"Start":"02:09.080 ","End":"02:16.055","Text":"because we know that there\u0027s a p series converges or before that,"},{"Start":"02:16.055 ","End":"02:19.115","Text":"the sum of 1 over n squared."},{"Start":"02:19.115 ","End":"02:21.335","Text":"It could be 2 over n squared,"},{"Start":"02:21.335 ","End":"02:30.135","Text":"but this series is a p series and this 1 converges."},{"Start":"02:30.135 ","End":"02:32.190","Text":"What do I mean by p series?"},{"Start":"02:32.190 ","End":"02:33.485","Text":"In case you forgotten,"},{"Start":"02:33.485 ","End":"02:37.835","Text":"if I have 1 over n to the power of p,"},{"Start":"02:37.835 ","End":"02:43.715","Text":"then this converges if and only if p is bigger than 1,"},{"Start":"02:43.715 ","End":"02:49.505","Text":"and certainly 2 is bigger than 1 because p is 2 in our case."},{"Start":"02:49.505 ","End":"02:53.270","Text":"This converges and if I put a 2 on the top,"},{"Start":"02:53.270 ","End":"02:55.040","Text":"it\u0027s also going to converge."},{"Start":"02:55.040 ","End":"02:56.290","Text":"I can put a 2 here,"},{"Start":"02:56.290 ","End":"03:01.000","Text":"it makes no difference with the constant and this is the first bit."},{"Start":"03:01.070 ","End":"03:06.260","Text":"What I\u0027m saying is that this part is 2 over n squared"},{"Start":"03:06.260 ","End":"03:12.755","Text":"and this part is less than 1 and you can be safe less than or equal to 1,"},{"Start":"03:12.755 ","End":"03:16.435","Text":"at least from n equals 3 onwards."},{"Start":"03:16.435 ","End":"03:20.480","Text":"It doesn\u0027t matter if n equals 1 or 2, I don\u0027t know,"},{"Start":"03:20.480 ","End":"03:24.890","Text":"but it doesn\u0027t matter and so what we can say is that"},{"Start":"03:24.890 ","End":"03:32.265","Text":"if I let this original series be a_n,"},{"Start":"03:32.265 ","End":"03:35.690","Text":"and I want it to be a_n because I want the convergent case"},{"Start":"03:35.690 ","End":"03:43.890","Text":"and if I let b_n be 2 over n squared,"},{"Start":"03:45.770 ","End":"03:51.880","Text":"n goes from 1 to infinity and call this thing, b_n."},{"Start":"03:51.880 ","End":"03:59.945","Text":"Then what we\u0027ve shown is that indeed a_n is less than or equal to b_n."},{"Start":"03:59.945 ","End":"04:07.720","Text":"Because I\u0027ve just thrown out something that\u0027s less than or equal to 1."},{"Start":"04:07.720 ","End":"04:11.510","Text":"This has to be less than or equal to this and b_n,"},{"Start":"04:11.510 ","End":"04:15.605","Text":"which is this I\u0027ve already shown, which is divergent."},{"Start":"04:15.605 ","End":"04:17.510","Text":"Sorry. Why did I say that?"},{"Start":"04:17.510 ","End":"04:18.530","Text":"Convergent."},{"Start":"04:18.530 ","End":"04:23.780","Text":"Convergent, sure, p series with p bigger than 1."},{"Start":"04:23.780 ","End":"04:25.085","Text":"We just said that."},{"Start":"04:25.085 ","End":"04:29.075","Text":"Because a_n is less than the b_n, which is convergent,"},{"Start":"04:29.075 ","End":"04:35.395","Text":"then it implies that a_n is also Sigma."},{"Start":"04:35.395 ","End":"04:40.760","Text":"This thing is also convergent to the series and that\u0027s all there is to it."},{"Start":"04:40.760 ","End":"04:42.450","Text":"We\u0027re done."}],"ID":10530},{"Watched":false,"Name":"Exercise 7 Part c","Duration":"5m 51s","ChapterTopicVideoID":10204,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10204.jpeg","UploadDate":"2017-08-22T04:38:01.3230000","DurationForVideoObject":"PT5M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.980","Text":"We have here an infinite series goes to infinity here."},{"Start":"00:05.980 ","End":"00:09.545","Text":"If you decide if it converges or diverges,"},{"Start":"00:09.545 ","End":"00:11.995","Text":"there could be several ways of doing it."},{"Start":"00:11.995 ","End":"00:16.285","Text":"I\u0027m going to use what is called the comparison test,"},{"Start":"00:16.285 ","End":"00:19.045","Text":"which is what stated in the box here,"},{"Start":"00:19.045 ","End":"00:24.865","Text":"which basically says that if I want to prove that a_n converges,"},{"Start":"00:24.865 ","End":"00:28.090","Text":"I find another simpler series,"},{"Start":"00:28.090 ","End":"00:31.530","Text":"b_n, which is greater or equal to a_n that I know"},{"Start":"00:31.530 ","End":"00:35.400","Text":"converges and then I\u0027ll deduce that a_n converges also,"},{"Start":"00:35.400 ","End":"00:38.255","Text":"and then there\u0027s a variant of that for the divergent case."},{"Start":"00:38.255 ","End":"00:42.915","Text":"But the first thing we have to do is at least in our heads,"},{"Start":"00:42.915 ","End":"00:48.530","Text":"decide whether we\u0027re going to go to try and prove it\u0027s convergent or divergent."},{"Start":"00:48.530 ","End":"00:56.290","Text":"To do this, the general rule of thumb is just only consider"},{"Start":"00:56.290 ","End":"01:00.260","Text":"the greatest exponent in each of the polynomials and to"},{"Start":"01:00.260 ","End":"01:04.715","Text":"say that this behaves roughly like that with a squiggly line,"},{"Start":"01:04.715 ","End":"01:14.120","Text":"not that it equals but it behaves like the series where I take here just the 2n cubed and"},{"Start":"01:14.120 ","End":"01:23.810","Text":"here just the square root of n^10 and here from 1 to infinity,"},{"Start":"01:23.810 ","End":"01:28.755","Text":"and that this behaves like,"},{"Start":"01:28.755 ","End":"01:33.575","Text":"I can throw a constant out and it still doesn\u0027t affect convergence,"},{"Start":"01:33.575 ","End":"01:40.400","Text":"n cubed over root"},{"Start":"01:40.400 ","End":"01:45.725","Text":"of n^10 is n cubed over n^5,"},{"Start":"01:45.725 ","End":"01:47.990","Text":"which is 1 over n squared."},{"Start":"01:47.990 ","End":"01:54.270","Text":"This behaves like, it\u0027s practically equal to except for the constant,"},{"Start":"01:54.730 ","End":"01:59.180","Text":"1 to infinity of 1 over n squared."},{"Start":"01:59.180 ","End":"02:01.460","Text":"Now we know all about this series,"},{"Start":"02:01.460 ","End":"02:03.475","Text":"we\u0027ve seen it before,"},{"Start":"02:03.475 ","End":"02:06.120","Text":"but just to remind you it\u0027s a p series."},{"Start":"02:06.120 ","End":"02:12.405","Text":"A p series is something of the form the infinite sum of 1 over"},{"Start":"02:12.405 ","End":"02:21.260","Text":"n^p and the condition for convergence on this is that p should be bigger than 1."},{"Start":"02:21.260 ","End":"02:26.750","Text":"In our case, p is 2 and 2 is definitely bigger than 1,"},{"Start":"02:26.750 ","End":"02:28.040","Text":"and so in this case,"},{"Start":"02:28.040 ","End":"02:34.060","Text":"this series is convergent."},{"Start":"02:35.570 ","End":"02:44.475","Text":"All I have to do is to call this b_n, no."},{"Start":"02:44.475 ","End":"02:47.320","Text":"This will be a_n"},{"Start":"02:50.210 ","End":"02:55.370","Text":"and I\u0027ll find b_n that somehow similar to the 1 over n squared,"},{"Start":"02:55.370 ","End":"03:00.270","Text":"but which is greater than or equal to a_n."},{"Start":"03:00.470 ","End":"03:04.955","Text":"I use the trick of when I have a fraction"},{"Start":"03:04.955 ","End":"03:11.825","Text":"of increasing the numerator and shrinking the denominator,"},{"Start":"03:11.825 ","End":"03:14.360","Text":"and then I get something bigger."},{"Start":"03:14.360 ","End":"03:20.720","Text":"This a_n is less than or equal to."},{"Start":"03:20.720 ","End":"03:30.195","Text":"Now if I increase the numerator and I write it as 2n cubed plus n cubed,"},{"Start":"03:30.195 ","End":"03:37.260","Text":"n cubed will always be bigger than n squared from 1 onwards in fact,"},{"Start":"03:37.260 ","End":"03:42.019","Text":"plus 4n cubed plus n cubed."},{"Start":"03:42.019 ","End":"03:46.100","Text":"I\u0027m making all these powers 3; I\u0027m only increasing."},{"Start":"03:46.100 ","End":"03:50.670","Text":"On the denominator, I\u0027m going to shrink,"},{"Start":"03:50.670 ","End":"03:54.830","Text":"which means that I\u0027m going to take the square root of something smaller,"},{"Start":"03:54.830 ","End":"03:59.040","Text":"just the n^10 and drop the rest."},{"Start":"03:59.360 ","End":"04:03.500","Text":"I take this, increase the numerator,"},{"Start":"04:03.500 ","End":"04:10.785","Text":"decrease the denominator so certainly I get something bigger or equal to this."},{"Start":"04:10.785 ","End":"04:12.785","Text":"This happens to equal,"},{"Start":"04:12.785 ","End":"04:17.749","Text":"let\u0027s say 2 plus 1 plus 4 plus 1 is 8,"},{"Start":"04:17.749 ","End":"04:26.420","Text":"so this is 8n cubed over n^5,"},{"Start":"04:26.420 ","End":"04:31.535","Text":"which is 8 over n squared."},{"Start":"04:31.535 ","End":"04:35.990","Text":"This, I\u0027m going to let now equal my b_n so"},{"Start":"04:35.990 ","End":"04:41.150","Text":"I really do have that a_n is less than or equal to b_n."},{"Start":"04:41.150 ","End":"04:46.130","Text":"I\u0027ve shown this that a_n is bigger or equal to 0."},{"Start":"04:46.130 ","End":"04:47.550","Text":"I should have mentioned also."},{"Start":"04:47.550 ","End":"04:55.500","Text":"If you look at this, everything is positive here and certainly bigger than 0."},{"Start":"04:56.180 ","End":"04:59.115","Text":"We\u0027ve met all the conditions."},{"Start":"04:59.115 ","End":"05:01.965","Text":"Now b_n converges."},{"Start":"05:01.965 ","End":"05:10.515","Text":"The sum of b_n is just 8 times the sum of 1 over n squared."},{"Start":"05:10.515 ","End":"05:18.905","Text":"This, we already know converges because we just said that there."},{"Start":"05:18.905 ","End":"05:24.195","Text":"Let me just write it properly, 1 to infinity."},{"Start":"05:24.195 ","End":"05:31.715","Text":"The 8 constant doesn\u0027t change convergence and so b_n converges and from the theorem,"},{"Start":"05:31.715 ","End":"05:34.630","Text":"therefore we get that,"},{"Start":"05:34.630 ","End":"05:36.160","Text":"this implies from the theorem,"},{"Start":"05:36.160 ","End":"05:38.330","Text":"that the sum of our original series,"},{"Start":"05:38.330 ","End":"05:45.305","Text":"which I called a_n is also convergent, it converges."},{"Start":"05:45.305 ","End":"05:48.060","Text":"n equals 1 to infinity,"},{"Start":"05:48.060 ","End":"05:51.520","Text":"and we are done."}],"ID":10531},{"Watched":false,"Name":"Exercise 7 Part d","Duration":"7m 22s","ChapterTopicVideoID":10205,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10205.jpeg","UploadDate":"2017-08-22T04:38:18.5670000","DurationForVideoObject":"PT7M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we have an infinite series."},{"Start":"00:03.720 ","End":"00:06.285","Text":"It\u0027s infinite because of the infinity here,"},{"Start":"00:06.285 ","End":"00:10.035","Text":"and as such, it will either converge or diverge."},{"Start":"00:10.035 ","End":"00:18.390","Text":"Our job is to decide which and we\u0027re going to use the comparison test,"},{"Start":"00:18.390 ","End":"00:21.585","Text":"which is written in the box here."},{"Start":"00:21.585 ","End":"00:27.830","Text":"That basically says that when we\u0027re talking about positive series,"},{"Start":"00:27.830 ","End":"00:31.940","Text":"if we\u0027re smaller than something convergent,"},{"Start":"00:31.940 ","End":"00:37.280","Text":"we converge, and if we\u0027re bigger than something divergent, then we diverge."},{"Start":"00:37.280 ","End":"00:41.210","Text":"We have to decide which case we are going for, 1 or 2."},{"Start":"00:41.210 ","End":"00:46.730","Text":"We do some approximation and remember that"},{"Start":"00:46.730 ","End":"00:49.625","Text":"roughly, polynomials behave pretty much"},{"Start":"00:49.625 ","End":"00:53.240","Text":"like the leading term when you\u0027re going to infinity."},{"Start":"00:53.240 ","End":"00:55.910","Text":"I\u0027m going to say that this behaves like,"},{"Start":"00:55.910 ","End":"01:03.840","Text":"I\u0027ll write a squiggly line to mean not equals but behaves like the sum of"},{"Start":"01:03.840 ","End":"01:12.790","Text":"just the leading term, 4n over the square root of n^4."},{"Start":"01:12.790 ","End":"01:18.205","Text":"Now, square root of n^4 is n squared."},{"Start":"01:18.205 ","End":"01:21.430","Text":"This is equal to"},{"Start":"01:21.430 ","End":"01:29.275","Text":"4 times the sum of 1 because it\u0027s n squared,"},{"Start":"01:29.275 ","End":"01:32.895","Text":"and the 4 makes no difference"},{"Start":"01:32.895 ","End":"01:37.755","Text":"so it behaves like the sum of 1,"},{"Start":"01:37.755 ","End":"01:41.325","Text":"which is the classic harmonic series."},{"Start":"01:41.325 ","End":"01:44.250","Text":"It\u0027s known to be divergent."},{"Start":"01:44.250 ","End":"01:47.925","Text":"I\u0027ll just write that this diverges."},{"Start":"01:47.925 ","End":"01:50.480","Text":"One way of seeing this,"},{"Start":"01:50.480 ","End":"01:51.950","Text":"in case you forgotten,"},{"Start":"01:51.950 ","End":"01:55.400","Text":"is just to apply the p-test,"},{"Start":"01:55.400 ","End":"02:05.570","Text":"which says that when we have the sum of 1^p from something to infinity,"},{"Start":"02:05.570 ","End":"02:11.705","Text":"then it converges if and only if p is bigger than 1,"},{"Start":"02:11.705 ","End":"02:14.345","Text":"and that is strictly bigger than 1."},{"Start":"02:14.345 ","End":"02:19.125","Text":"In our case, p equals 1, so that\u0027s not good."},{"Start":"02:19.125 ","End":"02:24.900","Text":"It doesn\u0027t fall into the convergent condition, so it\u0027s divergent."},{"Start":"02:24.900 ","End":"02:28.665","Text":"We\u0027re going to go for Case 2."},{"Start":"02:28.665 ","End":"02:33.430","Text":"What we\u0027re going to do is going to let this be our b_n."},{"Start":"02:33.800 ","End":"02:40.050","Text":"This here will be b_n,"},{"Start":"02:40.050 ","End":"02:45.260","Text":"and we\u0027re going to find some a_n which is less than or equal to it,"},{"Start":"02:45.260 ","End":"02:48.274","Text":"which diverges, but which is simpler."},{"Start":"02:48.274 ","End":"02:54.250","Text":"I want to somehow get it to be roughly like 1."},{"Start":"02:54.250 ","End":"03:02.720","Text":"The trick I\u0027m going to use in order to get this to be bigger or equal to something is to"},{"Start":"03:02.720 ","End":"03:12.940","Text":"decrease the numerator"},{"Start":"03:12.940 ","End":"03:15.205","Text":"and increase the denominator."},{"Start":"03:15.205 ","End":"03:21.855","Text":"Then I\u0027ll get something smaller possibly. Let me just write it."},{"Start":"03:21.855 ","End":"03:24.555","Text":"B_n is this thing here,"},{"Start":"03:24.555 ","End":"03:31.575","Text":"is 4n plus 5 over square root of all this stuff:"},{"Start":"03:31.575 ","End":"03:36.975","Text":"n^4 plus 2n cubed plus n squared plus 4n plus 1."},{"Start":"03:36.975 ","End":"03:39.310","Text":"Now this is going to be bigger or equal"},{"Start":"03:39.310 ","End":"03:44.350","Text":"to, making the numerator smaller can only decrease it,"},{"Start":"03:44.350 ","End":"03:46.975","Text":"so I\u0027ll write 4n."},{"Start":"03:46.975 ","End":"03:56.985","Text":"I\u0027m going to increase the denominator by writing, instead of this,"},{"Start":"03:56.985 ","End":"04:03.195","Text":"I\u0027ll increase because when n is bigger than 1,"},{"Start":"04:03.195 ","End":"04:05.680","Text":"or even when n equals 1,"},{"Start":"04:05.680 ","End":"04:08.770","Text":"if I put a higher power of n,"},{"Start":"04:08.770 ","End":"04:10.390","Text":"It can only increase the value."},{"Start":"04:10.390 ","End":"04:19.625","Text":"I\u0027m going to put 2n^4 plus n^4 plus 4n^4 plus n^4."},{"Start":"04:19.625 ","End":"04:21.760","Text":"Instead of all these powers 4,"},{"Start":"04:21.760 ","End":"04:23.110","Text":"3, 2, 1, 0,"},{"Start":"04:23.110 ","End":"04:28.590","Text":"I\u0027m going to write them all as powers of 4."},{"Start":"04:28.590 ","End":"04:32.040","Text":"I\u0027m only increasing the denominator."},{"Start":"04:32.040 ","End":"04:35.980","Text":"Again, that will make this possibly get smaller."},{"Start":"04:35.980 ","End":"04:41.450","Text":"Now, this thing is equal to"},{"Start":"04:41.450 ","End":"04:49.175","Text":"4n over the square root of,"},{"Start":"04:49.175 ","End":"04:52.655","Text":"let\u0027s see, 1 plus 2 plus 1 is 4,"},{"Start":"04:52.655 ","End":"04:56.060","Text":"plus 4 is 8,"},{"Start":"04:56.060 ","End":"05:04.480","Text":"plus 1 is 9, 9n^4."},{"Start":"05:04.480 ","End":"05:12.835","Text":"This is now equal to 4n/3n squared,"},{"Start":"05:12.835 ","End":"05:19.250","Text":"which is 4/3 times 1."},{"Start":"05:19.250 ","End":"05:27.080","Text":"The Sigma, the sum of the series b_ n, is just"},{"Start":"05:27.080 ","End":"05:36.050","Text":"equal to 4/3 of the sum of 1,"},{"Start":"05:36.050 ","End":"05:42.540","Text":"and this we know is divergent."},{"Start":"05:48.700 ","End":"05:52.350","Text":"Take 2 on the last bit."},{"Start":"05:53.480 ","End":"05:58.065","Text":"I will call this my a_n,"},{"Start":"05:58.065 ","End":"06:01.680","Text":"so we certainly have that b_n is bigger or equal"},{"Start":"06:01.680 ","End":"06:05.495","Text":"to a_ n because everywhere it was equal and here is a bigger or equal to,"},{"Start":"06:05.495 ","End":"06:08.900","Text":"so we have this condition."},{"Start":"06:08.900 ","End":"06:14.834","Text":"Now, we can say that the series a_n,"},{"Start":"06:14.834 ","End":"06:22.380","Text":"which is just 4/3 of 1,"},{"Start":"06:22.380 ","End":"06:31.310","Text":"I know that this diverges because we just discussed it here"},{"Start":"06:31.310 ","End":"06:33.770","Text":"that the harmonic series diverges."},{"Start":"06:33.770 ","End":"06:37.830","Text":"Let\u0027s just be precise, 1 to infinity."},{"Start":"06:38.650 ","End":"06:42.550","Text":"Now we have Condition 2."},{"Start":"06:42.550 ","End":"06:47.970","Text":"We have that b_n is bigger or equal to a_n, which we showed."},{"Start":"06:47.970 ","End":"06:52.950","Text":"Also, I should mention that a_n certainly bigger or equal to 0,"},{"Start":"06:52.950 ","End":"06:57.255","Text":"and just note that here certainly 1."},{"Start":"06:57.255 ","End":"07:01.815","Text":"Because this diverges using Case 2,"},{"Start":"07:01.815 ","End":"07:08.640","Text":"this implies that the sum of b_n also diverges,"},{"Start":"07:08.640 ","End":"07:13.200","Text":"but b_n is just our original series."},{"Start":"07:13.200 ","End":"07:21.510","Text":"We\u0027ve shown that it diverges and we are done."}],"ID":10532},{"Watched":false,"Name":"Exercise 7 Part e","Duration":"5m 20s","ChapterTopicVideoID":10206,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10206.jpeg","UploadDate":"2017-08-22T04:38:31.3970000","DurationForVideoObject":"PT5M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:04.380","Text":"In this exercise, we have an infinite series."},{"Start":"00:04.380 ","End":"00:06.780","Text":"You can see the infinity here."},{"Start":"00:06.780 ","End":"00:11.370","Text":"We have to decide if it converges or diverges."},{"Start":"00:11.370 ","End":"00:14.130","Text":"There are several ways 1 could do this,"},{"Start":"00:14.130 ","End":"00:17.610","Text":"but we are going to use the comparison test,"},{"Start":"00:17.610 ","End":"00:20.220","Text":"which is outlined in the box here,"},{"Start":"00:20.220 ","End":"00:24.389","Text":"which deals with the positive series,"},{"Start":"00:24.389 ","End":"00:30.430","Text":"and basically says that if a series is"},{"Start":"00:38.260 ","End":"00:42.860","Text":"less than or equal to a convergence series it converges,"},{"Start":"00:42.860 ","End":"00:48.380","Text":"and if it\u0027s bigger or equal to a divergent series, then it diverges."},{"Start":"00:48.380 ","End":"00:50.495","Text":"We have to decide for ourselves,"},{"Start":"00:50.495 ","End":"00:52.385","Text":"which we\u0027re going to aim for,"},{"Start":"00:52.385 ","End":"00:56.080","Text":"Case 1 or Case 2 and then go about proving it."},{"Start":"00:56.080 ","End":"01:01.470","Text":"Initially we do some informal mathematics."},{"Start":"01:01.470 ","End":"01:05.610","Text":"We just approximate what we expect."},{"Start":"01:05.610 ","End":"01:09.230","Text":"We look at the expression here"},{"Start":"01:09.230 ","End":"01:11.715","Text":"and see how does this roughly behave?"},{"Start":"01:11.715 ","End":"01:16.040","Text":"I say, well, what matters just like with polynomials,"},{"Start":"01:16.040 ","End":"01:17.450","Text":"we took the leading term,"},{"Start":"01:17.450 ","End":"01:20.510","Text":"in general, we take what is called the dominant term,"},{"Start":"01:20.510 ","End":"01:22.975","Text":"the 1 that goes to infinity the fastest."},{"Start":"01:22.975 ","End":"01:31.625","Text":"This behaves approximately at least as far as convergence or divergence like the series."},{"Start":"01:31.625 ","End":"01:34.580","Text":"2^n is the dominant thing here."},{"Start":"01:34.580 ","End":"01:39.035","Text":"I mean, this is a constant, this goes to infinity, its a stronger."},{"Start":"01:39.035 ","End":"01:47.240","Text":"On the denominator you have to remember that an exponent beats a polynomial."},{"Start":"01:47.240 ","End":"01:50.810","Text":"When you have your variable in the exponent,"},{"Start":"01:50.810 ","End":"01:53.390","Text":"the base being bigger than 1,"},{"Start":"01:53.390 ","End":"01:56.300","Text":"then it goes to infinity faster than any polynomial,"},{"Start":"01:56.300 ","End":"02:01.675","Text":"and in particular this, so this roughly behaves like 2^n over 3^n."},{"Start":"02:01.675 ","End":"02:04.305","Text":"Let me write 1 to infinity."},{"Start":"02:04.305 ","End":"02:15.805","Text":"This actually is a geometric series because it\u0027s the sum of 2/3^n."},{"Start":"02:15.805 ","End":"02:23.840","Text":"In general, when you have the sum from 1 to infinity of q^n,"},{"Start":"02:23.840 ","End":"02:28.670","Text":"then it\u0027s a geometric series and it converges provided that"},{"Start":"02:28.670 ","End":"02:34.100","Text":"or if and only if q is between minus 1 and 1 strictly."},{"Start":"02:34.100 ","End":"02:37.010","Text":"Now in our case, q is 2/3,"},{"Start":"02:37.010 ","End":"02:40.085","Text":"which is only between minus 1 and 1."},{"Start":"02:40.085 ","End":"02:46.290","Text":"We know that this series converges."},{"Start":"02:46.290 ","End":"02:51.045","Text":"We know that we\u0027re aiming for Case 1."},{"Start":"02:51.045 ","End":"02:58.030","Text":"In other words, this is going to be our a_n."},{"Start":"02:58.030 ","End":"03:06.520","Text":"We have to find a b_n which is bigger or equal to a_n and which is simpler."},{"Start":"03:06.950 ","End":"03:11.755","Text":"That we can more easily see that this converges."},{"Start":"03:11.755 ","End":"03:17.715","Text":"We\u0027re going to do a usual trick of let\u0027s say as follows."},{"Start":"03:17.715 ","End":"03:25.920","Text":"A_n is just the general term 2n minus 2 over 3^n plus 2n."},{"Start":"03:25.920 ","End":"03:31.280","Text":"A common trick that is used is to"},{"Start":"03:31.280 ","End":"03:35.790","Text":"increase the numerator and decrease the denominator"},{"Start":"03:35.790 ","End":"03:38.565","Text":"and then we can only get something bigger."},{"Start":"03:38.565 ","End":"03:41.030","Text":"This is less than or just to be safe,"},{"Start":"03:41.030 ","End":"03:44.255","Text":"I\u0027ll say less than or equal to, that\u0027s less than."},{"Start":"03:44.255 ","End":"03:46.850","Text":"Increase the numerator."},{"Start":"03:46.850 ","End":"03:49.015","Text":"I get 2^n."},{"Start":"03:49.015 ","End":"03:51.405","Text":"I can decrease the denominator."},{"Start":"03:51.405 ","End":"03:53.780","Text":"Again, I\u0027ll only get something possibly bigger"},{"Start":"03:53.780 ","End":"04:00.260","Text":"and just throw out the 2n and say 3^n."},{"Start":"04:00.260 ","End":"04:04.660","Text":"I will let this equal b_n."},{"Start":"04:04.660 ","End":"04:09.370","Text":"Now I have that a_n is less than or equal to b_n."},{"Start":"04:09.830 ","End":"04:16.825","Text":"Of course, I also should have shown that a_n is bigger or equal to 0."},{"Start":"04:16.825 ","End":"04:25.010","Text":"When n is 1, I\u0027ve got 2 minus 2 is 0, which is okay."},{"Start":"04:25.010 ","End":"04:26.760","Text":"Then n is bigger than 1."},{"Start":"04:26.760 ","End":"04:28.480","Text":"Certainly this is bigger than this,"},{"Start":"04:28.480 ","End":"04:32.295","Text":"and this is positive so we\u0027re definitely bigger than 0."},{"Start":"04:32.295 ","End":"04:35.100","Text":"This is true."},{"Start":"04:35.660 ","End":"04:39.960","Text":"We can apply the conclusion in Case 1 up."},{"Start":"04:39.960 ","End":"04:47.005","Text":"Now the sum of b_n converges."},{"Start":"04:47.005 ","End":"04:56.300","Text":"Because, this here we just showed that it converges and goes from 1 to infinity."},{"Start":"04:56.300 ","End":"05:00.920","Text":"Then because of this case and because we\u0027ve shown this inequality,"},{"Start":"05:00.920 ","End":"05:07.440","Text":"it implies that also the sum of a_n converges."},{"Start":"05:07.440 ","End":"05:14.190","Text":"But a_n is our original series and we had to show to decide."},{"Start":"05:14.190 ","End":"05:20.160","Text":"So the answer is that it converges and we are done."}],"ID":10533},{"Watched":false,"Name":"Exercise 7 Part f","Duration":"8m 25s","ChapterTopicVideoID":10207,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10207.jpeg","UploadDate":"2017-08-22T04:38:52.1300000","DurationForVideoObject":"PT8M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we\u0027re given this infinite series,"},{"Start":"00:03.630 ","End":"00:05.520","Text":"note the infinity here."},{"Start":"00:05.520 ","End":"00:12.690","Text":"An infinite series will converge or diverge and we have to decide which of these 2."},{"Start":"00:12.690 ","End":"00:14.640","Text":"There\u0027s many ways, but here,"},{"Start":"00:14.640 ","End":"00:22.170","Text":"we\u0027re going to use the comparison test."},{"Start":"00:22.170 ","End":"00:27.150","Text":"The comparison test is summarized in this box."},{"Start":"00:28.490 ","End":"00:31.640","Text":"You have to guess or make"},{"Start":"00:31.640 ","End":"00:34.940","Text":"an educated guess as to whether you think it converges or diverges"},{"Start":"00:34.940 ","End":"00:41.005","Text":"and then prove your guess using either 1 or 2."},{"Start":"00:41.005 ","End":"00:44.360","Text":"If you suspect it converges,"},{"Start":"00:44.360 ","End":"00:52.940","Text":"then you call your series a_n and find a series b_n,"},{"Start":"00:52.940 ","End":"00:57.770","Text":"which is simpler but greater than or equal to and which converges,"},{"Start":"00:57.770 ","End":"01:02.490","Text":"and then we conclude that this converges on a similar thing with divergent."},{"Start":"01:02.950 ","End":"01:07.225","Text":"The first thing we have to do is just see what we\u0027re going to shoot for;"},{"Start":"01:07.225 ","End":"01:09.215","Text":"number 1 or number 2."},{"Start":"01:09.215 ","End":"01:14.120","Text":"Let\u0027s see. We just do some approximate stuff."},{"Start":"01:17.900 ","End":"01:20.680","Text":"I\u0027ll write it here."},{"Start":"01:27.590 ","End":"01:29.200","Text":"Well, let"},{"Start":"01:29.200 ","End":"01:30.685","Text":"me start further back."},{"Start":"01:30.685 ","End":"01:33.295","Text":"The sine of anything, sine of n,"},{"Start":"01:33.295 ","End":"01:39.355","Text":"is between 1 and minus 1 because that\u0027s the sine function."},{"Start":"01:39.355 ","End":"01:40.780","Text":"If I square it,"},{"Start":"01:40.780 ","End":"01:47.935","Text":"it\u0027s going to be between 0 and 1 because the negative part goes to the positive."},{"Start":"01:47.935 ","End":"01:54.455","Text":"So basically, this thing is somewhere between 0 and 5."},{"Start":"01:54.455 ","End":"01:57.405","Text":"It\u0027s like a constant."},{"Start":"01:57.405 ","End":"02:00.870","Text":"At most, this numerator can be 5."},{"Start":"02:00.870 ","End":"02:05.370","Text":"Now, the n factorial part is n,"},{"Start":"02:05.370 ","End":"02:09.015","Text":"n minus 1, n minus 2,"},{"Start":"02:09.015 ","End":"02:18.440","Text":"and so on, down to 3 times 2 times 1 if n is bigger than 2 or 3,"},{"Start":"02:18.440 ","End":"02:24.210","Text":"let\u0027s say n is at least 3 so that all this make sense."},{"Start":"02:25.150 ","End":"02:29.840","Text":"This is bigger than a polynomial,"},{"Start":"02:29.840 ","End":"02:30.950","Text":"it\u0027s not a polynomial,"},{"Start":"02:30.950 ","End":"02:33.860","Text":"but certainly the first part of it, again,"},{"Start":"02:33.860 ","End":"02:36.080","Text":"assuming n is bigger than 3,"},{"Start":"02:36.080 ","End":"02:39.095","Text":"is a bit like n squared."},{"Start":"02:39.095 ","End":"02:44.585","Text":"It\u0027s like n squared and this thing is even like more than n squared."},{"Start":"02:44.585 ","End":"02:49.250","Text":"What we have is that this behaves like"},{"Start":"02:49.250 ","End":"02:55.180","Text":"5 over and something which is even bigger than n squared."},{"Start":"02:55.180 ","End":"02:58.880","Text":"When we have the sum of 5 squared,"},{"Start":"02:58.880 ","End":"03:00.545","Text":"you can forget about the 5."},{"Start":"03:00.545 ","End":"03:04.745","Text":"It\u0027s a p series with p equals,"},{"Start":"03:04.745 ","End":"03:08.710","Text":"so we know that this thing converges."},{"Start":"03:08.710 ","End":"03:12.680","Text":"This is just the approximate hand-waving part"},{"Start":"03:12.680 ","End":"03:16.010","Text":"where we make an educated guess as to which to go for,"},{"Start":"03:16.010 ","End":"03:19.355","Text":"1 or 2, so we\u0027re going to go for case 1."},{"Start":"03:19.355 ","End":"03:22.240","Text":"We\u0027re going to look for a b_n,"},{"Start":"03:22.240 ","End":"03:31.410","Text":"which is bigger or equal to a_n or at least for most n,"},{"Start":"03:31.410 ","End":"03:35.330","Text":"it will be bigger or equal to and then we\u0027ll be"},{"Start":"03:35.330 ","End":"03:40.280","Text":"able to conclude from b_n\u0027s convergence that a_n also converges."},{"Start":"03:40.280 ","End":"03:46.754","Text":"This is a_n, a_n is the general element,"},{"Start":"03:46.754 ","End":"03:54.520","Text":"5 sine squared n over n factorial"},{"Start":"03:54.770 ","End":"04:01.590","Text":"and this is equal to 5."},{"Start":"04:01.590 ","End":"04:05.130","Text":"Well, let me just rewrite it."},{"Start":"04:05.130 ","End":"04:13.739","Text":"It\u0027s 5 times sine squared n over n,"},{"Start":"04:13.739 ","End":"04:18.229","Text":"n minus 1 and then more stuff."},{"Start":"04:18.229 ","End":"04:19.400","Text":"I\u0027ll just put dot, dot,"},{"Start":"04:19.400 ","End":"04:22.170","Text":"dot. It goes down to 1."},{"Start":"04:22.390 ","End":"04:27.695","Text":"What I\u0027d like to do is find a b_n which is bigger or equal to this."},{"Start":"04:27.695 ","End":"04:37.075","Text":"What I\u0027m going to do is increase the numerator and decrease the denominator."},{"Start":"04:37.075 ","End":"04:45.930","Text":"This thing is certainly less than or equal to 5 times 1 over,"},{"Start":"04:45.930 ","End":"04:53.115","Text":"and here, I can write just n, n minus 1."},{"Start":"04:53.115 ","End":"04:54.960","Text":"I\u0027ve decreased the denominator,"},{"Start":"04:54.960 ","End":"04:57.225","Text":"I\u0027ve thrown out all this part,"},{"Start":"04:57.225 ","End":"05:01.430","Text":"and increased the denominator based on what we said here."},{"Start":"05:02.970 ","End":"05:07.295","Text":"This is not quite 1 squared;"},{"Start":"05:07.295 ","End":"05:12.585","Text":"in fact, this is equal to 5."},{"Start":"05:12.585 ","End":"05:15.230","Text":"I could throw out the 5, but never mind,"},{"Start":"05:15.230 ","End":"05:16.520","Text":"I\u0027ll carry it along."},{"Start":"05:16.520 ","End":"05:20.730","Text":"N squared minus n. Now,"},{"Start":"05:20.730 ","End":"05:23.990","Text":"there\u0027s all kinds of tricks we could do to get out of this,"},{"Start":"05:23.990 ","End":"05:26.555","Text":"to get it to be n squared,"},{"Start":"05:26.555 ","End":"05:33.975","Text":"but a trick that I thought of is that n squared minus n,"},{"Start":"05:33.975 ","End":"05:35.850","Text":"I\u0027m claiming that, well,"},{"Start":"05:35.850 ","End":"05:40.970","Text":"do that at the [inaudible] I say that n squared minus n is"},{"Start":"05:40.970 ","End":"05:47.730","Text":"at least 1/2 of n squared from a certain point on."},{"Start":"05:48.250 ","End":"05:53.015","Text":"In fact, I can actually solve this inequality."},{"Start":"05:53.015 ","End":"05:55.590","Text":"Let\u0027s see what it comes out to."},{"Start":"05:55.720 ","End":"05:58.275","Text":"If I bring this to the other side,"},{"Start":"05:58.275 ","End":"05:59.940","Text":"it\u0027s 1/2 n squared,"},{"Start":"05:59.940 ","End":"06:01.375","Text":"it\u0027s 1 minus 1/2,"},{"Start":"06:01.375 ","End":"06:04.880","Text":"bigger or equal to n, n is positive,"},{"Start":"06:04.880 ","End":"06:09.535","Text":"so 1/2 n is bigger or equal to 1."},{"Start":"06:09.535 ","End":"06:15.390","Text":"Then I multiply both sides by 2 and it\u0027s bigger or equal to 2."},{"Start":"06:15.390 ","End":"06:17.800","Text":"So as soon as n is bigger or equal to 2,"},{"Start":"06:17.800 ","End":"06:19.830","Text":"which means for almost all n,"},{"Start":"06:19.830 ","End":"06:21.480","Text":"except for n equals 1,"},{"Start":"06:21.480 ","End":"06:23.490","Text":"this thing is going to be true."},{"Start":"06:23.490 ","End":"06:25.844","Text":"If this is true, I can shrink"},{"Start":"06:25.844 ","End":"06:33.710","Text":"the denominator further and I can say that this is less than"},{"Start":"06:33.710 ","End":"06:41.150","Text":"or equal to 5 over 1/2 n"},{"Start":"06:41.150 ","End":"06:50.445","Text":"squared because this thing is bigger or equal to this over the smaller denominator."},{"Start":"06:50.445 ","End":"06:55.740","Text":"Basically, what we have is 10 over n squared."},{"Start":"06:55.740 ","End":"07:04.520","Text":"Now, if I take this as my b_n, then I know that"},{"Start":"07:04.520 ","End":"07:11.620","Text":"the sum of b_n"},{"Start":"07:11.620 ","End":"07:17.610","Text":"from n equals 1 to infinity is convergent."},{"Start":"07:18.740 ","End":"07:23.520","Text":"We\u0027ve done this before. Forget about the 10 even."},{"Start":"07:23.520 ","End":"07:25.140","Text":"1 over n squared is convergent."},{"Start":"07:25.140 ","End":"07:27.510","Text":"It\u0027s a p series with p equals 2,"},{"Start":"07:27.510 ","End":"07:29.770","Text":"which is bigger than 1."},{"Start":"07:42.000 ","End":"07:43.070","Text":"We\u0027ve seen this several times before."},{"Start":"07:43.070 ","End":"07:43.595","Text":"This is less than or equal to."},{"Start":"07:43.595 ","End":"07:45.560","Text":"We have 2 less than or equals,"},{"Start":"07:45.560 ","End":"07:54.390","Text":"basically because a_n is less than or equal to b_n and,"},{"Start":"07:54.390 ","End":"07:57.050","Text":"of course, a_n is also non-negative."},{"Start":"07:57.050 ","End":"07:58.865","Text":"I should have mentioned that in the beginning."},{"Start":"07:58.865 ","End":"08:00.440","Text":"Nothing here is negative,"},{"Start":"08:00.440 ","End":"08:01.730","Text":"it\u0027s bigger or equal to 0."},{"Start":"08:01.730 ","End":"08:08.035","Text":"We have this holds and we have that this holds now and now we can draw the conclusion."},{"Start":"08:08.035 ","End":"08:13.090","Text":"That\u0027s the last thing we needed to do to say that the sum of a_n"},{"Start":"08:13.090 ","End":"08:19.995","Text":"converges and a_n is our original series,"},{"Start":"08:19.995 ","End":"08:21.765","Text":"so we found the answer,"},{"Start":"08:21.765 ","End":"08:24.730","Text":"converges, and we\u0027re done."}],"ID":10534},{"Watched":false,"Name":"Exercise 7 Part g","Duration":"6m 41s","ChapterTopicVideoID":10208,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10208.jpeg","UploadDate":"2017-08-22T04:39:08.7030000","DurationForVideoObject":"PT6M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:04.710","Text":"In this exercise, we\u0027re given the following infinite series"},{"Start":"00:04.710 ","End":"00:08.820","Text":"and we have to decide if it converges or diverges."},{"Start":"00:08.820 ","End":"00:11.055","Text":"There are several ways one could do this."},{"Start":"00:11.055 ","End":"00:16.875","Text":"We\u0027re going to use the comparison test which is described in the box here."},{"Start":"00:16.875 ","End":"00:21.660","Text":"Basically, we have to compare it to another series,"},{"Start":"00:21.660 ","End":"00:25.540","Text":"which we know converges or diverges."},{"Start":"00:25.540 ","End":"00:29.280","Text":"We have to decide which one we are going for."},{"Start":"00:29.280 ","End":"00:33.380","Text":"Whether we\u0027re going to try and prove convergent or divergent."},{"Start":"00:33.380 ","End":"00:40.490","Text":"Let\u0027s just do some rough calculations just to get an idea of what we\u0027re expecting."},{"Start":"00:40.490 ","End":"00:45.950","Text":"This is different from most of the other exercises we\u0027ve encountered previously."},{"Start":"00:45.950 ","End":"00:51.980","Text":"Actually, we\u0027re going to have to do a bit of algebra here to change its form."},{"Start":"00:51.980 ","End":"00:54.560","Text":"You might remember that when dealing with square roots,"},{"Start":"00:54.560 ","End":"00:59.030","Text":"we often employ what is called the conjugate."},{"Start":"00:59.030 ","End":"01:02.675","Text":"I\u0027ll do an aside on this now."},{"Start":"01:02.675 ","End":"01:08.840","Text":"What I have here is something of the form a minus b, and in general,"},{"Start":"01:08.840 ","End":"01:12.275","Text":"when I have a minus b or a plus b,"},{"Start":"01:12.275 ","End":"01:15.500","Text":"then the thing with the opposite sign, in this case,"},{"Start":"01:15.500 ","End":"01:18.829","Text":"a plus b is called the conjugate."},{"Start":"01:18.829 ","End":"01:21.245","Text":"Each of these is the conjugate of the other,"},{"Start":"01:21.245 ","End":"01:23.890","Text":"and the property they have,"},{"Start":"01:23.890 ","End":"01:25.320","Text":"that makes them useful,"},{"Start":"01:25.320 ","End":"01:29.570","Text":"is that if you multiply an expression by its conjugate,"},{"Start":"01:29.570 ","End":"01:31.820","Text":"you get the difference of squares."},{"Start":"01:31.820 ","End":"01:34.475","Text":"In this case, a squared minus b squared,"},{"Start":"01:34.475 ","End":"01:38.275","Text":"which is very useful when you want to get rid of square roots."},{"Start":"01:38.275 ","End":"01:41.405","Text":"In our case, for example,"},{"Start":"01:41.405 ","End":"01:51.815","Text":"if I take the square root of n squared plus 1 minus n,"},{"Start":"01:51.815 ","End":"01:55.505","Text":"and then I multiply it by its conjugate,"},{"Start":"01:55.505 ","End":"02:01.050","Text":"the square root of n squared plus 1 plus n,"},{"Start":"02:01.050 ","End":"02:04.475","Text":"what I get according to this formula is a squared,"},{"Start":"02:04.475 ","End":"02:06.620","Text":"which is the square root squared,"},{"Start":"02:06.620 ","End":"02:15.435","Text":"which is n squared plus 1 less n squared."},{"Start":"02:15.435 ","End":"02:22.010","Text":"The n squared cancel out and we\u0027re just left with 1."},{"Start":"02:22.010 ","End":"02:28.730","Text":"This is very good for us because let me just,"},{"Start":"02:28.730 ","End":"02:32.374","Text":"I\u0027ll just copy it further down here,"},{"Start":"02:32.374 ","End":"02:37.595","Text":"and then I can scroll and get more space."},{"Start":"02:37.595 ","End":"02:41.750","Text":"Applying it to what we have here,"},{"Start":"02:41.750 ","End":"02:44.675","Text":"we can see that this expression,"},{"Start":"02:44.675 ","End":"02:47.150","Text":"just the general term."},{"Start":"02:47.150 ","End":"02:49.475","Text":"I\u0027ll do a division here."},{"Start":"02:49.475 ","End":"02:57.499","Text":"The square root of n squared plus 1 minus n is equal to 1 over,"},{"Start":"02:57.499 ","End":"02:59.435","Text":"I\u0027m just dividing by this thing,"},{"Start":"02:59.435 ","End":"03:04.550","Text":"the square root of n squared plus 1 plus n."},{"Start":"03:04.550 ","End":"03:11.480","Text":"I claim that this is more useful to me than the original form."},{"Start":"03:11.480 ","End":"03:14.870","Text":"Because here I can use my old techniques of saying, well,"},{"Start":"03:14.870 ","End":"03:18.365","Text":"it\u0027s the dominant exponent the power that counts,"},{"Start":"03:18.365 ","End":"03:25.700","Text":"that this behaves roughly like 1 over the square root of n squared plus n."},{"Start":"03:25.700 ","End":"03:29.345","Text":"Because n squared is dominant in this thing,"},{"Start":"03:29.345 ","End":"03:32.060","Text":"and the square root of n squared is n."},{"Start":"03:32.060 ","End":"03:35.450","Text":"Basically, what I get here is 1 over 2n,"},{"Start":"03:35.450 ","End":"03:41.160","Text":"and the sum of 1 over 2n is like the sum of 1 over n"},{"Start":"03:41.160 ","End":"03:44.525","Text":"with or without the 2 here, it doesn\u0027t matter."},{"Start":"03:44.525 ","End":"03:48.150","Text":"This is harmonic and it diverges."},{"Start":"03:49.580 ","End":"03:53.610","Text":"We\u0027re expecting this to diverge."},{"Start":"03:53.610 ","End":"03:57.305","Text":"What we\u0027ll do is we\u0027ll use the comparison test."},{"Start":"03:57.305 ","End":"04:02.900","Text":"If it diverges, then I want to"},{"Start":"04:02.900 ","End":"04:11.650","Text":"use part 2 and let this be b_n,"},{"Start":"04:11.650 ","End":"04:15.275","Text":"this general term is going to be my b_n,"},{"Start":"04:15.275 ","End":"04:21.395","Text":"and I\u0027m going to find some a_n which is still bigger or equal to 0,"},{"Start":"04:21.395 ","End":"04:23.330","Text":"but less than or equal to b_n."},{"Start":"04:23.330 ","End":"04:27.290","Text":"Something simpler that I know diverges."},{"Start":"04:27.290 ","End":"04:35.435","Text":"What I say is that b_n is equal to the general term,"},{"Start":"04:35.435 ","End":"04:43.010","Text":"and I\u0027m looking at the work I did over here is equal to 1 over up to here, I had equal,"},{"Start":"04:43.010 ","End":"04:45.110","Text":"the squiggly line is no longer an equal,"},{"Start":"04:45.110 ","End":"04:53.775","Text":"but it equal to 1 over the square root of n squared plus 1 plus n,"},{"Start":"04:53.775 ","End":"04:59.765","Text":"and now I want this to be bigger or equal to something."},{"Start":"04:59.765 ","End":"05:05.225","Text":"I can either increase the numerator,"},{"Start":"05:05.225 ","End":"05:07.175","Text":"or shrink the denominator."},{"Start":"05:07.175 ","End":"05:10.040","Text":"What I\u0027m going to do is actually put something"},{"Start":"05:10.040 ","End":"05:13.970","Text":"smaller in the denominator and this will be bigger than,"},{"Start":"05:13.970 ","End":"05:16.715","Text":"I like put bigger or equal to just cover myself."},{"Start":"05:16.715 ","End":"05:19.150","Text":"This is 1 over,"},{"Start":"05:19.150 ","End":"05:23.270","Text":"now, if I make this smaller,"},{"Start":"05:23.270 ","End":"05:28.250","Text":"if I put square root of n squared it\u0027s definitely smaller on the denominator,"},{"Start":"05:28.250 ","End":"05:33.300","Text":"and this thing is an equal to 1 over 2n,"},{"Start":"05:33.300 ","End":"05:34.820","Text":"and notice by the way,"},{"Start":"05:34.820 ","End":"05:40.490","Text":"that it\u0027s bigger or equal to 0 because I need this in order to use the theorem."},{"Start":"05:40.490 ","End":"05:47.270","Text":"If I let this bit be my a_n,"},{"Start":"05:47.270 ","End":"05:51.110","Text":"and all I have to show is that the sum of a_n diverges"},{"Start":"05:51.110 ","End":"05:53.525","Text":"and I\u0027ve got my original series diverges."},{"Start":"05:53.525 ","End":"06:01.730","Text":"Well, the sum of a_n is 1/5 the sum of 1 over n."},{"Start":"06:01.730 ","End":"06:06.060","Text":"Let me just write n goes from 1 to infinity."},{"Start":"06:06.310 ","End":"06:09.770","Text":"This is the classic harmonic series we said."},{"Start":"06:09.770 ","End":"06:14.015","Text":"We know that this diverges with or without the half,"},{"Start":"06:14.015 ","End":"06:18.265","Text":"the constants make no difference, and this diverges."},{"Start":"06:18.265 ","End":"06:23.120","Text":"Because this diverges, and because we\u0027ve shown this inequality,"},{"Start":"06:23.120 ","End":"06:29.400","Text":"then we conclude that the sum of b_n diverges."},{"Start":"06:29.400 ","End":"06:33.470","Text":"But this is exactly our original series,"},{"Start":"06:33.470 ","End":"06:40.830","Text":"and so we found the answer that our series is divergent and we\u0027re done."}],"ID":10535},{"Watched":false,"Name":"Exercise 7 Part h","Duration":"5m 36s","ChapterTopicVideoID":10209,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10209.jpeg","UploadDate":"2017-08-22T04:39:22.9970000","DurationForVideoObject":"PT5M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have an infinite series,"},{"Start":"00:04.320 ","End":"00:11.100","Text":"1 to infinity, and we have to decide if this converges or diverges."},{"Start":"00:11.100 ","End":"00:14.864","Text":"It\u0027s somewhat unusual."},{"Start":"00:14.864 ","End":"00:17.845","Text":"Usual tricks don\u0027t seem to work here."},{"Start":"00:17.845 ","End":"00:23.385","Text":"What we\u0027re going to do is use trigonometric identities."},{"Start":"00:23.385 ","End":"00:29.285","Text":"I\u0027ll just work on the we\u0027ll have more space down here."},{"Start":"00:29.285 ","End":"00:33.540","Text":"What I\u0027m going to do is modify the general term."},{"Start":"00:34.400 ","End":"00:37.790","Text":"I see it\u0027s 1 minus cosine something."},{"Start":"00:37.790 ","End":"00:41.600","Text":"Can I remember that there\u0027s a trigonometric formula that"},{"Start":"00:41.600 ","End":"00:49.175","Text":"1 minus cosine Alpha is twice sine squared Alpha over 2."},{"Start":"00:49.175 ","End":"01:00.455","Text":"In our case, what we have here is 1 minus cosine of 1 over n,"},{"Start":"01:00.455 ","End":"01:07.680","Text":"and that is going to equal twice sine squared Alpha\u0027s 1 over n,"},{"Start":"01:07.680 ","End":"01:10.650","Text":"so it\u0027s 1 over 2n."},{"Start":"01:10.650 ","End":"01:21.380","Text":"Now, I also need another result from trigonometry calculus,"},{"Start":"01:21.380 ","End":"01:26.585","Text":"that sine Alpha is less than or equal to Alpha."},{"Start":"01:26.585 ","End":"01:29.855","Text":"This is for non-negative,"},{"Start":"01:29.855 ","End":"01:34.610","Text":"which we are in the case of for all non-negative Alpha."},{"Start":"01:34.610 ","End":"01:37.465","Text":"If I use that here,"},{"Start":"01:37.465 ","End":"01:40.655","Text":"this general term is,"},{"Start":"01:40.655 ","End":"01:43.080","Text":"I\u0027ll continue over here."},{"Start":"01:43.220 ","End":"01:52.055","Text":"So 1 minus cosine of 1 over n is less than or equal to instead of sine,"},{"Start":"01:52.055 ","End":"01:54.065","Text":"I\u0027ll just throw out the sine,"},{"Start":"01:54.065 ","End":"01:56.680","Text":"and it will be twice,"},{"Start":"01:56.680 ","End":"02:01.155","Text":"1 over 2n squared,"},{"Start":"02:01.155 ","End":"02:03.910","Text":"and we\u0027re in positive numbers."},{"Start":"02:04.480 ","End":"02:06.800","Text":"If this is less than this,"},{"Start":"02:06.800 ","End":"02:09.575","Text":"then if I put a squared here,"},{"Start":"02:09.575 ","End":"02:15.005","Text":"do it in a different color, it\u0027s going to also be less than or equal to this,"},{"Start":"02:15.005 ","End":"02:16.850","Text":"because with positive numbers,"},{"Start":"02:16.850 ","End":"02:19.880","Text":"the square keeps the order."},{"Start":"02:19.880 ","End":"02:21.950","Text":"If we simplify this,"},{"Start":"02:21.950 ","End":"02:25.875","Text":"what we get is 1/2 of 1 over n squared,"},{"Start":"02:25.875 ","End":"02:32.630","Text":"and so this series now behaves like the 1 over n squared series,"},{"Start":"02:32.630 ","End":"02:34.340","Text":"and we know that\u0027s convergent."},{"Start":"02:34.340 ","End":"02:36.095","Text":"We\u0027ve done that several times."},{"Start":"02:36.095 ","End":"02:40.200","Text":"Typically I explained it that we have a p-series."},{"Start":"02:40.200 ","End":"02:47.120","Text":"The sum of 1 over n to the power of p converges when p is bigger than 1,"},{"Start":"02:47.120 ","End":"02:50.785","Text":"and in the case that p is 2 that\u0027s certainly bigger than 1."},{"Start":"02:50.785 ","End":"02:57.655","Text":"It looks like we\u0027re going to have a comparison with some variant of 1 over n squared,"},{"Start":"02:57.655 ","End":"03:02.105","Text":"and we\u0027re going to go for the convergent case."},{"Start":"03:02.105 ","End":"03:10.295","Text":"We\u0027ll let this be our a_n."},{"Start":"03:10.295 ","End":"03:18.300","Text":"In other words, we\u0027ll take a_n to be 1 minus cosine of 1 over n,"},{"Start":"03:18.300 ","End":"03:26.350","Text":"and we have by all the calculations above that this is equal"},{"Start":"03:26.350 ","End":"03:36.485","Text":"to 2 sine squared 1 over 2n."},{"Start":"03:36.485 ","End":"03:41.430","Text":"Then we said that the sine of an angle sine Alpha is less than or equal to Alpha."},{"Start":"03:41.430 ","End":"03:46.400","Text":"From this, we have that this is less than or equal to,"},{"Start":"03:46.400 ","End":"03:56.500","Text":"what we get in the end is 1/2 of 1 over n squared."},{"Start":"03:56.660 ","End":"04:01.765","Text":"This we\u0027re going to let equal to our b_n."},{"Start":"04:01.765 ","End":"04:08.970","Text":"Now, the sum of b_n 1 to infinity is"},{"Start":"04:08.970 ","End":"04:16.695","Text":"just 1/2 of the sum of the 1 over n squared series,"},{"Start":"04:16.695 ","End":"04:21.480","Text":"and we know that this converges."},{"Start":"04:21.480 ","End":"04:25.365","Text":"If the b_n converges,"},{"Start":"04:25.365 ","End":"04:27.420","Text":"so does the a_n,"},{"Start":"04:27.420 ","End":"04:34.150","Text":"because we\u0027ve seen that from this computation that a_n is less than or equal to b_n."},{"Start":"04:34.150 ","End":"04:42.295","Text":"We also see that I should\u0027ve mentioned that a_n is also bigger or equal to 0."},{"Start":"04:42.295 ","End":"04:48.520","Text":"This follows from the fact that in general,"},{"Start":"04:48.520 ","End":"04:52.470","Text":"cosine of anything, use a different letter,"},{"Start":"04:52.470 ","End":"04:54.330","Text":"instead of Alpha, I\u0027ll use Theta,"},{"Start":"04:54.330 ","End":"04:59.605","Text":"cosine Theta is always less than or equal to 1,"},{"Start":"04:59.605 ","End":"05:07.570","Text":"so 1 minus cosine of anything is always bigger or equal to 0,"},{"Start":"05:07.570 ","End":"05:11.290","Text":"in this case, 1 over n. We\u0027re okay with bigger or equal to 0,"},{"Start":"05:11.290 ","End":"05:12.745","Text":"we have this inequality."},{"Start":"05:12.745 ","End":"05:14.410","Text":"We have that Sigma,"},{"Start":"05:14.410 ","End":"05:18.905","Text":"the sum the series b_n converges."},{"Start":"05:18.905 ","End":"05:27.990","Text":"From all this, we conclude therefore that the sum of a_n also converges."},{"Start":"05:27.990 ","End":"05:31.700","Text":"This is our original series and this is what we were asked to show."},{"Start":"05:31.700 ","End":"05:36.630","Text":"The answer is, it converges. We\u0027re done."}],"ID":10536},{"Watched":false,"Name":"Exercise 7 Part i","Duration":"9m 47s","ChapterTopicVideoID":10201,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10201.jpeg","UploadDate":"2017-08-22T04:37:16.8000000","DurationForVideoObject":"PT9M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.110","Text":"In this exercise, we are given an infinite series."},{"Start":"00:04.110 ","End":"00:06.225","Text":"See the infinity here."},{"Start":"00:06.225 ","End":"00:09.960","Text":"An infinite series will either converge or diverge"},{"Start":"00:09.960 ","End":"00:14.380","Text":"and our job is to decide which of these two it is."},{"Start":"00:15.440 ","End":"00:18.660","Text":"There\u0027s many ways possibly to do this,"},{"Start":"00:18.660 ","End":"00:27.280","Text":"but we\u0027re going to use the comparison test where we compare one series with another."},{"Start":"00:28.340 ","End":"00:33.510","Text":"First, we have to get an intuition or educated guess"},{"Start":"00:33.510 ","End":"00:37.785","Text":"as to which we are going for whether we want to prove convergent or divergent."},{"Start":"00:37.785 ","End":"00:46.230","Text":"Then we do it more formally by producing another series and we compare it."},{"Start":"00:46.480 ","End":"00:50.645","Text":"I\u0027m not going to repeat the whole thing here just as needed."},{"Start":"00:50.645 ","End":"00:54.480","Text":"Let me just get some more space here."},{"Start":"00:55.570 ","End":"00:59.975","Text":"Now this is not something completely a routine."},{"Start":"00:59.975 ","End":"01:05.610","Text":"Normally what we do for the guess as to which of the two it is,"},{"Start":"01:05.610 ","End":"01:08.450","Text":"is we start approximating like polynomials."},{"Start":"01:08.450 ","End":"01:13.210","Text":"We take the leading term, the dominant term."},{"Start":"01:13.210 ","End":"01:16.650","Text":"I would say that this behaves like,"},{"Start":"01:16.650 ","End":"01:22.780","Text":"I used the symbol squiggly sign just to say behaves like."},{"Start":"01:23.210 ","End":"01:30.390","Text":"The sum of, here I would write n^1.5."},{"Start":"01:30.390 ","End":"01:32.915","Text":"Here I just take the dominant term,"},{"Start":"01:32.915 ","End":"01:38.360","Text":"n squared, but what to do with this natural log of n?"},{"Start":"01:38.360 ","End":"01:46.950","Text":"Well, it turns out that this behaves like n^ε,"},{"Start":"01:47.060 ","End":"01:51.230","Text":"where epsilon is just a mathematical symbol,"},{"Start":"01:51.230 ","End":"01:54.455","Text":"is customarily used as something very small."},{"Start":"01:54.455 ","End":"01:58.650","Text":"Maybe not 0, but as small as you like."},{"Start":"01:59.180 ","End":"02:03.080","Text":"In actual fact, there is"},{"Start":"02:03.080 ","End":"02:07.160","Text":"a more precise way and I\u0027ll write it in a moment of writing this,"},{"Start":"02:07.160 ","End":"02:11.375","Text":"but we\u0027re just at the first step of making the educated guess."},{"Start":"02:11.375 ","End":"02:13.715","Text":"If we compare this to this,"},{"Start":"02:13.715 ","End":"02:17.330","Text":"then n goes from 1 to infinity,"},{"Start":"02:17.330 ","End":"02:19.730","Text":"then what we get if we do the computation,"},{"Start":"02:19.730 ","End":"02:26.235","Text":"I mean so powers of n is that we have the sum of,"},{"Start":"02:26.235 ","End":"02:29.505","Text":"I want to put it as 1 over n to the power of something."},{"Start":"02:29.505 ","End":"02:34.920","Text":"If I bring all these to the denominator is 2 minus 1/2 minus an epsilon."},{"Start":"02:34.920 ","End":"02:43.440","Text":"It\u0027s n to the power of 1.5 minus epsilon."},{"Start":"02:44.030 ","End":"02:46.530","Text":"If epsilon is small,"},{"Start":"02:46.530 ","End":"02:49.010","Text":"in fact it doesn\u0027t even have to be very small."},{"Start":"02:49.010 ","End":"02:52.280","Text":"Epsilon could be say, 0.1,"},{"Start":"02:52.280 ","End":"02:57.680","Text":"could even take 0.4 as long as I get in the end n to the power of bigger than 1."},{"Start":"02:57.680 ","End":"03:03.319","Text":"Let\u0027s just say that epsilon equals 0.1, just for instance,"},{"Start":"03:03.319 ","End":"03:08.390","Text":"then I would get the sum"},{"Start":"03:08.390 ","End":"03:15.810","Text":"of 1 over n^1.4,"},{"Start":"03:15.810 ","End":"03:19.365","Text":"forgetting to write n goes from 1 to infinity."},{"Start":"03:19.365 ","End":"03:24.575","Text":"Then this is a p-series in case you\u0027ve forgotten,"},{"Start":"03:24.575 ","End":"03:29.995","Text":"the p-series is the sum of 1 over n^p."},{"Start":"03:29.995 ","End":"03:37.545","Text":"The convergence condition is that this converges if and only if p is bigger than 1."},{"Start":"03:37.545 ","End":"03:40.115","Text":"In this case 1.4,"},{"Start":"03:40.115 ","End":"03:49.275","Text":"I would say is bigger than 1 and so this 1 converges."},{"Start":"03:49.275 ","End":"03:56.750","Text":"I suspect that this is going to also converge and now let\u0027s get the more precise part."},{"Start":"03:56.750 ","End":"03:59.465","Text":"If we want to prove that something converges,"},{"Start":"03:59.465 ","End":"04:06.345","Text":"then I use rule number 1 and I call this series,"},{"Start":"04:06.345 ","End":"04:10.785","Text":"the original one a_n, the general term."},{"Start":"04:10.785 ","End":"04:14.295","Text":"I\u0027m going to produce something called b_n,"},{"Start":"04:14.295 ","End":"04:19.095","Text":"which is going to satisfy that b_n is bigger or equal to a_n."},{"Start":"04:19.095 ","End":"04:24.500","Text":"But b_n is easy to compute that it converges."},{"Start":"04:24.500 ","End":"04:26.420","Text":"When I\u0027ve shown that b_n converges,"},{"Start":"04:26.420 ","End":"04:30.295","Text":"then I\u0027ll be able to deduce that a_n converges, that\u0027s the strategy."},{"Start":"04:30.295 ","End":"04:38.465","Text":"I almost forgot, we also have to show that a_n is bigger or equal to 0."},{"Start":"04:38.465 ","End":"04:48.500","Text":"But just look at it, everything is positive so that\u0027s obvious it\u0027s bigger or equal to 0."},{"Start":"04:48.500 ","End":"04:50.515","Text":"Actually, when n equals 1,"},{"Start":"04:50.515 ","End":"04:53.385","Text":"it is 0 and for anything larger than 1,"},{"Start":"04:53.385 ","End":"04:55.490","Text":"it\u0027s strictly bigger than 0, but that doesn\u0027t matter."},{"Start":"04:55.490 ","End":"04:57.470","Text":"We have the bigger or equal to."},{"Start":"04:57.470 ","End":"05:00.210","Text":"What am I going to take as b_n?"},{"Start":"05:00.580 ","End":"05:05.740","Text":"Let\u0027s just start from a_n and say,"},{"Start":"05:05.740 ","End":"05:09.005","Text":"this equals square root of n,"},{"Start":"05:09.005 ","End":"05:15.200","Text":"natural log of n over n squared plus 1."},{"Start":"05:15.200 ","End":"05:21.235","Text":"Then I\u0027m going to say that this is less than or equal to."},{"Start":"05:21.235 ","End":"05:26.400","Text":"First of all, I can decrease the denominator and then things can only get larger."},{"Start":"05:26.400 ","End":"05:35.505","Text":"Just n squared, here I\u0027m going to leave it as n^1.5."},{"Start":"05:35.505 ","End":"05:44.860","Text":"I would like to say that natural log of n is less than n^ε."},{"Start":"05:44.860 ","End":"05:47.740","Text":"But this is not quite right,"},{"Start":"05:47.740 ","End":"05:52.720","Text":"or at least it requires some explanation and let me do this at the side."},{"Start":"05:52.720 ","End":"05:57.795","Text":"At this point, I just want to point out the magic words."},{"Start":"05:57.795 ","End":"05:59.450","Text":"Not for all n,"},{"Start":"05:59.450 ","End":"06:03.230","Text":"for almost all n. What does that mean?"},{"Start":"06:03.230 ","End":"06:07.040","Text":"That means that I can allow a finite number of exceptions."},{"Start":"06:07.040 ","End":"06:11.180","Text":"Or another way of saying it is from a certain point n onwards,"},{"Start":"06:11.180 ","End":"06:12.815","Text":"maybe the beginning not,"},{"Start":"06:12.815 ","End":"06:15.155","Text":"but the whole tail of the series."},{"Start":"06:15.155 ","End":"06:20.945","Text":"The reason I\u0027m emphasizing this is that it turns out that"},{"Start":"06:20.945 ","End":"06:29.030","Text":"natural log of n is less than n^ε."},{"Start":"06:29.030 ","End":"06:31.310","Text":"Even if epsilon is small,"},{"Start":"06:31.310 ","End":"06:34.879","Text":"say 0.1 in our case,"},{"Start":"06:34.879 ","End":"06:43.355","Text":"for example, but it\u0027s not true for all n. It is true from a certain point n onwards."},{"Start":"06:43.355 ","End":"06:47.210","Text":"I\u0027m not going to get heavily into it,"},{"Start":"06:47.210 ","End":"06:50.635","Text":"but I would like to give some justification."},{"Start":"06:50.635 ","End":"06:53.690","Text":"You can ignore this if you just trust me on this,"},{"Start":"06:53.690 ","End":"06:56.105","Text":"but I\u0027d like to give some explanation."},{"Start":"06:56.105 ","End":"07:02.690","Text":"It turns out that if you compute the limit of natural log of, let\u0027s say,"},{"Start":"07:02.690 ","End":"07:08.000","Text":"not n but a continuous variable log x over x^ε,"},{"Start":"07:08.000 ","End":"07:10.640","Text":"when x goes to infinity,"},{"Start":"07:10.640 ","End":"07:13.919","Text":"it\u0027s easy to show using"},{"Start":"07:18.830 ","End":"07:25.220","Text":"L\u0027Hopital\u0027s rule that this actually goes to 0."},{"Start":"07:25.220 ","End":"07:32.910","Text":"If it goes to 0, then from some point onwards, when x is big,"},{"Start":"07:32.910 ","End":"07:39.075","Text":"I\u0027ll just write this is for big x,"},{"Start":"07:39.075 ","End":"07:41.329","Text":"that means that from some point onwards,"},{"Start":"07:41.329 ","End":"07:44.970","Text":"this thing is less than 1."},{"Start":"07:47.380 ","End":"07:54.785","Text":"Which means that natural log of x is less than x^ε."},{"Start":"07:54.785 ","End":"07:57.635","Text":"This is a justification."},{"Start":"07:57.635 ","End":"08:00.870","Text":"This is less than or equal to this."},{"Start":"08:03.280 ","End":"08:13.200","Text":"For most n or for almost all n,"},{"Start":"08:13.480 ","End":"08:17.010","Text":"I just rewrote it, it wasn\u0027t clear,"},{"Start":"08:17.390 ","End":"08:21.540","Text":"and the explanation I gave over here."},{"Start":"08:21.860 ","End":"08:26.330","Text":"I\u0027m just going to simplify this, just like above,"},{"Start":"08:26.330 ","End":"08:32.580","Text":"this comes out to be 1 over, here it is."},{"Start":"08:32.580 ","End":"08:37.790","Text":"We\u0027ll just write here, n^1.5 minus epsilon."},{"Start":"08:37.790 ","End":"08:43.795","Text":"I can take epsilon as 0.1, n^1.4."},{"Start":"08:43.795 ","End":"08:47.190","Text":"This is what I\u0027m going to call b_n."},{"Start":"08:47.190 ","End":"08:51.305","Text":"Now we\u0027ve done everything, we\u0027ve shown that a_n is bigger or equal to 0."},{"Start":"08:51.305 ","End":"08:56.400","Text":"We have here that a_n is less than or equal to b, not forever the n,"},{"Start":"08:56.400 ","End":"09:02.210","Text":"but for almost every n. Now using 1 because b_n"},{"Start":"09:02.210 ","End":"09:09.055","Text":"converges and we already talked about that here because the p-series, this one converges."},{"Start":"09:09.055 ","End":"09:14.890","Text":"Well, not it, but the sum converges."},{"Start":"09:16.400 ","End":"09:20.840","Text":"We can conclude that because this converges,"},{"Start":"09:20.840 ","End":"09:27.600","Text":"this will give us that the sum of a_n also converges."},{"Start":"09:28.550 ","End":"09:36.840","Text":"To be precise I\u0027ll write it here and get lazy after awhile."},{"Start":"09:36.840 ","End":"09:41.825","Text":"This is our original series and it converges and that\u0027s what we were asked to say."},{"Start":"09:41.825 ","End":"09:47.190","Text":"To say it converges or diverges. We are done."}],"ID":10537},{"Watched":false,"Name":"Exercise 7 Part j","Duration":"6m 53s","ChapterTopicVideoID":10211,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10211.jpeg","UploadDate":"2017-08-22T04:40:17.3670000","DurationForVideoObject":"PT6M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.409","Text":"In this exercise, we have an infinite series,"},{"Start":"00:04.409 ","End":"00:06.915","Text":"you can see it from 1 to infinity."},{"Start":"00:06.915 ","End":"00:12.315","Text":"We have to decide whether it converges or diverges."},{"Start":"00:12.315 ","End":"00:16.395","Text":"There\u0027s more than 1 way to do this,"},{"Start":"00:16.395 ","End":"00:22.840","Text":"but what we\u0027re going to use is the limit comparison test."},{"Start":"00:23.330 ","End":"00:27.000","Text":"The whole theorem is written out here in the box,"},{"Start":"00:27.000 ","End":"00:29.160","Text":"I won\u0027t read all of it,"},{"Start":"00:29.160 ","End":"00:34.320","Text":"but 1 of the things is, to compare it to another series,"},{"Start":"00:34.320 ","End":"00:38.880","Text":"which is simpler, let\u0027s say this is our series a_n,"},{"Start":"00:38.880 ","End":"00:41.440","Text":"we have to find another series b_n,"},{"Start":"00:41.440 ","End":"00:43.595","Text":"and then look at the ratios,"},{"Start":"00:43.595 ","End":"00:50.215","Text":"but we have to initially anticipate whether we expect it to converge or diverge."},{"Start":"00:50.215 ","End":"00:53.640","Text":"I\u0027m just going to get some more space here."},{"Start":"00:55.780 ","End":"01:01.640","Text":"The initial guess as to whether it converges or diverges is a rule of thumb,"},{"Start":"01:01.640 ","End":"01:06.035","Text":"that when you have polynomials or rational functions,"},{"Start":"01:06.035 ","End":"01:09.500","Text":"the idea is to say that this behaves approximately,"},{"Start":"01:09.500 ","End":"01:12.500","Text":"I\u0027ll write a squiggly line, not that it\u0027s equal to,"},{"Start":"01:12.500 ","End":"01:16.030","Text":"but it roughly behaves as if I had"},{"Start":"01:16.030 ","End":"01:24.230","Text":"5n squared over 14n^5."},{"Start":"01:24.230 ","End":"01:27.575","Text":"The highest terms are what really count in these things,"},{"Start":"01:27.575 ","End":"01:30.095","Text":"and constants don\u0027t much matter."},{"Start":"01:30.095 ","End":"01:33.920","Text":"Really this series behaves in the sense of converge or diverge,"},{"Start":"01:33.920 ","End":"01:36.130","Text":"like n squared over n^5,"},{"Start":"01:36.130 ","End":"01:40.960","Text":"like the sum of 1 cubed."},{"Start":"01:42.980 ","End":"01:49.055","Text":"I\u0027ll write the n goes from 1 to infinity just to be correct here."},{"Start":"01:49.055 ","End":"01:51.785","Text":"This is actually a p-series."},{"Start":"01:51.785 ","End":"01:57.590","Text":"Remember p-series is Sigma 1^p,"},{"Start":"01:57.590 ","End":"02:02.785","Text":"and it converges if p is bigger than 1."},{"Start":"02:02.785 ","End":"02:06.795","Text":"In this case, p is bigger than 1,"},{"Start":"02:06.795 ","End":"02:11.310","Text":"and so we know that this is"},{"Start":"02:11.310 ","End":"02:16.700","Text":"a convergent series, and because this behaves roughly like this,"},{"Start":"02:16.700 ","End":"02:19.870","Text":"we\u0027re expecting this to converge."},{"Start":"02:19.870 ","End":"02:25.740","Text":"What we\u0027re going to do is take this as the a_n,"},{"Start":"02:25.740 ","End":"02:27.830","Text":"I don\u0027t want to copy it again,"},{"Start":"02:27.830 ","End":"02:32.465","Text":"but this bit will be a_n equals this,"},{"Start":"02:32.465 ","End":"02:39.675","Text":"and b_n will equal 1 cubed."},{"Start":"02:39.675 ","End":"02:46.620","Text":"Now what we have to do is look at the limit of a_n/b_n."},{"Start":"02:46.760 ","End":"02:50.630","Text":"If this goes to something which is"},{"Start":"02:50.630 ","End":"02:54.630","Text":"either finite"},{"Start":"02:56.600 ","End":"03:01.770","Text":"or 0,"},{"Start":"03:01.770 ","End":"03:04.355","Text":"I mean, cases 1 and 2 are both good for us."},{"Start":"03:04.355 ","End":"03:07.625","Text":"If I have some finite number but not 0,"},{"Start":"03:07.625 ","End":"03:10.170","Text":"then this is a double arrow,"},{"Start":"03:10.170 ","End":"03:11.430","Text":"and if b_n converges,"},{"Start":"03:11.430 ","End":"03:15.920","Text":"which it does, then a_n will converge and the 0 case is also good for us."},{"Start":"03:15.920 ","End":"03:21.335","Text":"If we get anything greater or equal to 0 but not infinity,"},{"Start":"03:21.335 ","End":"03:22.835","Text":"so let\u0027s compute that."},{"Start":"03:22.835 ","End":"03:25.400","Text":"What we need now is the limit,"},{"Start":"03:25.400 ","End":"03:28.085","Text":"as n goes to infinity."},{"Start":"03:28.085 ","End":"03:32.480","Text":"I\u0027ll just write a_n/b_n here first,"},{"Start":"03:32.480 ","End":"03:35.240","Text":"because that\u0027s what the theorem is."},{"Start":"03:35.240 ","End":"03:40.370","Text":"The theorem, by the way, deals with the positive series,"},{"Start":"03:40.370 ","End":"03:46.895","Text":"at least the b_n have to be strictly positive and the a_n\u0027s could also be 0."},{"Start":"03:46.895 ","End":"03:54.804","Text":"Anyway, a_n/b_n is equal to the limit."},{"Start":"03:54.804 ","End":"03:59.450","Text":"Now if I divide this by this,"},{"Start":"03:59.450 ","End":"04:04.760","Text":"dividing by 1 cubed means I put an n cubed in the numerator."},{"Start":"04:04.760 ","End":"04:10.230","Text":"This is the limit as n goes to infinity of,"},{"Start":"04:10.230 ","End":"04:12.425","Text":"that\u0027s a dividing line."},{"Start":"04:12.425 ","End":"04:15.305","Text":"Multiply everything by n cubed."},{"Start":"04:15.305 ","End":"04:23.285","Text":"5n^5 multiplying by n cubed, that\u0027s better,"},{"Start":"04:23.285 ","End":"04:29.459","Text":"plus 4n^4 plus 8n cubed,"},{"Start":"04:29.459 ","End":"04:31.680","Text":"and denominator stays the same."},{"Start":"04:31.680 ","End":"04:38.960","Text":"14n^5 plus 10n cubed,"},{"Start":"04:38.960 ","End":"04:45.040","Text":"plus 4n squared, plus 10n plus 1."},{"Start":"04:45.040 ","End":"04:48.860","Text":"Now, this limit is familiar,"},{"Start":"04:48.860 ","End":"04:54.205","Text":"we divide top and bottom by n^5."},{"Start":"04:54.205 ","End":"04:56.960","Text":"Let me just scroll here."},{"Start":"04:56.960 ","End":"05:02.419","Text":"We have the limit as n goes to infinity,"},{"Start":"05:02.419 ","End":"05:06.335","Text":"dividing everything by n^5 we have 5,"},{"Start":"05:06.335 ","End":"05:09.140","Text":"plus 4,"},{"Start":"05:09.140 ","End":"05:11.375","Text":"because n^4^5,"},{"Start":"05:11.375 ","End":"05:20.990","Text":"plus 8 squared divided by 14 plus,"},{"Start":"05:20.990 ","End":"05:24.810","Text":"here we have 10n cubed"},{"Start":"05:28.390 ","End":"05:35.040","Text":"divided by n^5 means 10 squared,"},{"Start":"05:35.040 ","End":"05:47.445","Text":"that\u0027s right, plus n squared over n^5 is 4 cubed plus 10^4,"},{"Start":"05:47.445 ","End":"05:51.250","Text":"plus 1^5."},{"Start":"05:51.550 ","End":"05:59.180","Text":"Now this limit is straightforward because this goes to 0,"},{"Start":"05:59.180 ","End":"06:00.800","Text":"this goes to 0,"},{"Start":"06:00.800 ","End":"06:02.360","Text":"denominators will go to infinity,"},{"Start":"06:02.360 ","End":"06:07.460","Text":"this goes to 0, this goes to 0, this and this."},{"Start":"06:07.460 ","End":"06:11.425","Text":"The limit is 5/14,"},{"Start":"06:11.425 ","End":"06:16.300","Text":"and it satisfies the condition"},{"Start":"06:16.330 ","End":"06:23.840","Text":"of being between 0 and infinity, and b_n, we know converges because like I said,"},{"Start":"06:23.840 ","End":"06:33.715","Text":"it\u0027s the p-series, with p being bigger than 1."},{"Start":"06:33.715 ","End":"06:40.080","Text":"Because 3 is bigger than 1. That\u0027s the b_n."},{"Start":"06:40.080 ","End":"06:45.105","Text":"B_n converges and therefore a_n also converges."},{"Start":"06:45.105 ","End":"06:53.440","Text":"We\u0027ll write that the original series, converges. That\u0027s the answer."}],"ID":10538},{"Watched":false,"Name":"Exercise 7 Part k","Duration":"5m 52s","ChapterTopicVideoID":10212,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10212.jpeg","UploadDate":"2017-08-22T04:40:32.9130000","DurationForVideoObject":"PT5M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"Here we have an infinite series."},{"Start":"00:02.580 ","End":"00:05.895","Text":"We see it from 1 to infinity."},{"Start":"00:05.895 ","End":"00:12.645","Text":"This expression, we have to decide if it converges or diverges."},{"Start":"00:12.645 ","End":"00:14.610","Text":"Not that it really matters,"},{"Start":"00:14.610 ","End":"00:17.745","Text":"but we could actually write this in a shorthand way."},{"Start":"00:17.745 ","End":"00:21.435","Text":"The numerator is actually n factorial,"},{"Start":"00:21.435 ","End":"00:25.695","Text":"and the denominator is n^n."},{"Start":"00:25.695 ","End":"00:34.470","Text":"I know it\u0027s the power of n because each n goes with its corresponding counterpart here."},{"Start":"00:34.470 ","End":"00:35.520","Text":"There\u0027s n terms here,"},{"Start":"00:35.520 ","End":"00:36.990","Text":"there\u0027s going to be n^n here."},{"Start":"00:36.990 ","End":"00:41.660","Text":"Anyway, we\u0027re going to use the limit comparison test."},{"Start":"00:41.660 ","End":"00:45.830","Text":"There are other limit tests for doing this,"},{"Start":"00:45.830 ","End":"00:48.920","Text":"but in this case, we\u0027re going to practice a limit comparison test,"},{"Start":"00:48.920 ","End":"00:50.750","Text":"which is written here,"},{"Start":"00:50.750 ","End":"00:58.700","Text":"which basically involves comparing our series with another series,"},{"Start":"00:58.700 ","End":"01:01.090","Text":"but we have to know in advance."},{"Start":"01:01.090 ","End":"01:05.750","Text":"We have to make a guess as whether to expect it to converge or diverge."},{"Start":"01:05.750 ","End":"01:07.400","Text":"Occasionally, we\u0027ll make the wrong guess,"},{"Start":"01:07.400 ","End":"01:09.860","Text":"and we\u0027ll get stuck, and then we\u0027ll try the other way,"},{"Start":"01:09.860 ","End":"01:12.525","Text":"and I, from experience,"},{"Start":"01:12.525 ","End":"01:16.450","Text":"believe that this actually will converge."},{"Start":"01:16.670 ","End":"01:19.370","Text":"1 way of looking at it is,"},{"Start":"01:19.370 ","End":"01:25.965","Text":"if you just look at the first 2 terms,"},{"Start":"01:25.965 ","End":"01:29.385","Text":"top and bottom, what we have is,"},{"Start":"01:29.385 ","End":"01:32.680","Text":"here is 2 over n squared."},{"Start":"01:32.900 ","End":"01:39.710","Text":"The completion only makes"},{"Start":"01:39.710 ","End":"01:44.285","Text":"it smaller because each thing above is less than or equal to it,"},{"Start":"01:44.285 ","End":"01:47.490","Text":"the corresponding term below."},{"Start":"01:48.530 ","End":"01:58.180","Text":"By the way, I can assume that n is bigger or equal to 3 because,"},{"Start":"02:03.980 ","End":"02:10.380","Text":"well, the limit doesn\u0027t make any difference if I start n from 3."},{"Start":"02:10.380 ","End":"02:13.355","Text":"When I take the limit of something,"},{"Start":"02:13.355 ","End":"02:19.635","Text":"I\u0027m going to assume that n starts from 3."},{"Start":"02:19.635 ","End":"02:25.700","Text":"Now, if this is what I want to compare it to so that\u0027s going to be my comparison series."},{"Start":"02:25.700 ","End":"02:28.450","Text":"This is going to be my a_n,"},{"Start":"02:28.450 ","End":"02:30.730","Text":"is this expression here,"},{"Start":"02:30.730 ","End":"02:34.150","Text":"and I\u0027m going to compare it to the b_n."},{"Start":"02:34.150 ","End":"02:37.175","Text":"It will be 1 times 2 over n times n,"},{"Start":"02:37.175 ","End":"02:40.730","Text":"which is 2 over n squared."},{"Start":"02:40.730 ","End":"02:43.920","Text":"That\u0027s going to be my b_n."},{"Start":"02:44.300 ","End":"02:54.525","Text":"This is the comparison series and what we\u0027re going to show is that, well,"},{"Start":"02:54.525 ","End":"02:58.730","Text":"a_n over b_n is going to be just the tail bit of this,"},{"Start":"02:58.730 ","End":"03:00.560","Text":"and it\u0027s less than or equal to 1,"},{"Start":"03:00.560 ","End":"03:04.520","Text":"so the limit will certainly be less than 1,"},{"Start":"03:04.520 ","End":"03:13.755","Text":"and then we\u0027ll be in good shape because if b_n we know converges,"},{"Start":"03:13.755 ","End":"03:16.125","Text":"and so we\u0027ll get that a_n converges."},{"Start":"03:16.125 ","End":"03:19.275","Text":"Ah, you might ask, \"How do I know that b_n converges?\""},{"Start":"03:19.275 ","End":"03:22.005","Text":"Because the p series,"},{"Start":"03:22.005 ","End":"03:28.780","Text":"the sum of 1 over n^p,"},{"Start":"03:28.780 ","End":"03:32.735","Text":"it converges provided that p is bigger than 1."},{"Start":"03:32.735 ","End":"03:34.490","Text":"For example, in our case,"},{"Start":"03:34.490 ","End":"03:36.265","Text":"if p equals 2,"},{"Start":"03:36.265 ","End":"03:38.190","Text":"then it\u0027s certainly bigger than 1,"},{"Start":"03:38.190 ","End":"03:39.510","Text":"so this series converges."},{"Start":"03:39.510 ","End":"03:41.685","Text":"The constant 2 makes no difference."},{"Start":"03:41.685 ","End":"03:44.520","Text":"This is what I\u0027m going to take as the b_n,"},{"Start":"03:44.520 ","End":"03:50.200","Text":"and now we have to look at the limit of a_n over b_n."},{"Start":"03:50.200 ","End":"03:58.015","Text":"The limit as n goes to infinity of a_n over b_n."},{"Start":"03:58.015 ","End":"03:59.900","Text":"Like I said, I\u0027m assuming that n is"},{"Start":"03:59.900 ","End":"04:06.090","Text":"at least 3 because the limit of finite number of terms doesn\u0027t matter,"},{"Start":"04:06.090 ","End":"04:15.120","Text":"then it makes sense to say that a_n over b_n is just 3 times whatever,"},{"Start":"04:15.120 ","End":"04:17.475","Text":"I\u0027ll write 4 times and so on,"},{"Start":"04:17.475 ","End":"04:26.580","Text":"up to n over n times n times dot dot dot up to n because it\u0027s divided by this."},{"Start":"04:26.580 ","End":"04:30.420","Text":"The 1 times 2 over n times n just cancels out."},{"Start":"04:31.870 ","End":"04:39.060","Text":"I don\u0027t know what the limit actually is,"},{"Start":"04:39.060 ","End":"04:41.940","Text":"but it\u0027s certainly less than or equal"},{"Start":"04:41.940 ","End":"04:50.785","Text":"to 1"},{"Start":"04:50.785 ","End":"04:52.555","Text":"because the numerator"},{"Start":"04:52.555 ","End":"04:57.710","Text":"is less than or equal to the denominator."},{"Start":"04:58.350 ","End":"05:01.915","Text":"I seem to remember from somewhere that this limit,"},{"Start":"05:01.915 ","End":"05:04.960","Text":"or at least this limit in fact,"},{"Start":"05:04.960 ","End":"05:07.450","Text":"of this a_n is actually 0,"},{"Start":"05:07.450 ","End":"05:10.315","Text":"and so it would be for b_n also,"},{"Start":"05:10.315 ","End":"05:16.875","Text":"but whether it\u0027s 0 or bigger than 0 in both cases, it\u0027s okay."},{"Start":"05:16.875 ","End":"05:19.380","Text":"Because in each case, if b_n converges,"},{"Start":"05:19.380 ","End":"05:21.760","Text":"then a_n converges similarly here."},{"Start":"05:21.760 ","End":"05:26.240","Text":"This could be some number less than 1,"},{"Start":"05:26.240 ","End":"05:27.845","Text":"or it could even be 0."},{"Start":"05:27.845 ","End":"05:29.404","Text":"But in both cases,"},{"Start":"05:29.404 ","End":"05:37.675","Text":"it implies that the series"},{"Start":"05:37.675 ","End":"05:42.840","Text":"a_n converges because the series b_n converges."},{"Start":"05:42.840 ","End":"05:51.640","Text":"It\u0027s a p series with p bigger than 1. We\u0027re done."}],"ID":10539},{"Watched":false,"Name":"Exercise 7 Part l","Duration":"6m 26s","ChapterTopicVideoID":10213,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10213.jpeg","UploadDate":"2017-08-22T04:40:49.1230000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.520","Text":"We have here this infinite series."},{"Start":"00:02.520 ","End":"00:08.340","Text":"Notice that it goes from 1 to infinity and we have to decide if it converges or diverges."},{"Start":"00:08.340 ","End":"00:10.140","Text":"There might be many ways to do this,"},{"Start":"00:10.140 ","End":"00:13.680","Text":"but we\u0027re going to use the limit comparison test,"},{"Start":"00:13.680 ","End":"00:15.660","Text":"which is summarized in the box here."},{"Start":"00:15.660 ","End":"00:18.375","Text":"I\u0027m not going to go through the whole thing again."},{"Start":"00:18.375 ","End":"00:21.795","Text":"The idea is to compare it to another series."},{"Start":"00:21.795 ","End":"00:25.320","Text":"This will be say, an and the n will find another series"},{"Start":"00:25.320 ","End":"00:32.040","Text":"bn and tie the convergence or divergence of this to the other series."},{"Start":"00:32.040 ","End":"00:36.690","Text":"Ideally, we have to have a good guess as to what we expect,"},{"Start":"00:36.690 ","End":"00:39.965","Text":"whether it\u0027s going to converge or diverge."},{"Start":"00:39.965 ","End":"00:42.605","Text":"1 rule of thumb for this,"},{"Start":"00:42.605 ","End":"00:44.285","Text":"especially when you have polynomials,"},{"Start":"00:44.285 ","End":"00:49.070","Text":"is just to look at the highest powers and say, this behaves like,"},{"Start":"00:49.070 ","End":"00:54.500","Text":"I\u0027ll write that with a squiggly line, behaves roughly like,"},{"Start":"00:54.500 ","End":"00:58.905","Text":"you can even forget the constant,"},{"Start":"00:58.905 ","End":"01:05.870","Text":"n cubed over square root of n to the 10,"},{"Start":"01:05.870 ","End":"01:08.045","Text":"with a sigma if you like."},{"Start":"01:08.045 ","End":"01:11.280","Text":"This series roughly behaves like this."},{"Start":"01:11.280 ","End":"01:14.540","Text":"If we can see if this converges or diverges,"},{"Start":"01:14.540 ","End":"01:16.315","Text":"we\u0027ll say something about this."},{"Start":"01:16.315 ","End":"01:20.620","Text":"This square root of n to the 10 is n to the 5."},{"Start":"01:20.620 ","End":"01:26.400","Text":"This behaves like roughly like a sum of 1 over n squared."},{"Start":"01:26.400 ","End":"01:28.470","Text":"This happens to be a p-series."},{"Start":"01:28.470 ","End":"01:32.340","Text":"A p-series is the sum of 1 over n to the"},{"Start":"01:32.340 ","End":"01:39.720","Text":"p. We know it converges if p is bigger than 1."},{"Start":"01:39.720 ","End":"01:41.930","Text":"In our case, p is 2,"},{"Start":"01:41.930 ","End":"01:44.585","Text":"which is certainly bigger than 1."},{"Start":"01:44.585 ","End":"01:48.260","Text":"We expect this 1 to be convergent."},{"Start":"01:48.260 ","End":"01:56.030","Text":"If we apply these ratios here to this over this,"},{"Start":"01:56.030 ","End":"01:58.940","Text":"we should be able to conclude this is convergent to."},{"Start":"01:58.940 ","End":"02:06.570","Text":"I\u0027ll let this be an and this will be bn."},{"Start":"02:06.570 ","End":"02:11.960","Text":"What we need to do is to look at the limit as n goes to"},{"Start":"02:11.960 ","End":"02:17.630","Text":"infinity of an over bn."},{"Start":"02:17.630 ","End":"02:20.375","Text":"Because bn is convergent."},{"Start":"02:20.375 ","End":"02:27.709","Text":"If this limit is either 0 or some finite number between 0 and infinity,"},{"Start":"02:27.709 ","End":"02:29.405","Text":"both cases are good."},{"Start":"02:29.405 ","End":"02:30.755","Text":"If we get this,"},{"Start":"02:30.755 ","End":"02:35.530","Text":"then because bn converges and this is a double arrow, then an converges."},{"Start":"02:35.530 ","End":"02:39.890","Text":"If we get 0, then also because bn converges an converges,"},{"Start":"02:39.890 ","End":"02:43.130","Text":"so if this limit is 0 or positive,"},{"Start":"02:43.130 ","End":"02:45.215","Text":"as long as it\u0027s not infinite then we\u0027re all right."},{"Start":"02:45.215 ","End":"02:47.950","Text":"Let\u0027s see what this limit is."},{"Start":"02:47.950 ","End":"02:50.580","Text":"Now, an over bn."},{"Start":"02:50.580 ","End":"02:53.970","Text":"When we divide by bn,"},{"Start":"02:53.970 ","End":"02:56.715","Text":"we\u0027re dividing by 1 over n squared."},{"Start":"02:56.715 ","End":"03:01.720","Text":"Dividing by 1 over n squared is like putting n squared in the numerator."},{"Start":"03:01.720 ","End":"03:07.150","Text":"What I have is 2 and I\u0027ll just raise all the powers by 2."},{"Start":"03:07.150 ","End":"03:11.220","Text":"2n to the 5 plus n to the 4,"},{"Start":"03:11.220 ","End":"03:19.985","Text":"plus 4n cubed plus n squared over same thing."},{"Start":"03:19.985 ","End":"03:22.170","Text":"Boring, but I\u0027ll write it out."},{"Start":"03:22.170 ","End":"03:26.625","Text":"N the 10 plus 4n to the 4,"},{"Start":"03:26.625 ","End":"03:31.460","Text":"plus n squared plus n plus 1."},{"Start":"03:31.460 ","End":"03:39.520","Text":"I guess I should have mentioned that the theorem applies to non-negative series."},{"Start":"03:39.520 ","End":"03:40.975","Text":"All the terms are non-negative."},{"Start":"03:40.975 ","End":"03:43.570","Text":"Actually for B, they have to be strictly positive"},{"Start":"03:43.570 ","End":"03:48.445","Text":"and 1 over n squared is, of course, positive."},{"Start":"03:48.445 ","End":"03:51.595","Text":"Whenever we\u0027re from 1 to infinity,"},{"Start":"03:51.595 ","End":"03:53.380","Text":"I should really be writing these things."},{"Start":"03:53.380 ","End":"03:58.040","Text":"You get lazy after awhile and stop writing the limits."},{"Start":"03:59.220 ","End":"04:01.775","Text":"What do we do here?"},{"Start":"04:01.775 ","End":"04:08.450","Text":"The standard trick, divide top and bottom by n to the 5."},{"Start":"04:08.700 ","End":"04:12.225","Text":"If we do that,"},{"Start":"04:12.225 ","End":"04:17.890","Text":"we\u0027ll get the limit as n goes to infinity of,"},{"Start":"04:19.010 ","End":"04:22.465","Text":"on the top, dividing by n to the 5,"},{"Start":"04:22.465 ","End":"04:30.450","Text":"I get 2 plus 1 over n plus 4 over n squared,"},{"Start":"04:30.450 ","End":"04:33.320","Text":"see, n cubed over n to the 5 is 1 over n squared,"},{"Start":"04:33.320 ","End":"04:37.325","Text":"and so on, 1 over n cubed."},{"Start":"04:37.325 ","End":"04:41.335","Text":"On denominator, because it\u0027s a square root,"},{"Start":"04:41.335 ","End":"04:47.360","Text":"when I divide by n to the 5 I really have to divide by n to the 10,"},{"Start":"04:47.360 ","End":"04:50.705","Text":"because n to the 5 is the square root of n to the 10th."},{"Start":"04:50.705 ","End":"04:57.980","Text":"What I get under the square root sign is 1 plus 4 over 4 minus 10 is minus 6,"},{"Start":"04:57.980 ","End":"05:00.990","Text":"so it\u0027s 4 over n to the 6,"},{"Start":"05:00.990 ","End":"05:05.260","Text":"plus 1 over n to the 8."},{"Start":"05:05.260 ","End":"05:07.940","Text":"That\u0027s n squared over n to the 10,"},{"Start":"05:07.940 ","End":"05:12.005","Text":"plus 1 over n to the 9th,"},{"Start":"05:12.005 ","End":"05:15.110","Text":"plus 1 over n to the 10."},{"Start":"05:15.110 ","End":"05:25.520","Text":"Just extend these lines here and here. 10 here."},{"Start":"05:25.520 ","End":"05:30.710","Text":"Now, everything\u0027s easy because all these things go to 0."},{"Start":"05:30.710 ","End":"05:34.640","Text":"They have an n or a power of n in the denominator."},{"Start":"05:34.640 ","End":"05:45.480","Text":"This limit is going to equal just 2 over the square root of 1, which is 2."},{"Start":"05:45.480 ","End":"05:53.370","Text":"We are in the case where the limit is finite,"},{"Start":"05:53.370 ","End":"05:56.060","Text":"could be 0 as long as it\u0027s less than infinity."},{"Start":"05:56.060 ","End":"05:58.940","Text":"Then because bn converges,"},{"Start":"05:58.940 ","End":"06:00.545","Text":"remember the p series,"},{"Start":"06:00.545 ","End":"06:02.960","Text":"then an converges too,"},{"Start":"06:02.960 ","End":"06:11.005","Text":"so we have that the sum of an converges."},{"Start":"06:11.005 ","End":"06:16.560","Text":"That\u0027s because the sum of bn converges."},{"Start":"06:16.560 ","End":"06:23.960","Text":"The limit comparison test tells us that this is so."},{"Start":"06:23.960 ","End":"06:26.310","Text":"We are done."}],"ID":10540},{"Watched":false,"Name":"Exercise 7 Part m","Duration":"7m 53s","ChapterTopicVideoID":10214,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10214.jpeg","UploadDate":"2017-08-22T04:41:09.5630000","DurationForVideoObject":"PT7M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.230","Text":"In this exercise, we are given an infinite series,"},{"Start":"00:04.230 ","End":"00:06.945","Text":"infinite because you can see the infinity there"},{"Start":"00:06.945 ","End":"00:12.660","Text":"and we just have to decide if it converges or diverges."},{"Start":"00:12.660 ","End":"00:15.540","Text":"There may be more than 1 way of doing this,"},{"Start":"00:15.540 ","End":"00:18.990","Text":"but here we\u0027re going to use the limit ratio test,"},{"Start":"00:18.990 ","End":"00:22.810","Text":"which is summarized inside the box."},{"Start":"00:22.880 ","End":"00:25.650","Text":"I\u0027m not going to go over the whole thing."},{"Start":"00:25.650 ","End":"00:30.710","Text":"But basically, we have to find another simpler theories that we"},{"Start":"00:30.710 ","End":"00:37.500","Text":"know converges or diverges and compute the limit of the ratios."},{"Start":"00:37.510 ","End":"00:41.980","Text":"We have to expect, initially,"},{"Start":"00:41.980 ","End":"00:46.535","Text":"make a good guess as to whether it does converge or diverge."},{"Start":"00:46.535 ","End":"00:48.290","Text":"The way to do this,"},{"Start":"00:48.290 ","End":"00:55.590","Text":"I\u0027m just going to get some more space here, is to look at the leading terms,"},{"Start":"00:55.590 ","End":"00:57.800","Text":"because really they are what count when you have"},{"Start":"00:57.800 ","End":"01:01.295","Text":"a rational or something similar, polynomials,"},{"Start":"01:01.295 ","End":"01:02.960","Text":"the leading term is what counts."},{"Start":"01:02.960 ","End":"01:05.330","Text":"So I write a little squiggly sign,"},{"Start":"01:05.330 ","End":"01:08.760","Text":"meaning it behaves like"},{"Start":"01:08.760 ","End":"01:15.700","Text":"the series where I just take the leading term,"},{"Start":"01:15.700 ","End":"01:21.950","Text":"I can say, 4 n over the square root of n to the 4th."},{"Start":"01:22.490 ","End":"01:26.860","Text":"Also, constants don\u0027t affect convergence or divergence."},{"Start":"01:26.860 ","End":"01:33.040","Text":"What I really have here is n over n squared, so it behaves as far as"},{"Start":"01:33.040 ","End":"01:40.520","Text":"convergence or divergence, like 1 over n. Now, sigma."},{"Start":"01:40.710 ","End":"01:45.055","Text":"Now this series is going to be the comparison."},{"Start":"01:45.055 ","End":"01:48.720","Text":"Well, the bn, if this is here,"},{"Start":"01:48.720 ","End":"01:54.420","Text":"is an, this is going to be our bn."},{"Start":"01:54.420 ","End":"01:57.650","Text":"Now, this is a harmonic series,"},{"Start":"01:57.650 ","End":"01:59.300","Text":"the classic harmonic series,"},{"Start":"01:59.300 ","End":"02:06.240","Text":"the sum of 1 over n and it is known that this 1 diverges."},{"Start":"02:07.780 ","End":"02:13.145","Text":"Like I said, this is the classic harmonic series."},{"Start":"02:13.145 ","End":"02:15.990","Text":"But in case you\u0027ve forgotten about it,"},{"Start":"02:15.990 ","End":"02:21.815","Text":"I can give you 1 explanation of why it diverges using the p-series test."},{"Start":"02:21.815 ","End":"02:27.665","Text":"For example, if we have the sum of 1 over n to the power of p,"},{"Start":"02:27.665 ","End":"02:31.610","Text":"where n goes from whatever to infinities,"},{"Start":"02:31.610 ","End":"02:36.325","Text":"say 1 to infinity, then this converges"},{"Start":"02:36.325 ","End":"02:43.060","Text":"when p is bigger than 1 and diverges when p is less than or equal,"},{"Start":"02:43.060 ","End":"02:49.580","Text":"and here is the convergence, and here divergence."},{"Start":"02:50.330 ","End":"02:53.310","Text":"P is 1, so diverges."},{"Start":"02:53.310 ","End":"02:57.070","Text":"Now we know what to expect,"},{"Start":"02:57.070 ","End":"03:00.940","Text":"we have to actually prove it using some formal method,"},{"Start":"03:00.940 ","End":"03:02.823","Text":"which is in this case this theorem."},{"Start":"03:02.823 ","End":"03:06.655","Text":"Can\u0027t just leave the answer and say, yeah, it diverges."},{"Start":"03:06.655 ","End":"03:12.120","Text":"Let\u0027s take the limit as n goes to"},{"Start":"03:12.120 ","End":"03:19.660","Text":"infinity of an over bn and see what we get."},{"Start":"03:19.760 ","End":"03:30.620","Text":"If we get that a is between 0 and infinity,"},{"Start":"03:30.620 ","End":"03:34.700","Text":"then this converges if and only if this"},{"Start":"03:34.700 ","End":"03:41.760","Text":"converges which implies that if this doesn\u0027t converge, this doesn\u0027t converge."},{"Start":"03:42.890 ","End":"03:48.110","Text":"Also, if we get the answer of infinity,"},{"Start":"03:48.110 ","End":"03:50.240","Text":"that\u0027s also going to be good for us."},{"Start":"03:50.240 ","End":"03:54.155","Text":"As long as we don\u0027t get the 0 case, we\u0027re okay."},{"Start":"03:54.155 ","End":"03:55.780","Text":"Let\u0027s see then."},{"Start":"03:55.780 ","End":"04:00.730","Text":"This is equal to the limit as n goes to infinity."},{"Start":"04:00.730 ","End":"04:05.340","Text":"Now an over bn is"},{"Start":"04:05.340 ","End":"04:10.945","Text":"this thing over 1 over n means you multiply by n because dividing by 1 over n,"},{"Start":"04:10.945 ","End":"04:13.720","Text":"so I put an extra n in the numerator."},{"Start":"04:13.720 ","End":"04:19.630","Text":"It\u0027s n times 4n plus"},{"Start":"04:19.630 ","End":"04:25.990","Text":"5 over same thing on the denominator."},{"Start":"04:25.990 ","End":"04:35.680","Text":"Square root of n to the 4th plus 2n cubed plus n squared plus 4n plus 1."},{"Start":"04:35.680 ","End":"04:39.755","Text":"Yes, this is a rather tedious denominator."},{"Start":"04:39.755 ","End":"04:44.040","Text":"Let\u0027s see what this equals,"},{"Start":"04:44.040 ","End":"04:46.970","Text":"so we\u0027ll continue over here."},{"Start":"04:46.970 ","End":"04:50.000","Text":"Tell you what,"},{"Start":"04:50.000 ","End":"04:55.240","Text":"let me just rewrite the numerator,"},{"Start":"04:55.240 ","End":"04:57.470","Text":"don\u0027t want to copy the whole thing again,"},{"Start":"04:57.470 ","End":"05:05.280","Text":"save a step and rewrite this as 4n squared plus 5n, okay?"},{"Start":"05:06.940 ","End":"05:14.645","Text":"What, I\u0027m going to do now is divide numerator and denominator by the same thing."},{"Start":"05:14.645 ","End":"05:17.580","Text":"That\u0027s this highest power n squared,"},{"Start":"05:17.580 ","End":"05:22.715","Text":"if I multiply the numerator by 1 over n squared,"},{"Start":"05:22.715 ","End":"05:27.380","Text":"and I also multiply the denominator by 1 over n squared,"},{"Start":"05:27.380 ","End":"05:32.945","Text":"then the fraction is unchanged and it\u0027s just algebra."},{"Start":"05:32.945 ","End":"05:37.630","Text":"This is equal to the limit,"},{"Start":"05:37.630 ","End":"05:39.860","Text":"as n goes to infinity."},{"Start":"05:39.860 ","End":"05:42.740","Text":"Now, if I divide the top by n squared,"},{"Start":"05:42.740 ","End":"05:50.120","Text":"I\u0027m left with 4 plus 5 over n. Each term is divided by n squared."},{"Start":"05:50.120 ","End":"05:57.810","Text":"On the bottom, this 1 over n squared is actually the same as,"},{"Start":"05:57.810 ","End":"06:04.025","Text":"I want to put it inside so this is the same as the square root of n to the 4th,"},{"Start":"06:04.025 ","End":"06:09.570","Text":"and then if I divide inside by n to the 4th,"},{"Start":"06:09.570 ","End":"06:11.535","Text":"that will be the same thing."},{"Start":"06:11.535 ","End":"06:16.100","Text":"What I get is here, the square root of,"},{"Start":"06:16.100 ","End":"06:18.195","Text":"everything in here divided by n to the 4th,"},{"Start":"06:18.195 ","End":"06:26.735","Text":"so it\u0027s 1 plus 2 over n because n cubed over n to the 4th, and so on,"},{"Start":"06:26.735 ","End":"06:31.850","Text":"plus 1 over n squared plus 4"},{"Start":"06:31.850 ","End":"06:37.900","Text":"over n cubed plus 1 over n to the 4th."},{"Start":"06:37.900 ","End":"06:41.025","Text":"Because n goes to infinity,"},{"Start":"06:41.025 ","End":"06:44.685","Text":"5 over n goes to 0,"},{"Start":"06:44.685 ","End":"06:47.540","Text":"2 over n goes to 0,"},{"Start":"06:47.540 ","End":"06:49.040","Text":"this goes to 0,"},{"Start":"06:49.040 ","End":"06:50.660","Text":"this goes to 0,"},{"Start":"06:50.660 ","End":"06:52.714","Text":"and this goes to 0."},{"Start":"06:52.714 ","End":"06:59.420","Text":"So the answer to this limit is 4 over the square root of 1,"},{"Start":"06:59.420 ","End":"07:02.180","Text":"and that\u0027s equal to 4."},{"Start":"07:02.180 ","End":"07:15.720","Text":"4 is in this case, that\u0027s the a from the theorem,"},{"Start":"07:15.720 ","End":"07:18.260","Text":"and it\u0027s between 0 and infinity,"},{"Start":"07:18.260 ","End":"07:20.960","Text":"which means that, like I said,"},{"Start":"07:20.960 ","End":"07:23.750","Text":"they both converge or they both don\u0027t converge,"},{"Start":"07:23.750 ","End":"07:25.955","Text":"and since bn doesn\u0027t converge,"},{"Start":"07:25.955 ","End":"07:30.620","Text":"an doesn\u0027t converge, so the answer"},{"Start":"07:30.620 ","End":"07:38.050","Text":"is that the original series diverges."},{"Start":"07:38.720 ","End":"07:44.230","Text":"That\u0027s our original sum of an,"},{"Start":"07:45.790 ","End":"07:47.960","Text":"this is the 1 here,"},{"Start":"07:47.960 ","End":"07:52.980","Text":"and that\u0027s the answer. Diverges. We\u0027re done."}],"ID":10541},{"Watched":false,"Name":"Exercise 7 Part n","Duration":"10m 8s","ChapterTopicVideoID":10215,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10215.jpeg","UploadDate":"2017-08-22T04:41:34.4700000","DurationForVideoObject":"PT10M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.409","Text":"In this exercise, we\u0027re given an infinite series."},{"Start":"00:04.409 ","End":"00:07.545","Text":"You can see it\u0027s infinite because of the infinity there,"},{"Start":"00:07.545 ","End":"00:12.390","Text":"and you have to decide if it converges or diverges."},{"Start":"00:12.390 ","End":"00:15.495","Text":"There are probably several ways to do this,"},{"Start":"00:15.495 ","End":"00:18.420","Text":"but I\u0027m going to use the limit comparison test,"},{"Start":"00:18.420 ","End":"00:21.030","Text":"which is summarized in this box,"},{"Start":"00:21.030 ","End":"00:24.180","Text":"and I\u0027m not going to go over the whole thing again."},{"Start":"00:24.180 ","End":"00:32.790","Text":"But the idea is to use the original series and call that say a_n,"},{"Start":"00:32.790 ","End":"00:36.550","Text":"I\u0027ll just get some more space here."},{"Start":"00:38.000 ","End":"00:41.395","Text":"This is going to be my a_n,"},{"Start":"00:41.395 ","End":"00:46.670","Text":"and then I look for a comparison series b_n"},{"Start":"00:46.670 ","End":"00:52.515","Text":"and use b_n to determine whether a_n converges or not,"},{"Start":"00:52.515 ","End":"00:54.890","Text":"b_n will be something much simpler,"},{"Start":"00:54.890 ","End":"00:57.425","Text":"that will be easy to spot."},{"Start":"00:57.425 ","End":"01:00.440","Text":"Now how do I find such a b_n?"},{"Start":"01:00.440 ","End":"01:07.925","Text":"What I do, so first I have to decide to myself whether it\u0027s going to converge or diverge."},{"Start":"01:07.925 ","End":"01:15.779","Text":"The thing to do when you have an expression like a fraction,"},{"Start":"01:15.779 ","End":"01:21.815","Text":"is to look at the most dominant term on the top and on the bottom."},{"Start":"01:21.815 ","End":"01:24.905","Text":"Dominant means goes to infinity fastest."},{"Start":"01:24.905 ","End":"01:28.215","Text":"Obviously, 2^n goes to infinity,"},{"Start":"01:28.215 ","End":"01:30.185","Text":"2 doesn\u0027t go to infinity at all."},{"Start":"01:30.185 ","End":"01:33.770","Text":"So what I\u0027m saying is that this behaves like,"},{"Start":"01:33.770 ","End":"01:40.580","Text":"I\u0027ve just put a squiggly line to mean behaves like or converges like 2^n."},{"Start":"01:40.580 ","End":"01:43.610","Text":"On the denominator, it\u0027s well-known that"},{"Start":"01:43.610 ","End":"01:48.995","Text":"an exponent goes to infinity much faster than just a polynomial."},{"Start":"01:48.995 ","End":"01:51.425","Text":"This is somehow dominant over this."},{"Start":"01:51.425 ","End":"01:54.630","Text":"Even if you don\u0027t understand this"},{"Start":"01:54.630 ","End":"02:00.079","Text":"exactly and you just accepted that exponents go faster than polynomials,"},{"Start":"02:00.079 ","End":"02:02.330","Text":"ultimately, we\u0027re not using this as a proof,"},{"Start":"02:02.330 ","End":"02:05.420","Text":"we\u0027re just using this as our initial guess as to"},{"Start":"02:05.420 ","End":"02:08.720","Text":"whether we have convergence or divergence."},{"Start":"02:08.720 ","End":"02:14.060","Text":"I\u0027m saying this series roughly behaves like this,"},{"Start":"02:14.060 ","End":"02:22.460","Text":"and this in fact is a geometric series because if I let q equals 2/3,"},{"Start":"02:22.460 ","End":"02:29.660","Text":"I can see that this thing is the series q^n,"},{"Start":"02:29.660 ","End":"02:38.690","Text":"and we know that if q is between minus 1 and 1, the series converges."},{"Start":"02:38.690 ","End":"02:41.120","Text":"Now here, our q is 2/3,"},{"Start":"02:41.120 ","End":"02:43.745","Text":"certainly is between minus 1 and 1."},{"Start":"02:43.745 ","End":"02:49.435","Text":"So I know that this series converges."},{"Start":"02:49.435 ","End":"02:53.100","Text":"I\u0027ll just write that down. This 1 converges,"},{"Start":"02:53.100 ","End":"02:57.650","Text":"and I\u0027ll use this as my comparison series."},{"Start":"02:57.650 ","End":"03:00.125","Text":"I didn\u0027t leave room to write. I\u0027ll write it here."},{"Start":"03:00.125 ","End":"03:05.495","Text":"This will be the b_n series."},{"Start":"03:05.495 ","End":"03:07.790","Text":"Now, this is all informal."},{"Start":"03:07.790 ","End":"03:12.620","Text":"The formal way, the formal thing I have to do is to look at"},{"Start":"03:12.620 ","End":"03:19.930","Text":"the limit as n goes to infinity of a_n/b_n."},{"Start":"03:21.110 ","End":"03:25.405","Text":"I just forgot to mention that if this thing works for positive,"},{"Start":"03:25.405 ","End":"03:27.670","Text":"the elements all have to be positive,"},{"Start":"03:27.670 ","End":"03:33.600","Text":"and in fact the a_n could be 0 or positive,"},{"Start":"03:33.600 ","End":"03:35.830","Text":"but b_n has to be strictly positive because it"},{"Start":"03:35.830 ","End":"03:38.410","Text":"appears in the denominator, and this is true."},{"Start":"03:38.410 ","End":"03:43.120","Text":"2/3 to the power of n is certainly positive,"},{"Start":"03:43.120 ","End":"03:49.425","Text":"and 2 to the n minus 2,"},{"Start":"03:49.425 ","End":"03:55.425","Text":"well, yeah, it\u0027s non-negative, certainly."},{"Start":"03:55.425 ","End":"03:57.460","Text":"Even when n equals 1,"},{"Start":"03:57.460 ","End":"04:00.125","Text":"we get 2 minus 2 is 0."},{"Start":"04:00.125 ","End":"04:02.325","Text":"Denominator is positive."},{"Start":"04:02.325 ","End":"04:04.500","Text":"When n is bigger than 1,"},{"Start":"04:04.500 ","End":"04:06.750","Text":"this is certainly positive,"},{"Start":"04:06.750 ","End":"04:08.290","Text":"2^n is bigger than 2."},{"Start":"04:08.290 ","End":"04:14.695","Text":"So yeah, 0 once and then probably positive, and that\u0027s okay."},{"Start":"04:14.695 ","End":"04:18.755","Text":"Let\u0027s just compute this limit and find what this a is,"},{"Start":"04:18.755 ","End":"04:19.970","Text":"which is the limit,"},{"Start":"04:19.970 ","End":"04:23.930","Text":"and see which range it is and what conclusion we draw."},{"Start":"04:23.930 ","End":"04:29.990","Text":"This is equal to the limit as n goes to infinity."},{"Start":"04:29.990 ","End":"04:36.595","Text":"Now, a_n/b_n is going to be"},{"Start":"04:36.595 ","End":"04:40.520","Text":"2^n minus 2 over"},{"Start":"04:40.580 ","End":"04:47.150","Text":"3^n"},{"Start":"04:47.150 ","End":"04:56.220","Text":"plus 2n."},{"Start":"04:56.220 ","End":"05:00.770","Text":"Well, instead of dividing by a fraction,"},{"Start":"05:00.770 ","End":"05:03.425","Text":"I can multiply by the inverse fraction."},{"Start":"05:03.425 ","End":"05:06.875","Text":"So b_n is 2/3 to the power of n,"},{"Start":"05:06.875 ","End":"05:09.890","Text":"so I\u0027ll multiply by the reverse."},{"Start":"05:09.890 ","End":"05:12.305","Text":"Actually, I prefer to use this form."},{"Start":"05:12.305 ","End":"05:14.120","Text":"This is an equals here."},{"Start":"05:14.120 ","End":"05:19.180","Text":"I\u0027m going to just multiply by 3^n/2^n,"},{"Start":"05:19.180 ","End":"05:21.450","Text":"because this is also b_n,"},{"Start":"05:21.450 ","End":"05:24.490","Text":"and dividing, let\u0027s say by a fraction,"},{"Start":"05:24.490 ","End":"05:26.794","Text":"multiplied by the reciprocal."},{"Start":"05:26.794 ","End":"05:29.105","Text":"Let\u0027s see what we can do with this."},{"Start":"05:29.105 ","End":"05:31.350","Text":"I need more space."},{"Start":"05:31.760 ","End":"05:35.970","Text":"I\u0027ll split it up into 2 bits."},{"Start":"05:35.970 ","End":"05:39.100","Text":"Still, the limit is n goes to infinity."},{"Start":"05:39.100 ","End":"05:41.240","Text":"I\u0027ll take this bit with this bit."},{"Start":"05:41.240 ","End":"05:47.240","Text":"So I have 2^n minus 2 over 2^n,"},{"Start":"05:47.240 ","End":"05:50.340","Text":"and the other bit is"},{"Start":"05:50.690 ","End":"05:59.845","Text":"3^n over 3^n plus 2n."},{"Start":"05:59.845 ","End":"06:03.310","Text":"Now we need to do some simplification."},{"Start":"06:03.310 ","End":"06:08.870","Text":"I\u0027m going to divide top and bottom by the exponents."},{"Start":"06:08.870 ","End":"06:13.655","Text":"I\u0027ve got the limit as n goes to infinity."},{"Start":"06:13.655 ","End":"06:16.070","Text":"Now, if I divide it out,"},{"Start":"06:16.070 ","End":"06:23.870","Text":"here I have 2^n/2^n is 1,"},{"Start":"06:23.870 ","End":"06:28.470","Text":"minus, I\u0027ll just leave it as 2/2^n,"},{"Start":"06:28.470 ","End":"06:31.400","Text":"although we could perhaps simplify it."},{"Start":"06:31.400 ","End":"06:33.215","Text":"That\u0027s the first bit."},{"Start":"06:33.215 ","End":"06:39.350","Text":"The second bit, divide top and bottom by 3^n."},{"Start":"06:39.350 ","End":"06:45.400","Text":"I\u0027ve got 1 over 1 plus"},{"Start":"06:45.400 ","End":"06:55.670","Text":"2n/3^n."},{"Start":"06:55.670 ","End":"06:58.770","Text":"Now let\u0027s see what this is."},{"Start":"07:00.320 ","End":"07:02.850","Text":"When n goes to infinity,"},{"Start":"07:02.850 ","End":"07:05.970","Text":"2^n goes to infinity even faster,"},{"Start":"07:05.970 ","End":"07:08.390","Text":"and 2 over infinity is 0."},{"Start":"07:08.390 ","End":"07:13.960","Text":"So this part here goes to 0."},{"Start":"07:14.690 ","End":"07:19.580","Text":"The question is, what happens to this part here?"},{"Start":"07:19.580 ","End":"07:23.390","Text":"Both numerator and denominator go to infinity."},{"Start":"07:23.390 ","End":"07:27.225","Text":"But remember we talked about dominant before?"},{"Start":"07:27.225 ","End":"07:33.300","Text":"Somehow 3^n goes to infinity faster than 2n."},{"Start":"07:33.490 ","End":"07:40.835","Text":"There are several ways to show that this actually also goes to 0."},{"Start":"07:40.835 ","End":"07:46.715","Text":"1 way to do this would be to use L\u0027Hopital\u0027s rule."},{"Start":"07:46.715 ","End":"07:48.860","Text":"I\u0027ll do this at the side."},{"Start":"07:48.860 ","End":"07:55.180","Text":"I could replace the discrete variable n by the continuous variable x,"},{"Start":"07:55.180 ","End":"07:56.820","Text":"and look at the limit,"},{"Start":"07:56.820 ","End":"07:58.460","Text":"notice n goes to infinity,"},{"Start":"07:58.460 ","End":"08:02.820","Text":"but x goes to infinity of 2x/3^x."},{"Start":"08:07.130 ","End":"08:09.360","Text":"Now, as I say,"},{"Start":"08:09.360 ","End":"08:11.795","Text":"because infinity over infinity,"},{"Start":"08:11.795 ","End":"08:20.645","Text":"I can use L\u0027Hopital\u0027s rule and say that this is equal to the limit as x goes to infinity."},{"Start":"08:20.645 ","End":"08:26.285","Text":"Differentiate both numerator and denominator separately."},{"Start":"08:26.285 ","End":"08:30.440","Text":"This is according to L\u0027Hopital,"},{"Start":"08:30.440 ","End":"08:32.405","Text":"I\u0027ll write his name."},{"Start":"08:32.405 ","End":"08:33.980","Text":"It\u0027s a French name."},{"Start":"08:33.980 ","End":"08:38.460","Text":"I think it has a little hat over the o."},{"Start":"08:39.200 ","End":"08:43.370","Text":"We\u0027re talking about the infinity over infinity case."},{"Start":"08:43.370 ","End":"08:46.970","Text":"Differentiate the top, and we get 2."},{"Start":"08:46.970 ","End":"08:51.420","Text":"Differentiate the denominator. I\u0027ll just tell you the answer."},{"Start":"08:51.420 ","End":"08:54.960","Text":"You might have forgotten the rule for a^x,"},{"Start":"08:54.960 ","End":"09:00.095","Text":"it\u0027s just a^x times natural log of a,"},{"Start":"09:00.095 ","End":"09:02.585","Text":"in this case, natural log of 3."},{"Start":"09:02.585 ","End":"09:06.485","Text":"This is a constant, this goes to infinity,"},{"Start":"09:06.485 ","End":"09:12.140","Text":"so this whole thing is equal to 0, this limit."},{"Start":"09:12.140 ","End":"09:15.155","Text":"Now this is also going to 0,"},{"Start":"09:15.155 ","End":"09:18.640","Text":"so what are we left with?"},{"Start":"09:18.640 ","End":"09:22.035","Text":"What we have is, well, this thing\u0027s 0,"},{"Start":"09:22.035 ","End":"09:24.510","Text":"it\u0027s just 1, this is 1,"},{"Start":"09:24.510 ","End":"09:26.415","Text":"the answer is 1."},{"Start":"09:26.415 ","End":"09:32.340","Text":"This is our a from the theorem,"},{"Start":"09:32.340 ","End":"09:37.820","Text":"so if a is between 0 and infinity,"},{"Start":"09:37.820 ","End":"09:41.745","Text":"and if 1of them converges,"},{"Start":"09:41.745 ","End":"09:43.500","Text":"so does the other, b_n,"},{"Start":"09:43.500 ","End":"09:45.195","Text":"in this case converges."},{"Start":"09:45.195 ","End":"09:52.560","Text":"We said b_n converges geometric series and therefore, a_n also converges."},{"Start":"09:53.900 ","End":"09:58.380","Text":"This is our original series, converges."},{"Start":"09:58.380 ","End":"10:00.125","Text":"Now we\u0027ve shown it,"},{"Start":"10:00.125 ","End":"10:01.895","Text":"we just suspected it before,"},{"Start":"10:01.895 ","End":"10:07.530","Text":"but now we\u0027ve demonstrated it and we are done."}],"ID":10542},{"Watched":false,"Name":"Exercise 7 Part o","Duration":"11m 17s","ChapterTopicVideoID":10216,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10216.jpeg","UploadDate":"2017-08-22T04:42:04.1970000","DurationForVideoObject":"PT11M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.445","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.445 ","End":"00:07.195","Text":"an infinite series and now it\u0027s infinite because I see infinity here."},{"Start":"00:07.195 ","End":"00:11.130","Text":"To decide if this series converges or diverges,"},{"Start":"00:11.130 ","End":"00:12.975","Text":"probably many ways to do this,"},{"Start":"00:12.975 ","End":"00:16.005","Text":"I\u0027m going to use the limit comparison test,"},{"Start":"00:16.005 ","End":"00:18.060","Text":"which is summarized here."},{"Start":"00:18.060 ","End":"00:21.285","Text":"I\u0027m not going to go over the whole thing but basically,"},{"Start":"00:21.285 ","End":"00:23.430","Text":"what happens is that we\u0027re going to"},{"Start":"00:23.430 ","End":"00:33.615","Text":"let our original series"},{"Start":"00:33.615 ","End":"00:38.975","Text":"be the a_n in the theorem."},{"Start":"00:38.975 ","End":"00:42.770","Text":"Notice that a_n is bigger or equal to 0,"},{"Start":"00:42.770 ","End":"00:45.515","Text":"by the way, it\u0027s fairly obvious."},{"Start":"00:45.515 ","End":"00:48.245","Text":"We\u0027re going to find some b_n,"},{"Start":"00:48.245 ","End":"00:57.840","Text":"which is a lot simpler than a_n that we know about its convergence or divergence."},{"Start":"00:58.330 ","End":"01:03.185","Text":"We\u0027re going to do this using a limit of a ratio,"},{"Start":"01:03.185 ","End":"01:06.905","Text":"but we\u0027ll get to that but before that,"},{"Start":"01:06.905 ","End":"01:09.080","Text":"we have to have a rough idea what to expect."},{"Start":"01:09.080 ","End":"01:10.970","Text":"We have to decide whether we\u0027re going for"},{"Start":"01:10.970 ","End":"01:15.950","Text":"the converges or diverges because that affects a whole strategy."},{"Start":"01:17.660 ","End":"01:24.840","Text":"We\u0027ll do something approximate just to give us an idea about what to expect."},{"Start":"01:25.160 ","End":"01:29.685","Text":"Let\u0027s look at the sine squared n part."},{"Start":"01:29.685 ","End":"01:39.530","Text":"Now, because the sine function is always between minus 1 and 1,"},{"Start":"01:39.530 ","End":"01:45.080","Text":"I know that sine of n is between minus 1 and 1."},{"Start":"01:45.080 ","End":"01:47.435","Text":"If I square it,"},{"Start":"01:47.435 ","End":"01:50.230","Text":"it\u0027s going to be between 0 and 1,"},{"Start":"01:50.230 ","End":"01:56.015","Text":"so 0 is going to be less than or equal to sine squared n,"},{"Start":"01:56.015 ","End":"02:00.120","Text":"less than or equal to 1 but before we write the 1,"},{"Start":"02:00.120 ","End":"02:01.860","Text":"let\u0027s take care of the 5 also,"},{"Start":"02:01.860 ","End":"02:03.825","Text":"so I put 5 here,"},{"Start":"02:03.825 ","End":"02:09.430","Text":"5 sine squared n is bounded between 0 and 5."},{"Start":"02:09.520 ","End":"02:11.720","Text":"That\u0027s really what I care about."},{"Start":"02:11.720 ","End":"02:16.590","Text":"I\u0027ll just make a note that this is 5 sine squared and is bounded."},{"Start":"02:16.870 ","End":"02:21.020","Text":"In some ways, it could be like a constant as far as"},{"Start":"02:21.020 ","End":"02:26.570","Text":"this approximate math that I\u0027m doing now but let\u0027s just leave the word bounded."},{"Start":"02:26.570 ","End":"02:28.040","Text":"Now, that\u0027s the numerator."},{"Start":"02:28.040 ","End":"02:30.005","Text":"What about the denominator?"},{"Start":"02:30.005 ","End":"02:35.210","Text":"The denominator, which is n factorial,"},{"Start":"02:35.210 ","End":"02:38.975","Text":"is n, n minus 1."},{"Start":"02:38.975 ","End":"02:43.880","Text":"Let\u0027s assume that n is bigger than 3 or 4."},{"Start":"02:43.880 ","End":"02:50.255","Text":"Let\u0027s even say that n is bigger than 10 because first few terms don\u0027t matter."},{"Start":"02:50.255 ","End":"02:55.520","Text":"Just want that to be enough that if I say that this is n, n minus 1,"},{"Start":"02:55.520 ","End":"02:59.300","Text":"n minus 2 times and so times,"},{"Start":"02:59.300 ","End":"03:04.470","Text":"and then times 3 times 2 times 1."},{"Start":"03:04.470 ","End":"03:12.200","Text":"I\u0027m just saying that let\u0027s say n is bigger than 10 or something and this makes sense."},{"Start":"03:12.200 ","End":"03:16.790","Text":"Otherwise, if n is too small I can\u0027t, do you see what I mean?"},{"Start":"03:16.790 ","End":"03:25.090","Text":"Now this, I can break up into 2 parts."},{"Start":"03:27.770 ","End":"03:32.705","Text":"I\u0027ll just put a dividing line here and say,"},{"Start":"03:32.705 ","End":"03:39.245","Text":"this thing is n squared minus n. Remember with polynomials,"},{"Start":"03:39.245 ","End":"03:41.345","Text":"we use the dominant term."},{"Start":"03:41.345 ","End":"03:47.735","Text":"This is roughly like n squared and this thing,"},{"Start":"03:47.735 ","End":"03:51.520","Text":"n minus 2 times more stuff."},{"Start":"03:51.520 ","End":"03:54.290","Text":"It certainly goes to infinity."},{"Start":"03:54.290 ","End":"03:58.940","Text":"Just the n minus 2 alone goes to infinity when n goes to infinity,"},{"Start":"03:58.940 ","End":"04:01.370","Text":"and if there\u0027s more stuff that\u0027s bigger than 1,"},{"Start":"04:01.370 ","End":"04:02.840","Text":"everything is bigger than 1."},{"Start":"04:02.840 ","End":"04:07.690","Text":"Certainly this thing goes to infinity."},{"Start":"04:08.210 ","End":"04:12.725","Text":"What I\u0027m going to do is, first of all,"},{"Start":"04:12.725 ","End":"04:19.550","Text":"I\u0027ll take my b_n to be 1 squared."},{"Start":"04:19.550 ","End":"04:26.705","Text":"I\u0027m expecting it to behave like 1 squared."},{"Start":"04:26.705 ","End":"04:29.270","Text":"Well, because my original series,"},{"Start":"04:29.270 ","End":"04:32.360","Text":"if I rewrite it in this informal,"},{"Start":"04:32.360 ","End":"04:35.470","Text":"I\u0027ve got the numerator which is bounded,"},{"Start":"04:35.470 ","End":"04:41.615","Text":"that\u0027s the a_n, bounded over"},{"Start":"04:41.615 ","End":"04:49.010","Text":"something that behaves like n squared over something that goes to infinity."},{"Start":"04:49.010 ","End":"04:50.585","Text":"This is just informal,"},{"Start":"04:50.585 ","End":"04:57.150","Text":"so I\u0027m going to compare this to 1 squared and the other"},{"Start":"04:57.150 ","End":"05:03.635","Text":"stuff\u0027s only going to help as far as the limit goes, you will see."},{"Start":"05:03.635 ","End":"05:07.860","Text":"You just going to get a feel for this."},{"Start":"05:08.810 ","End":"05:13.940","Text":"Increasing the denominator is only going to make it more convergent."},{"Start":"05:13.940 ","End":"05:16.025","Text":"I\u0027m going to let my b_n,"},{"Start":"05:16.025 ","End":"05:24.325","Text":"I\u0027m going to compare this to the sum n goes from 1 to infinity of b_n,"},{"Start":"05:24.325 ","End":"05:31.270","Text":"which will be just 1 squared and so this will cool my b_n,"},{"Start":"05:31.270 ","End":"05:34.530","Text":"which is strictly positive,"},{"Start":"05:34.530 ","End":"05:38.080","Text":"so that\u0027s okay as far as that goes."},{"Start":"05:38.810 ","End":"05:43.190","Text":"What we\u0027re going to do now is this was imprecise."},{"Start":"05:43.190 ","End":"05:47.000","Text":"This is just to get an idea of what to expect."},{"Start":"05:47.000 ","End":"05:55.200","Text":"I\u0027m going to show you that this is convergent and this behaves like this,"},{"Start":"05:55.200 ","End":"06:01.040","Text":"and behaves like this means that the limit will go to something finite."},{"Start":"06:01.040 ","End":"06:04.040","Text":"Then we\u0027ll say that a_n is convergent too."},{"Start":"06:04.040 ","End":"06:07.520","Text":"Now why is b_n convergent?"},{"Start":"06:07.520 ","End":"06:09.845","Text":"Let me write that."},{"Start":"06:09.845 ","End":"06:12.950","Text":"Say up here, I\u0027ve got some space,"},{"Start":"06:12.950 ","End":"06:17.230","Text":"the sum of 1 squared,"},{"Start":"06:17.230 ","End":"06:22.315","Text":"many ways to see it, what easiest probably is looking at it as a p series."},{"Start":"06:22.315 ","End":"06:27.690","Text":"Compare it to look at the sum of 1 to"},{"Start":"06:27.690 ","End":"06:32.310","Text":"the power of p. You might want to refresh your memory,"},{"Start":"06:32.310 ","End":"06:34.690","Text":"look up your notes on p series,"},{"Start":"06:34.690 ","End":"06:41.360","Text":"that this thing converges provided that p is bigger than 1 and in our case,"},{"Start":"06:41.360 ","End":"06:46.880","Text":"p is 2, so it\u0027s certainly bigger than 1, this converges."},{"Start":"06:46.880 ","End":"06:51.295","Text":"My series b_n converges."},{"Start":"06:51.295 ","End":"06:56.330","Text":"When I do this limit of a_n/b_n I\u0027m expecting to get"},{"Start":"06:56.330 ","End":"07:01.790","Text":"something that will show that a_n is also convergent."},{"Start":"07:01.790 ","End":"07:03.739","Text":"In fact, there\u0027s 2 good cases."},{"Start":"07:03.739 ","End":"07:07.550","Text":"If I get a number between 0 and infinity, in other words,"},{"Start":"07:07.550 ","End":"07:09.455","Text":"a finite positive number,"},{"Start":"07:09.455 ","End":"07:15.755","Text":"then because b_n converges and I can use the arrow this way, then a_n converges."},{"Start":"07:15.755 ","End":"07:18.710","Text":"If I get that this limit is 0,"},{"Start":"07:18.710 ","End":"07:23.630","Text":"that\u0027s also good for me because if I get 0 and then this converges,"},{"Start":"07:23.630 ","End":"07:25.820","Text":"I can also use this arrow here,"},{"Start":"07:25.820 ","End":"07:27.695","Text":"so both of these cases."},{"Start":"07:27.695 ","End":"07:30.365","Text":"As long as I get something finite,"},{"Start":"07:30.365 ","End":"07:31.830","Text":"I\u0027m going to be okay."},{"Start":"07:31.830 ","End":"07:39.365","Text":"Now, what remains to do is the formal step of computing this limit."},{"Start":"07:39.365 ","End":"07:47.090","Text":"Do the limit as n goes to infinity of a_n/b_n,"},{"Start":"07:47.090 ","End":"07:51.275","Text":"which is the limit of n goes to infinity."},{"Start":"07:51.275 ","End":"07:52.835","Text":"Now, this over this,"},{"Start":"07:52.835 ","End":"07:57.620","Text":"when I divide by 1 squared is like multiplying by n squared."},{"Start":"07:57.620 ","End":"08:08.055","Text":"I can put the n squared in the numerator and then I have 5 sine squared n over,"},{"Start":"08:08.055 ","End":"08:10.760","Text":"and the n factorial breakup,"},{"Start":"08:10.760 ","End":"08:14.660","Text":"just like I did here, into n,"},{"Start":"08:14.660 ","End":"08:20.895","Text":"n minus 1 times and then leave a space,"},{"Start":"08:20.895 ","End":"08:22.710","Text":"so there\u0027s a gap here,"},{"Start":"08:22.710 ","End":"08:25.595","Text":"then n minus 2,"},{"Start":"08:25.595 ","End":"08:28.595","Text":"n minus 3 and so on,"},{"Start":"08:28.595 ","End":"08:32.280","Text":"down to 2 times 1."},{"Start":"08:34.390 ","End":"08:41.430","Text":"This path and there\u0027s like a separator here."},{"Start":"08:41.660 ","End":"08:45.270","Text":"Good, I need a bit more space here."},{"Start":"08:45.270 ","End":"08:49.230","Text":"Let\u0027s see what this limit is."},{"Start":"08:49.230 ","End":"08:55.655","Text":"Now, this is equal to the limit."},{"Start":"08:55.655 ","End":"08:58.220","Text":"Again, n goes to infinity."},{"Start":"08:58.220 ","End":"09:01.595","Text":"Now this part, we can compute,"},{"Start":"09:01.595 ","End":"09:06.755","Text":"if I divide top and bottom by n squared, here I\u0027ll get 1."},{"Start":"09:06.755 ","End":"09:09.170","Text":"If I divide this product by n squared,"},{"Start":"09:09.170 ","End":"09:17.285","Text":"1 of the ends will cancel and n minus 1 will just be 1 minus 1."},{"Start":"09:17.285 ","End":"09:19.055","Text":"That\u0027s this part."},{"Start":"09:19.055 ","End":"09:21.110","Text":"The second part,"},{"Start":"09:21.110 ","End":"09:23.765","Text":"what I can do is write this."},{"Start":"09:23.765 ","End":"09:29.340","Text":"I\u0027ll just use the informal term bounded."},{"Start":"09:31.810 ","End":"09:34.660","Text":"I could have just copied it as is,"},{"Start":"09:34.660 ","End":"09:45.270","Text":"and remember that it\u0027s between 0 and 5 and this thing which tends to infinity."},{"Start":"09:48.400 ","End":"09:51.455","Text":"I\u0027ll just write it symbolically,"},{"Start":"09:51.455 ","End":"09:54.510","Text":"something that goes to infinity."},{"Start":"09:55.910 ","End":"10:01.550","Text":"I could copy it as is but the important thing about it is not the actual expression,"},{"Start":"10:01.550 ","End":"10:04.670","Text":"but the fact that it goes to infinity and we\u0027ve already seen that."},{"Start":"10:04.670 ","End":"10:11.300","Text":"So what we get basically, we know about this,"},{"Start":"10:11.300 ","End":"10:19.415","Text":"we know about this, I just have to say that this part here goes to 0."},{"Start":"10:19.415 ","End":"10:26.870","Text":"We have essentially bounded over something that goes to infinity."},{"Start":"10:26.870 ","End":"10:28.670","Text":"This actually goes to 0,"},{"Start":"10:28.670 ","End":"10:30.050","Text":"this goes to 1."},{"Start":"10:30.050 ","End":"10:34.940","Text":"Actually the limit is 0 because this is 1."},{"Start":"10:34.940 ","End":"10:44.940","Text":"This limit is 1 for the first bit times bounded between 0 and 5 over infinity is 0."},{"Start":"10:44.940 ","End":"10:47.370","Text":"Altogether, it\u0027s 0."},{"Start":"10:47.370 ","End":"10:49.685","Text":"If we go back up a bit,"},{"Start":"10:49.685 ","End":"10:55.265","Text":"we are actually in this case where our limit,"},{"Start":"10:55.265 ","End":"10:58.730","Text":"this is our a, a is 0,"},{"Start":"10:58.730 ","End":"11:01.535","Text":"then if b_n converges,"},{"Start":"11:01.535 ","End":"11:03.685","Text":"which it does we saw,"},{"Start":"11:03.685 ","End":"11:06.885","Text":"then also as we have,"},{"Start":"11:06.885 ","End":"11:09.240","Text":"now we have that a_n converges,"},{"Start":"11:09.240 ","End":"11:10.940","Text":"so we just write the answer,"},{"Start":"11:10.940 ","End":"11:16.740","Text":"the original series converges and we\u0027re done."}],"ID":10543},{"Watched":false,"Name":"Exercise 7 Part p","Duration":"10m 32s","ChapterTopicVideoID":10217,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10217.jpeg","UploadDate":"2017-08-22T04:42:32.1000000","DurationForVideoObject":"PT10M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"In this exercise, we have an infinite series."},{"Start":"00:03.645 ","End":"00:06.030","Text":"You can see the infinity here."},{"Start":"00:06.030 ","End":"00:10.755","Text":"We have to decide if it converges or diverges."},{"Start":"00:10.755 ","End":"00:13.650","Text":"There\u0027s many ways to do this."},{"Start":"00:13.650 ","End":"00:19.755","Text":"I\u0027m going to use the limit comparison test."},{"Start":"00:19.755 ","End":"00:22.860","Text":"It\u0027s written in the box here."},{"Start":"00:22.860 ","End":"00:25.709","Text":"I\u0027m not going to go over the whole thing,"},{"Start":"00:25.709 ","End":"00:29.175","Text":"but the idea is to"},{"Start":"00:29.175 ","End":"00:36.855","Text":"find another series to call the original series a_n."},{"Start":"00:36.855 ","End":"00:40.045","Text":"A_n is this bit."},{"Start":"00:40.045 ","End":"00:45.710","Text":"We have to come up with another series which is simpler and use"},{"Start":"00:45.710 ","End":"00:52.250","Text":"its convergence or divergence to infer about the original series."},{"Start":"00:52.250 ","End":"00:55.729","Text":"But we really have to have an idea of what to expect,"},{"Start":"00:55.729 ","End":"00:57.770","Text":"whether we\u0027re going to go for the convergence or"},{"Start":"00:57.770 ","End":"01:00.790","Text":"whether we\u0027re going to go for the divergence."},{"Start":"01:00.790 ","End":"01:05.480","Text":"It\u0027s certainly not immediately clear from this expression,"},{"Start":"01:05.480 ","End":"01:08.750","Text":"it\u0027s 1 of those infinity minus infinity thing,"},{"Start":"01:08.750 ","End":"01:12.975","Text":"anyway, not immediately clear."},{"Start":"01:12.975 ","End":"01:16.265","Text":"There\u0027s a standard trick which is used."},{"Start":"01:16.265 ","End":"01:19.965","Text":"What I\u0027m going to do is simplify a_n or anyway,"},{"Start":"01:19.965 ","End":"01:22.410","Text":"modify it to be more useful."},{"Start":"01:22.410 ","End":"01:24.105","Text":"I\u0027ll do this at the side."},{"Start":"01:24.105 ","End":"01:27.710","Text":"Our a_n, which is"},{"Start":"01:27.710 ","End":"01:35.430","Text":"the square root of n squared plus 1 minus n,"},{"Start":"01:35.430 ","End":"01:41.690","Text":"what I can do is I can use the concept of a conjugate."},{"Start":"01:41.690 ","End":"01:43.910","Text":"When you have a difference or a sum,"},{"Start":"01:43.910 ","End":"01:48.380","Text":"the conjugate is the same thing with the opposite sign."},{"Start":"01:48.380 ","End":"01:53.045","Text":"I\u0027m going to take this and multiply it by the conjugate,"},{"Start":"01:53.045 ","End":"01:59.180","Text":"which is the square root of n squared plus 1,"},{"Start":"01:59.180 ","End":"02:04.230","Text":"with the opposite sign meaning plus n. But,"},{"Start":"02:04.230 ","End":"02:06.500","Text":"of course, I can\u0027t just leave it like that,"},{"Start":"02:06.500 ","End":"02:11.130","Text":"I also have to divide by the same thing,"},{"Start":"02:11.780 ","End":"02:19.785","Text":"so square root n squared plus 1 plus n. Now,"},{"Start":"02:19.785 ","End":"02:21.270","Text":"I\u0027ve multiplied by 1,"},{"Start":"02:21.270 ","End":"02:23.310","Text":"our top and bottom by the same thing,"},{"Start":"02:23.310 ","End":"02:27.740","Text":"so this is algebraically equal to a_n."},{"Start":"02:27.740 ","End":"02:31.400","Text":"Now, why did I do this?"},{"Start":"02:31.400 ","End":"02:34.820","Text":"You might remember the use of conjugate, you might not,"},{"Start":"02:34.820 ","End":"02:41.930","Text":"I\u0027ll just remind you that the idea is to use the difference of squares formula."},{"Start":"02:41.930 ","End":"02:47.680","Text":"In this case, we have something like a minus b, a plus b."},{"Start":"02:47.680 ","End":"02:52.550","Text":"These 2 expressions are conjugates and it\u0027s equal to the difference of the squares,"},{"Start":"02:52.550 ","End":"02:56.040","Text":"a squared minus b squared."},{"Start":"02:56.050 ","End":"02:59.230","Text":"If we apply that here,"},{"Start":"02:59.230 ","End":"03:01.774","Text":"we get a squared,"},{"Start":"03:01.774 ","End":"03:07.270","Text":"which is the first thing squared is just n squared plus 1 without the square root."},{"Start":"03:07.270 ","End":"03:09.125","Text":"The square root squared,"},{"Start":"03:09.125 ","End":"03:11.300","Text":"the square root disappears,"},{"Start":"03:11.300 ","End":"03:20.100","Text":"and then minus b squared is minus n squared over the same thing,"},{"Start":"03:20.100 ","End":"03:29.280","Text":"square root of n squared plus 1 plus n. I can cancel this with this,"},{"Start":"03:29.280 ","End":"03:36.230","Text":"and so what we are left with is just 1"},{"Start":"03:36.230 ","End":"03:45.080","Text":"over the square root of n squared plus 1 plus n. Now,"},{"Start":"03:45.080 ","End":"03:47.590","Text":"how does this help us?"},{"Start":"03:47.590 ","End":"03:56.380","Text":"When we have polynomials and we use them in limits very often, typically,"},{"Start":"03:56.380 ","End":"04:00.200","Text":"the thing that matters is the leading term and"},{"Start":"04:00.200 ","End":"04:04.220","Text":"the rest of it is negligible in comparison,"},{"Start":"04:04.220 ","End":"04:06.380","Text":"that\u0027s when we\u0027re going to infinity."},{"Start":"04:06.380 ","End":"04:10.085","Text":"Really, this is roughly like n squared."},{"Start":"04:10.085 ","End":"04:13.170","Text":"The n squared plus 1 behaves like that."},{"Start":"04:13.170 ","End":"04:16.290","Text":"What I have here, as we continue,"},{"Start":"04:16.290 ","End":"04:21.735","Text":"as far as the limit behavior,"},{"Start":"04:21.735 ","End":"04:28.670","Text":"is roughly 1 over the square root of n squared plus n,"},{"Start":"04:28.670 ","End":"04:34.830","Text":"which is actually 1 over 2n."},{"Start":"04:37.210 ","End":"04:46.820","Text":"Getting back here, I\u0027m just using the term squiggly line to say that this behaves,"},{"Start":"04:46.820 ","End":"04:48.695","Text":"as far as convergence,"},{"Start":"04:48.695 ","End":"04:53.265","Text":"roughly like the sum of,"},{"Start":"04:53.265 ","End":"04:55.725","Text":"instead of a_n I use b_n."},{"Start":"04:55.725 ","End":"04:57.899","Text":"Constants make no difference,"},{"Start":"04:57.899 ","End":"05:07.580","Text":"so I can just say that this behaves like the series 1 over n. This is fuzzy mathematics."},{"Start":"05:07.580 ","End":"05:11.960","Text":"I mean, I\u0027m just wanting to know what to expect,"},{"Start":"05:11.960 ","End":"05:16.620","Text":"and then we do it more precisely using this limit."},{"Start":"05:16.620 ","End":"05:18.780","Text":"But we first have to make"},{"Start":"05:18.780 ","End":"05:22.580","Text":"an educated guess as whether we\u0027re expecting convergence or divergence."},{"Start":"05:22.580 ","End":"05:29.220","Text":"Now, this is the classical harmonic series and we know that this diverges."},{"Start":"05:29.360 ","End":"05:35.040","Text":"I\u0027m going to use this as my b_n and use 1"},{"Start":"05:35.040 ","End":"05:40.990","Text":"of the divergence cases."},{"Start":"05:40.990 ","End":"05:46.445","Text":"In our case, if we take the limit of a_n over b_n,"},{"Start":"05:46.445 ","End":"05:53.180","Text":"what will be good for us is that if a is infinity,"},{"Start":"05:53.180 ","End":"05:57.860","Text":"that will be good because then if b_n diverges, then a_n diverges."},{"Start":"05:57.860 ","End":"06:04.490","Text":"But also any finite number bigger than 0 will also"},{"Start":"06:04.490 ","End":"06:12.180","Text":"work because this double arrow means these 2 converge or diverge together,"},{"Start":"06:12.180 ","End":"06:16.440","Text":"meaning if b_n diverges,"},{"Start":"06:16.440 ","End":"06:19.425","Text":"then a_n also diverges"},{"Start":"06:19.425 ","End":"06:24.030","Text":"because if it converges then b_n would converge too which it doesn\u0027t."},{"Start":"06:24.030 ","End":"06:28.865","Text":"What I\u0027m going to do is compute this limit and hope that a is"},{"Start":"06:28.865 ","End":"06:35.040","Text":"either infinity or a finite number bigger than 0,"},{"Start":"06:35.040 ","End":"06:36.705","Text":"and then I\u0027ll be done."},{"Start":"06:36.705 ","End":"06:43.320","Text":"All it needs to be computed now is this limit."},{"Start":"06:43.990 ","End":"06:49.295","Text":"I\u0027ll also mention that I\u0027m supposed to check,"},{"Start":"06:49.295 ","End":"06:53.705","Text":"mentally make a note that these conditions also hold true."},{"Start":"06:53.705 ","End":"07:01.860","Text":"That for the b_n, the b_n is a positive series and the a_n is also positive"},{"Start":"07:01.860 ","End":"07:03.360","Text":"although we can allow"},{"Start":"07:03.360 ","End":"07:10.815","Text":"non-negative where even when n is 1 it\u0027s already positive in nature."},{"Start":"07:10.815 ","End":"07:13.200","Text":"These conditions hold."},{"Start":"07:13.200 ","End":"07:18.240","Text":"Let\u0027s see what is this a equal to,"},{"Start":"07:18.240 ","End":"07:26.465","Text":"which is the limit as n goes to infinity of a_n over b_n."},{"Start":"07:26.465 ","End":"07:29.374","Text":"Now, b_n is 1 over n,"},{"Start":"07:29.374 ","End":"07:38.145","Text":"so instead of dividing by 1 over n,"},{"Start":"07:38.145 ","End":"07:43.935","Text":"I can multiply by n. It\u0027s n times a_n,"},{"Start":"07:43.935 ","End":"07:48.735","Text":"but a_n is equal to this."},{"Start":"07:48.735 ","End":"07:51.470","Text":"Here we had a squiggly line,"},{"Start":"07:51.470 ","End":"07:55.050","Text":"but up to this point, we had equality."},{"Start":"07:55.090 ","End":"08:03.405","Text":"I\u0027m going to take my a_n as this expression here."},{"Start":"08:03.405 ","End":"08:09.009","Text":"Actually, I can put the n in the numerator and in the denominator,"},{"Start":"08:09.009 ","End":"08:15.475","Text":"just copy this thing which is square root of n squared plus 1"},{"Start":"08:15.475 ","End":"08:23.950","Text":"plus n. Now what I\u0027m going to do is divide top and bottom by n,"},{"Start":"08:23.950 ","End":"08:26.735","Text":"which won\u0027t change the fraction,"},{"Start":"08:26.735 ","End":"08:30.290","Text":"but I\u0027m still working with the limit."},{"Start":"08:30.290 ","End":"08:34.615","Text":"If I divide the top by n, I\u0027ve got 1."},{"Start":"08:34.615 ","End":"08:37.525","Text":"If I divide the bottom by n,"},{"Start":"08:37.525 ","End":"08:40.250","Text":"actually it\u0027s difficult to last."},{"Start":"08:40.250 ","End":"08:50.260","Text":"Here I have plus 1 also and I divide this by n. How do I divide this by n?"},{"Start":"08:50.720 ","End":"08:53.924","Text":"I\u0027ll do that at the side."},{"Start":"08:53.924 ","End":"09:00.110","Text":"The square root of n squared plus 1 divided by n is"},{"Start":"09:00.110 ","End":"09:06.755","Text":"the square root of n squared plus 1 over the square root of n squared."},{"Start":"09:06.755 ","End":"09:09.230","Text":"I can write n that way."},{"Start":"09:09.230 ","End":"09:17.955","Text":"Then this is equal to the square root of n squared plus 1 over n squared."},{"Start":"09:17.955 ","End":"09:24.125","Text":"From here I can go back here and write it as the square root."},{"Start":"09:24.125 ","End":"09:28.050","Text":"This is 1 plus 1 over n squared."},{"Start":"09:30.710 ","End":"09:36.690","Text":"Now, this limit is straightforward because this thing,"},{"Start":"09:36.690 ","End":"09:38.425","Text":"1 over n squared,"},{"Start":"09:38.425 ","End":"09:42.065","Text":"goes to 0 as n goes to infinity,"},{"Start":"09:42.065 ","End":"09:52.360","Text":"so what we\u0027re left with is 1 over the square root of 1 plus 1,"},{"Start":"09:52.360 ","End":"09:57.170","Text":"and this is equal to 1/2"},{"Start":"09:57.170 ","End":"10:04.260","Text":"and 1/2 for a falls into this case,"},{"Start":"10:04.260 ","End":"10:10.680","Text":"which is between 0 and infinity and because b_n diverges,"},{"Start":"10:10.680 ","End":"10:13.700","Text":"a_n must also diverge."},{"Start":"10:13.700 ","End":"10:18.125","Text":"That\u0027s logically equivalent because of the if and only if."},{"Start":"10:18.125 ","End":"10:26.190","Text":"The answer is that the original series diverges."},{"Start":"10:28.460 ","End":"10:32.170","Text":"That\u0027s the answer and we\u0027re done."}],"ID":10544},{"Watched":false,"Name":"Exercise 7 Part q","Duration":"10m 22s","ChapterTopicVideoID":10218,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10218.jpeg","UploadDate":"2017-08-22T04:42:59.6170000","DurationForVideoObject":"PT10M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:04.680","Text":"In this exercise, we\u0027re given an infinite series."},{"Start":"00:04.680 ","End":"00:07.740","Text":"It\u0027s infinite because it goes up to infinity,"},{"Start":"00:07.740 ","End":"00:12.135","Text":"and we have to decide if it converges or diverges."},{"Start":"00:12.135 ","End":"00:14.370","Text":"There might be many ways of doing this,"},{"Start":"00:14.370 ","End":"00:19.005","Text":"but we\u0027re going to do this with the limit comparison test,"},{"Start":"00:19.005 ","End":"00:21.720","Text":"which is summarized in this box."},{"Start":"00:21.720 ","End":"00:24.930","Text":"I\u0027m not going to go over it all again,"},{"Start":"00:24.930 ","End":"00:27.195","Text":"you should be familiar with it."},{"Start":"00:27.195 ","End":"00:32.355","Text":"The main idea is that our original series,"},{"Start":"00:32.355 ","End":"00:35.880","Text":"I\u0027m just to get some more room here."},{"Start":"00:35.880 ","End":"00:43.790","Text":"The original series is going to be the a_n in the theorem,"},{"Start":"00:43.790 ","End":"00:51.470","Text":"and notice that it is indeed bigger or equal to 0 because the cosine can never exceed 1,"},{"Start":"00:51.470 ","End":"00:55.625","Text":"so 1 minus this will not drop below 0."},{"Start":"00:55.625 ","End":"00:59.480","Text":"We\u0027ve got to find another series b_n,"},{"Start":"00:59.480 ","End":"01:06.275","Text":"that\u0027s somehow easier to deal with or to decide if it converges or diverges."},{"Start":"01:06.275 ","End":"01:12.560","Text":"Using the convergence or divergence of b_ n to make conclusions"},{"Start":"01:12.560 ","End":"01:19.500","Text":"about a_n according to this test."},{"Start":"01:19.980 ","End":"01:24.085","Text":"Now, it\u0027s not immediately obvious what to do,"},{"Start":"01:24.085 ","End":"01:30.460","Text":"but there\u0027s a little, wouldn\u0027t say trick,"},{"Start":"01:30.460 ","End":"01:36.565","Text":"but we can use some trigonometrical identities to make this easier to deal with,"},{"Start":"01:36.565 ","End":"01:38.875","Text":"and I\u0027ll do that at the side."},{"Start":"01:38.875 ","End":"01:45.610","Text":"There is a formula in trigonometry that says that 1 minus cosine"},{"Start":"01:45.610 ","End":"01:53.345","Text":"Alpha is twice sine squared Alpha over 2."},{"Start":"01:53.345 ","End":"02:01.240","Text":"If I use this here with Alpha being 1 over n,"},{"Start":"02:01.400 ","End":"02:09.559","Text":"then what we will get is that 1 minus cosine of 1 over n,"},{"Start":"02:09.559 ","End":"02:16.695","Text":"is equal to 2 sine squared 1 over n over 2,"},{"Start":"02:16.695 ","End":"02:21.310","Text":"which means 1 over 2n."},{"Start":"02:24.530 ","End":"02:28.475","Text":"If I plug this back here,"},{"Start":"02:28.475 ","End":"02:37.865","Text":"I will get that this is equal to the 2 can come out in front twice the sum of,"},{"Start":"02:37.865 ","End":"02:45.120","Text":"from n equals 1 to infinity of sine squared 1 over 2n."},{"Start":"02:45.120 ","End":"02:48.425","Text":"I just would like to emphasize the squared."},{"Start":"02:48.425 ","End":"02:52.350","Text":"I\u0027m going to put that outside."},{"Start":"02:52.350 ","End":"02:57.150","Text":"Same thing, I just find it more convenient this way."},{"Start":"02:58.360 ","End":"03:07.475","Text":"This, I\u0027m going to compare it to and afterwards I\u0027ll tell you my motivation."},{"Start":"03:07.475 ","End":"03:09.815","Text":"I\u0027m going to compare this,"},{"Start":"03:09.815 ","End":"03:15.880","Text":"say this roughly behaves like twice,"},{"Start":"03:15.880 ","End":"03:18.410","Text":"well, we can drop the 2,"},{"Start":"03:18.410 ","End":"03:21.050","Text":"doesn\u0027t matter, we can keep it also."},{"Start":"03:21.050 ","End":"03:24.080","Text":"Constants are not going to make any difference."},{"Start":"03:24.080 ","End":"03:30.920","Text":"What I want to do is throw out the sine and say 1 over 2n squared."},{"Start":"03:30.920 ","End":"03:33.950","Text":"Now, what do I mean by this?"},{"Start":"03:33.950 ","End":"03:45.184","Text":"I\u0027m looking for a b_n that behaves pretty much like the a_ n,"},{"Start":"03:45.184 ","End":"03:51.420","Text":"as far as convergence goes on ignoring constants."},{"Start":"03:52.270 ","End":"03:55.950","Text":"That\u0027s simpler as an expression."},{"Start":"03:55.950 ","End":"03:57.170","Text":"Certainly, this is simpler than this,"},{"Start":"03:57.170 ","End":"03:59.375","Text":"but what makes me get from here to here?"},{"Start":"03:59.375 ","End":"04:09.690","Text":"Well, I happen to remember there\u0027s another theorem that the limit as,"},{"Start":"04:09.690 ","End":"04:15.330","Text":"let\u0027s use Alpha, as Alpha goes to 0 of"},{"Start":"04:15.330 ","End":"04:21.840","Text":"sine Alpha over Alpha is equal to 1."},{"Start":"04:21.840 ","End":"04:24.255","Text":"What it means is that,"},{"Start":"04:24.255 ","End":"04:26.325","Text":"when Alpha\u0027s very small,"},{"Start":"04:26.325 ","End":"04:30.420","Text":"sine Alpha is approximately equal to Alpha."},{"Start":"04:30.760 ","End":"04:33.650","Text":"Or even just looking at this way,"},{"Start":"04:33.650 ","End":"04:39.900","Text":"I\u0027m going to get a ratio between sine of something small and the small thing itself,"},{"Start":"04:39.900 ","End":"04:47.290","Text":"and that\u0027s what gave me the inspiration to replace the sine with just the angle because,"},{"Start":"04:47.290 ","End":"04:50.375","Text":"I mean, n is going to infinity,"},{"Start":"04:50.375 ","End":"04:53.510","Text":"sure, but 1 over 2n is going to 0."},{"Start":"04:53.510 ","End":"04:59.360","Text":"That\u0027s why this will also work for if I take 0 from the right, for example,"},{"Start":"04:59.360 ","End":"05:08.060","Text":"works 2-sided limit, but n is going to plus infinity and 1 over n is going to plus 0."},{"Start":"05:08.060 ","End":"05:18.900","Text":"What I suggest is that we take this as our b_n."},{"Start":"05:19.010 ","End":"05:25.020","Text":"Now b_n is certainly strictly positive."},{"Start":"05:25.020 ","End":"05:31.395","Text":"Let\u0027s see if we can figure out the limit."},{"Start":"05:31.395 ","End":"05:40.040","Text":"Before that, I happen to know that this series converges because this is"},{"Start":"05:40.040 ","End":"05:49.970","Text":"exactly equal to not just behave like if I just pull out 1 over 2 squared from here,"},{"Start":"05:49.970 ","End":"05:51.920","Text":"the 1/2 comes out in front,"},{"Start":"05:51.920 ","End":"05:54.959","Text":"it\u0027s 2 over 2 squared."},{"Start":"05:55.610 ","End":"06:07.540","Text":"Here\u0027s 1/2, and then I have the sum of 1 over n squared."},{"Start":"06:07.580 ","End":"06:10.680","Text":"It was getting a bit crowded here,"},{"Start":"06:10.680 ","End":"06:12.720","Text":"let me do some rearranging."},{"Start":"06:12.720 ","End":"06:14.345","Text":"Move that up there,"},{"Start":"06:14.345 ","End":"06:17.540","Text":"get rid of this, straighten this up."},{"Start":"06:17.540 ","End":"06:20.350","Text":"What we have here is,"},{"Start":"06:20.350 ","End":"06:22.130","Text":"this should be familiar to you,"},{"Start":"06:22.130 ","End":"06:25.370","Text":"the sum of 1 over n squared and we know that it converges."},{"Start":"06:25.370 ","End":"06:30.830","Text":"But I can remind you why at least 1 way of looking at it is because"},{"Start":"06:30.830 ","End":"06:37.265","Text":"the sum of 1 over n to the power of p is what we call the p series,"},{"Start":"06:37.265 ","End":"06:41.030","Text":"and it converges provided that p is bigger than 1."},{"Start":"06:41.030 ","End":"06:43.535","Text":"In our case, p is 2,"},{"Start":"06:43.535 ","End":"06:46.220","Text":"which is certainly bigger than 1,"},{"Start":"06:46.220 ","End":"06:50.740","Text":"and so this thing converges."},{"Start":"06:51.260 ","End":"06:54.615","Text":"I\u0027ve got b_n which is convergent,"},{"Start":"06:54.615 ","End":"07:01.770","Text":"and then if this limit is equal to A,"},{"Start":"07:01.770 ","End":"07:11.385","Text":"that is, could be 0 or it could be bigger than 0, but finite."},{"Start":"07:11.385 ","End":"07:13.715","Text":"If A is finite,"},{"Start":"07:13.715 ","End":"07:19.680","Text":"then when b converges b_n, so does an."},{"Start":"07:19.680 ","End":"07:20.915","Text":"Also here, I mean,"},{"Start":"07:20.915 ","End":"07:24.355","Text":"we either you\u0027re going to use this arrow or this arrow,"},{"Start":"07:24.355 ","End":"07:27.890","Text":"and so 0 is good and bigger than 0 is good,"},{"Start":"07:27.890 ","End":"07:29.945","Text":"but infinity won\u0027t tell us anything."},{"Start":"07:29.945 ","End":"07:31.520","Text":"Let\u0027s see what this limit is,"},{"Start":"07:31.520 ","End":"07:33.560","Text":"what A comes out to be."},{"Start":"07:33.560 ","End":"07:39.380","Text":"Then hopefully, we\u0027ll be able to conclude that this one is convergent too."},{"Start":"07:39.380 ","End":"07:48.180","Text":"Our A is the limit as n goes to infinity of our a _n."},{"Start":"07:52.340 ","End":"07:55.920","Text":"Well, I\u0027ll take the other form,"},{"Start":"07:55.920 ","End":"08:02.520","Text":"a_n is, I took the 2 out,"},{"Start":"08:02.520 ","End":"08:05.610","Text":"I\u0027ll put it back in just to be strict even though constants don\u0027t matter,"},{"Start":"08:05.610 ","End":"08:09.300","Text":"but just for precision, we\u0027ll say,"},{"Start":"08:09.300 ","End":"08:13.400","Text":"a_n is 2 sine squared,"},{"Start":"08:13.400 ","End":"08:14.525","Text":"just what we did here,"},{"Start":"08:14.525 ","End":"08:22.050","Text":"1 over 2 n and that divided by b_n."},{"Start":"08:22.450 ","End":"08:32.360","Text":"I said that will take b_n as 1 over 2n squared."},{"Start":"08:32.360 ","End":"08:33.980","Text":"Look, the constants really don\u0027t matter."},{"Start":"08:33.980 ","End":"08:39.105","Text":"I\u0027ll just erase that"},{"Start":"08:39.105 ","End":"08:43.460","Text":"to re-simplify this and not have to worry whether we take or don\u0027t take the 2."},{"Start":"08:43.460 ","End":"08:52.500","Text":"We have here over 1 over 2n squared."},{"Start":"08:52.500 ","End":"08:54.950","Text":"Just for emphasis, just like I did before,"},{"Start":"08:54.950 ","End":"08:56.980","Text":"I\u0027ll put the 2 outside,"},{"Start":"08:56.980 ","End":"09:01.350","Text":"so erase it here and put brackets,"},{"Start":"09:01.350 ","End":"09:05.850","Text":"and that way I can see that it\u0027s the limit."},{"Start":"09:08.470 ","End":"09:14.775","Text":"If I let Alpha equals 1 over 2n,"},{"Start":"09:14.775 ","End":"09:16.995","Text":"then when n goes to infinity,"},{"Start":"09:16.995 ","End":"09:19.470","Text":"Alpha goes to 0,"},{"Start":"09:19.470 ","End":"09:30.580","Text":"so I get sine Alpha squared over Alpha squared."},{"Start":"09:31.250 ","End":"09:34.484","Text":"Since this limit is 1,"},{"Start":"09:34.484 ","End":"09:36.629","Text":"when you square something,"},{"Start":"09:36.629 ","End":"09:39.165","Text":"you just square the limit."},{"Start":"09:39.165 ","End":"09:44.085","Text":"Well, this over this all squared is 1 squared is 1."},{"Start":"09:44.085 ","End":"09:51.255","Text":"We are in the case of the A between 0 and infinity."},{"Start":"09:51.255 ","End":"09:55.690","Text":"The constants here didn\u0027t matter if I put it times 2 or divided by 2,"},{"Start":"09:55.690 ","End":"09:58.675","Text":"it\u0027s still going to be a number, it\u0027s finite."},{"Start":"09:58.675 ","End":"10:01.209","Text":"We conclude from this arrow,"},{"Start":"10:01.209 ","End":"10:06.670","Text":"from this direction that because b_n converges, so does a_n."},{"Start":"10:06.670 ","End":"10:11.420","Text":"The answer is that the original series also converges."},{"Start":"10:11.420 ","End":"10:15.770","Text":"It\u0027s a convergent series, that\u0027s the original,"},{"Start":"10:15.770 ","End":"10:17.670","Text":"that\u0027s the sum of a_n,"},{"Start":"10:17.670 ","End":"10:19.455","Text":"which is our original series,"},{"Start":"10:19.455 ","End":"10:22.510","Text":"and that\u0027s it. We\u0027re done."}],"ID":10545},{"Watched":false,"Name":"Exercise 7 Part r","Duration":"12m 59s","ChapterTopicVideoID":10210,"CourseChapterTopicPlaylistID":52095,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10210.jpeg","UploadDate":"2017-08-22T04:39:58.3400000","DurationForVideoObject":"PT12M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we\u0027re given an infinite series."},{"Start":"00:04.050 ","End":"00:05.845","Text":"See the infinity here,"},{"Start":"00:05.845 ","End":"00:07.110","Text":"so it\u0027s an infinite series,"},{"Start":"00:07.110 ","End":"00:10.820","Text":"so it might converge or it might diverge,"},{"Start":"00:10.820 ","End":"00:13.525","Text":"and we have to decide which."},{"Start":"00:13.525 ","End":"00:16.950","Text":"There\u0027s probably more than 1 way to do it."},{"Start":"00:16.950 ","End":"00:21.450","Text":"Here, we\u0027re going to do it with the limit comparison test."},{"Start":"00:21.450 ","End":"00:25.515","Text":"In the limit comparison test, basically,"},{"Start":"00:25.515 ","End":"00:31.860","Text":"our series is what we would call a_n."},{"Start":"00:31.860 ","End":"00:35.280","Text":"This is the typical a_n,"},{"Start":"00:35.280 ","End":"00:38.905","Text":"and we have to compare it to find another series b_n,"},{"Start":"00:38.905 ","End":"00:44.555","Text":"which is simpler, and whose convergence we know about that it does or doesn\u0027t,"},{"Start":"00:44.555 ","End":"00:45.950","Text":"and with its help,"},{"Start":"00:45.950 ","End":"00:49.220","Text":"we\u0027ll conclude about a_n."},{"Start":"00:49.220 ","End":"00:55.715","Text":"But we first have to decide whether we\u0027re going to try to prove it converges or diverges."},{"Start":"00:55.715 ","End":"01:00.200","Text":"We want to see roughly what this series behaves like."},{"Start":"01:00.200 ","End":"01:03.590","Text":"Now, with polynomials we already know what to do."},{"Start":"01:03.590 ","End":"01:09.395","Text":"I would say that this is roughly like the denominator,"},{"Start":"01:09.395 ","End":"01:12.290","Text":"just the highest power counts,"},{"Start":"01:12.290 ","End":"01:14.375","Text":"so that\u0027s like n squared."},{"Start":"01:14.375 ","End":"01:19.250","Text":"Here we have the square root of n, which is n^1/2."},{"Start":"01:19.250 ","End":"01:22.595","Text":"But what to do with this natural log of n?"},{"Start":"01:22.595 ","End":"01:31.830","Text":"I\u0027m going to have to borrow a result from calculus and the several ways of phrasing it."},{"Start":"01:31.830 ","End":"01:38.690","Text":"1 way of phrasing it is to say that the natural log of x is less"},{"Start":"01:38.690 ","End":"01:45.890","Text":"than any positive power of x. x to the power of a small positive number,"},{"Start":"01:45.890 ","End":"01:49.145","Text":"sometimes use the Greek letter Epsilon."},{"Start":"01:49.145 ","End":"01:53.090","Text":"Epsilon is bigger than 0,"},{"Start":"01:53.090 ","End":"01:54.605","Text":"but it could be very small,"},{"Start":"01:54.605 ","End":"01:56.750","Text":"like a millionth or something."},{"Start":"01:56.750 ","End":"02:02.180","Text":"But this is only true from a certain point x on wards,"},{"Start":"02:02.180 ","End":"02:05.240","Text":"only for large values of x as 1 way of phrasing it."},{"Start":"02:05.240 ","End":"02:11.690","Text":"Another way of phrasing it is to say that the limit as x goes to"},{"Start":"02:11.690 ","End":"02:20.075","Text":"infinity of the natural log of x over x^Epsilon,"},{"Start":"02:20.075 ","End":"02:21.890","Text":"for any positive Epsilon,"},{"Start":"02:21.890 ","End":"02:24.170","Text":"but we\u0027re mostly interested in very,"},{"Start":"02:24.170 ","End":"02:26.180","Text":"very small values of Epsilon,"},{"Start":"02:26.180 ","End":"02:29.930","Text":"that this thing is equal to 0."},{"Start":"02:29.930 ","End":"02:33.059","Text":"I\u0027m going to have to borrow this result."},{"Start":"02:33.760 ","End":"02:37.760","Text":"It\u0027s not difficult to prove."},{"Start":"02:37.760 ","End":"02:41.795","Text":"We actually could prove it using L\u0027Hopital\u0027s theorem."},{"Start":"02:41.795 ","End":"02:46.355","Text":"In fact, I could just briefly demonstrate that for you."},{"Start":"02:46.355 ","End":"02:49.910","Text":"You can skip this part if you take my word for it,"},{"Start":"02:49.910 ","End":"02:52.235","Text":"but I feel I should show you why."},{"Start":"02:52.235 ","End":"02:56.105","Text":"This goes to infinity when x goes to infinity,"},{"Start":"02:56.105 ","End":"02:58.025","Text":"and this goes to infinity."},{"Start":"02:58.025 ","End":"03:02.000","Text":"We have a case of infinity over infinity,"},{"Start":"03:02.000 ","End":"03:09.245","Text":"and we\u0027re going to use, I\u0027ll pay him some respect and write his name, L\u0027Hopital."},{"Start":"03:09.245 ","End":"03:12.395","Text":"He has this rule that says that, when this happens,"},{"Start":"03:12.395 ","End":"03:15.740","Text":"if differentiate the numerator and the denominator,"},{"Start":"03:15.740 ","End":"03:18.965","Text":"you\u0027ll get something that goes to the same limit."},{"Start":"03:18.965 ","End":"03:23.960","Text":"We get the limit as x goes to infinity,"},{"Start":"03:23.960 ","End":"03:26.920","Text":"derivative of this is 1 over x,"},{"Start":"03:26.920 ","End":"03:36.070","Text":"derivative of the denominator is Epsilon x^Epsilon minus 1."},{"Start":"03:38.810 ","End":"03:42.555","Text":"Well, the constant can come out in front."},{"Start":"03:42.555 ","End":"03:45.675","Text":"This is 1 over x."},{"Start":"03:45.675 ","End":"03:48.240","Text":"This is x^minus 1,"},{"Start":"03:48.240 ","End":"03:51.050","Text":"and this is x^Epsilon minus 1."},{"Start":"03:51.050 ","End":"04:00.095","Text":"If I subtract, I basically just get x^minus Epsilon."},{"Start":"04:00.095 ","End":"04:03.420","Text":"The minus 1 is like 1 over x,"},{"Start":"04:03.420 ","End":"04:05.100","Text":"and it cancels out with the 1 over x,"},{"Start":"04:05.100 ","End":"04:08.060","Text":"so I\u0027m left with 1 over x^Epsilon."},{"Start":"04:08.060 ","End":"04:11.020","Text":"Perhaps that would be a better way of writing it."},{"Start":"04:11.020 ","End":"04:15.940","Text":"1 over x^Epsilon, the limit of course,"},{"Start":"04:15.940 ","End":"04:19.000","Text":"as x goes to infinity."},{"Start":"04:19.000 ","End":"04:23.575","Text":"Now, when x goes to infinity,"},{"Start":"04:23.575 ","End":"04:29.645","Text":"even x to the power of a millionth also goes to infinity."},{"Start":"04:29.645 ","End":"04:32.020","Text":"This still goes to infinity,"},{"Start":"04:32.020 ","End":"04:34.900","Text":"and therefore we get this equals 0,"},{"Start":"04:34.900 ","End":"04:39.670","Text":"so that justifies this. That\u0027s on aside."},{"Start":"04:39.670 ","End":"04:41.630","Text":"Let\u0027s get back to this."},{"Start":"04:41.630 ","End":"04:51.995","Text":"We can say that natural log of n behaves a bit like n^Epsilon,"},{"Start":"04:51.995 ","End":"04:56.975","Text":"where Epsilon is some very small number,"},{"Start":"04:56.975 ","End":"05:02.420","Text":"and I guess I should have put the Sigma n. Essentially this is going"},{"Start":"05:02.420 ","End":"05:07.625","Text":"to be my b_n series."},{"Start":"05:07.625 ","End":"05:11.030","Text":"But let\u0027s see, does this converge or diverge?"},{"Start":"05:11.030 ","End":"05:15.050","Text":"Well, if we do the computation,"},{"Start":"05:15.050 ","End":"05:22.820","Text":"this is the sum of let\u0027s say"},{"Start":"05:22.820 ","End":"05:32.260","Text":"0.5 plus Epsilon minus 2 from this,"},{"Start":"05:33.440 ","End":"05:38.140","Text":"and n goes from 1 to infinity."},{"Start":"05:38.840 ","End":"05:41.925","Text":"Sorry, I messed that up."},{"Start":"05:41.925 ","End":"05:45.060","Text":"I forgot the n just a second."},{"Start":"05:45.060 ","End":"05:46.770","Text":"Yeah that\u0027s right."},{"Start":"05:46.770 ","End":"05:48.910","Text":"Here\u0027s the n, sorry."},{"Start":"05:50.510 ","End":"05:54.095","Text":"If I let Epsilon equal,"},{"Start":"05:54.095 ","End":"05:57.380","Text":"I could even let it equals 0.1."},{"Start":"05:57.380 ","End":"06:04.310","Text":"I just played with the numbers bit at the side so I know what to say, what to take."},{"Start":"06:04.310 ","End":"06:06.455","Text":"I\u0027ll take Epsilon 0.1,"},{"Start":"06:06.455 ","End":"06:07.855","Text":"and then what do I get?"},{"Start":"06:07.855 ","End":"06:15.520","Text":"0.5 plus 0.1 minus 2 is going to be,"},{"Start":"06:15.520 ","End":"06:20.360","Text":"0.6 minus 2 is minus 1.4."},{"Start":"06:20.360 ","End":"06:26.560","Text":"It\u0027s going to be equal to the sum from 1 to infinity,"},{"Start":"06:26.560 ","End":"06:33.580","Text":"n^minus 1.4, 1 over n^1.4."},{"Start":"06:33.580 ","End":"06:37.895","Text":"I could have even taken Epsilon to be a bit bigger,"},{"Start":"06:37.895 ","End":"06:44.420","Text":"even 0.4 as long as I end up with something here that\u0027s bigger than 1."},{"Start":"06:44.420 ","End":"06:46.985","Text":"Why do I want this to be bigger than 1?"},{"Start":"06:46.985 ","End":"06:49.999","Text":"Because I\u0027m going to compare it to the p series."},{"Start":"06:49.999 ","End":"06:51.980","Text":"Remember the p series?"},{"Start":"06:51.980 ","End":"06:58.270","Text":"Remember we have the sum from 1 to infinity of 1 over n^p."},{"Start":"06:58.270 ","End":"07:04.380","Text":"It converges if p is bigger than 1,"},{"Start":"07:04.380 ","End":"07:06.900","Text":"and in our case, p is 1.4,"},{"Start":"07:06.900 ","End":"07:09.540","Text":"is certainly bigger than 1."},{"Start":"07:09.540 ","End":"07:15.300","Text":"I just chose a value of Epsilon that would make this bigger than 1,"},{"Start":"07:15.300 ","End":"07:19.290","Text":"and as I say anything up to 0.5,"},{"Start":"07:19.290 ","End":"07:22.095","Text":"but less than 0.5 would do."},{"Start":"07:22.095 ","End":"07:28.050","Text":"Now that we know that this is convergent,"},{"Start":"07:28.050 ","End":"07:31.020","Text":"let\u0027s write it, this is convergent,"},{"Start":"07:31.020 ","End":"07:33.900","Text":"as I say, using the p series."},{"Start":"07:33.900 ","End":"07:42.755","Text":"Now we are going to formally do the limit."},{"Start":"07:42.755 ","End":"07:47.270","Text":"I also should have mentioned that we satisfied the conditions,"},{"Start":"07:47.270 ","End":"07:51.770","Text":"is that a_n is bigger or equal to 0."},{"Start":"07:51.770 ","End":"07:56.470","Text":"Certainly everything here is positive."},{"Start":"07:56.470 ","End":"08:00.215","Text":"Well, when n equals 1,"},{"Start":"08:00.215 ","End":"08:02.510","Text":"natural log of n is actually equal to 0,"},{"Start":"08:02.510 ","End":"08:05.360","Text":"but that\u0027s fine because we allow bigger or equal to,"},{"Start":"08:05.360 ","End":"08:08.015","Text":"and the b_n, which is this,"},{"Start":"08:08.015 ","End":"08:10.955","Text":"is actually just bigger than 0."},{"Start":"08:10.955 ","End":"08:14.495","Text":"In fact, b_n is just this."},{"Start":"08:14.495 ","End":"08:17.130","Text":"This is in fact what I\u0027m going to take."},{"Start":"08:18.500 ","End":"08:25.510","Text":"I\u0027ll highlight this, and I\u0027ll say not this,"},{"Start":"08:25.510 ","End":"08:29.690","Text":"but this here that I\u0027ve shaded, that\u0027s the b_n."},{"Start":"08:29.690 ","End":"08:34.965","Text":"Now let\u0027s check what this limit is. This will help us."},{"Start":"08:34.965 ","End":"08:37.575","Text":"We know that b_n is convergent,"},{"Start":"08:37.575 ","End":"08:41.219","Text":"so we want to use either 1 of these 2 cases."},{"Start":"08:41.219 ","End":"08:49.010","Text":"If we take the limit of a_n over b_n and compute it."},{"Start":"08:49.010 ","End":"08:51.350","Text":"If it comes out to be 0, we\u0027re okay,"},{"Start":"08:51.350 ","End":"08:56.870","Text":"and if it comes out to be bigger than 0 but not infinity, that\u0027s also okay."},{"Start":"08:56.870 ","End":"09:00.950","Text":"Then we\u0027ll be able to conclude either using this arrow or this arrow"},{"Start":"09:00.950 ","End":"09:05.120","Text":"that our original series converges. Let\u0027s see what we get."},{"Start":"09:05.120 ","End":"09:06.830","Text":"The limit of a_n over b_n,"},{"Start":"09:06.830 ","End":"09:11.570","Text":"n goes to infinity,"},{"Start":"09:11.570 ","End":"09:20.180","Text":"is the limit as n goes to infinity of this thing, the original thing,"},{"Start":"09:20.180 ","End":"09:21.695","Text":"square root of n,"},{"Start":"09:21.695 ","End":"09:26.630","Text":"natural log of n over n squared plus 1,"},{"Start":"09:26.630 ","End":"09:29.390","Text":"and divide it by b_n."},{"Start":"09:29.390 ","End":"09:34.550","Text":"b_n is this, so dividing by 1 over means,"},{"Start":"09:34.550 ","End":"09:36.860","Text":"I can multiply by the reciprocal."},{"Start":"09:36.860 ","End":"09:46.125","Text":"So I\u0027ll put n^1.4 in the numerator and figure out this limit."},{"Start":"09:46.125 ","End":"09:56.310","Text":"Let\u0027s see. What we get is the square root of n times n^1.4 is"},{"Start":"09:56.310 ","End":"10:04.140","Text":"actually n^1.9 times natural log"},{"Start":"10:04.140 ","End":"10:10.200","Text":"of n over n squared plus 1."},{"Start":"10:10.200 ","End":"10:18.565","Text":"Now let\u0027s divide top and bottom by n squared."},{"Start":"10:18.565 ","End":"10:20.590","Text":"Well, not exactly."},{"Start":"10:20.590 ","End":"10:24.384","Text":"What I mean to say is that I can write this"},{"Start":"10:24.384 ","End":"10:31.405","Text":"as n^1.9 natural log"},{"Start":"10:31.405 ","End":"10:37.380","Text":"of n over n squared, that\u0027s on the bottom."},{"Start":"10:37.380 ","End":"10:39.750","Text":"Then I put another n squared on the numerator,"},{"Start":"10:39.750 ","End":"10:43.870","Text":"and here I\u0027ll put the n squared plus 1."},{"Start":"10:46.820 ","End":"10:50.570","Text":"I forgot to write limit, didn\u0027t I?"},{"Start":"10:50.570 ","End":"10:54.965","Text":"Here, limit, and here, limit."},{"Start":"10:54.965 ","End":"10:57.935","Text":"Of course, n goes to infinity,"},{"Start":"10:57.935 ","End":"11:00.155","Text":"n goes to infinity."},{"Start":"11:00.155 ","End":"11:10.535","Text":"Now, what I claim is that this part goes to 1."},{"Start":"11:10.535 ","End":"11:14.900","Text":"It\u0027s easy to see because if I divide top and bottom by n squared,"},{"Start":"11:14.900 ","End":"11:21.035","Text":"well I\u0027ll show you it\u0027s 1 over 1 plus 1 over n squared."},{"Start":"11:21.035 ","End":"11:23.900","Text":"This part goes to 0,"},{"Start":"11:23.900 ","End":"11:25.195","Text":"so it\u0027s 1 over 1,"},{"Start":"11:25.195 ","End":"11:27.930","Text":"so this first part goes to 1."},{"Start":"11:27.930 ","End":"11:33.090","Text":"This part here, if I rewrite it,"},{"Start":"11:33.090 ","End":"11:36.590","Text":"it divide both top and bottom by n^1.9,"},{"Start":"11:36.590 ","End":"11:43.690","Text":"this is natural log of n over n^0.1."},{"Start":"11:43.690 ","End":"11:47.135","Text":"Going back to what I wrote earlier,"},{"Start":"11:47.135 ","End":"11:51.920","Text":"this limit of the natural log of x over x to"},{"Start":"11:51.920 ","End":"11:56.660","Text":"the power of something small or anything positive is 0."},{"Start":"11:56.660 ","End":"11:59.495","Text":"If I take Epsilon of 0.1,"},{"Start":"11:59.495 ","End":"12:07.950","Text":"this thing will actually go to 0."},{"Start":"12:07.950 ","End":"12:10.170","Text":"This is when n goes to infinity,"},{"Start":"12:10.170 ","End":"12:12.450","Text":"this is when n goes to infinity."},{"Start":"12:12.450 ","End":"12:16.065","Text":"This is not an arrow, this is an equality, don\u0027t mind."},{"Start":"12:16.065 ","End":"12:23.100","Text":"Anyway, altogether we get basically that this thing goes to 0 times 1."},{"Start":"12:25.670 ","End":"12:28.595","Text":"The first part is 0,"},{"Start":"12:28.595 ","End":"12:30.365","Text":"second part is 1,"},{"Start":"12:30.365 ","End":"12:33.635","Text":"0 times 1 is 0."},{"Start":"12:33.635 ","End":"12:36.875","Text":"We are in this case, case 2,"},{"Start":"12:36.875 ","End":"12:40.190","Text":"where if b_n converges, the Sigma,"},{"Start":"12:40.190 ","End":"12:44.525","Text":"then so the series a_n converge."},{"Start":"12:44.525 ","End":"12:46.715","Text":"This is convergent."},{"Start":"12:46.715 ","End":"12:50.715","Text":"We know that Sigma a_n also converges,"},{"Start":"12:50.715 ","End":"12:53.630","Text":"and that is exactly what we were asked to determine."},{"Start":"12:53.630 ","End":"12:58.620","Text":"The answer is converges. We\u0027re done."}],"ID":10546}],"Thumbnail":null,"ID":52095},{"Name":"The Ratio Test","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 8 Part a","Duration":"","ChapterTopicVideoID":10223,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we\u0027re given an infinite series as an infinity here."},{"Start":"00:04.770 ","End":"00:06.660","Text":"We have to decide,"},{"Start":"00:06.660 ","End":"00:10.125","Text":"does it converge or does it diverge?"},{"Start":"00:10.125 ","End":"00:12.915","Text":"Because of the factorial,"},{"Start":"00:12.915 ","End":"00:17.805","Text":"it\u0027s a good sign that we should be using the ratio test."},{"Start":"00:17.805 ","End":"00:20.400","Text":"The ratio test for a positive series,"},{"Start":"00:20.400 ","End":"00:22.560","Text":"and note that this is positive,"},{"Start":"00:22.560 ","End":"00:25.815","Text":"is to take the quotient,"},{"Start":"00:25.815 ","End":"00:28.710","Text":"the ratio of 2 successive terms,"},{"Start":"00:28.710 ","End":"00:33.210","Text":"and then take that to infinity and see what we get and accordingly decide."},{"Start":"00:33.210 ","End":"00:38.230","Text":"To get more space, I\u0027ve copied it down here."},{"Start":"00:38.230 ","End":"00:44.000","Text":"Let\u0027s straight away jump into computing this limit."},{"Start":"00:44.000 ","End":"00:51.525","Text":"What we need to compute is the limit as n goes to infinity."},{"Start":"00:51.525 ","End":"00:53.970","Text":"Now, this is our a_n,"},{"Start":"00:53.970 ","End":"01:00.165","Text":"this part here besides the Sigma, this is a_n."},{"Start":"01:00.165 ","End":"01:04.830","Text":"Now, we need a_n plus 1 over a_n,"},{"Start":"01:04.830 ","End":"01:09.030","Text":"which is the limit."},{"Start":"01:09.030 ","End":"01:18.530","Text":"Now, a_n plus 1 is what we get when we replace n by n plus 1 here,"},{"Start":"01:18.530 ","End":"01:28.790","Text":"so we get n plus 1 factorial cubed over"},{"Start":"01:28.790 ","End":"01:33.370","Text":"3 n"},{"Start":"01:33.370 ","End":"01:39.700","Text":"plus 1 factorial."},{"Start":"01:39.700 ","End":"01:42.525","Text":"That\u0027s the a_n plus 1."},{"Start":"01:42.525 ","End":"01:45.080","Text":"A_n is just what\u0027s written here."},{"Start":"01:45.080 ","End":"01:49.415","Text":"But instead of dividing by the fraction,"},{"Start":"01:49.415 ","End":"01:53.570","Text":"we can multiply by the upside down fraction, the reciprocal."},{"Start":"01:53.570 ","End":"01:57.390","Text":"I\u0027m going to multiply by 1 over a_n,"},{"Start":"01:57.390 ","End":"02:01.440","Text":"which is 3n factorial"},{"Start":"02:01.440 ","End":"02:09.120","Text":"over n factorial cubed."},{"Start":"02:09.120 ","End":"02:16.180","Text":"Now, this is equal to the limit."},{"Start":"02:16.880 ","End":"02:25.065","Text":"What I want to do is take this bit with this bit and see what cancels,"},{"Start":"02:25.065 ","End":"02:27.640","Text":"and then this bit with this bit."},{"Start":"02:27.640 ","End":"02:30.365","Text":"Let me just write it out first."},{"Start":"02:30.365 ","End":"02:33.035","Text":"What I have here is"},{"Start":"02:33.035 ","End":"02:43.960","Text":"3n factorial over and this part here is 3n plus 3 factorial."},{"Start":"02:46.130 ","End":"02:50.825","Text":"Then this bit is"},{"Start":"02:50.825 ","End":"02:58.400","Text":"n plus 1 factorial cubed over n factorial cubed."},{"Start":"02:58.400 ","End":"03:01.340","Text":"I can take the cubed just 1 time."},{"Start":"03:01.340 ","End":"03:07.520","Text":"I can put over n factorial and then this whole thing cubed."},{"Start":"03:07.520 ","End":"03:10.880","Text":"Let me do some side calculations."},{"Start":"03:10.880 ","End":"03:14.015","Text":"The first ratio or quotient,"},{"Start":"03:14.015 ","End":"03:17.210","Text":"I\u0027ll call this bit asterisk,"},{"Start":"03:17.210 ","End":"03:19.235","Text":"and the second part,"},{"Start":"03:19.235 ","End":"03:22.010","Text":"I\u0027ll call this part double asterisk."},{"Start":"03:22.010 ","End":"03:24.230","Text":"Then I\u0027m going to go over here."},{"Start":"03:24.230 ","End":"03:27.230","Text":"I\u0027ll first work on the asterisk part,"},{"Start":"03:27.230 ","End":"03:35.340","Text":"which is 3n factorial over."},{"Start":"03:35.340 ","End":"03:39.825","Text":"Now, 3n plus 3 factorial,"},{"Start":"03:39.825 ","End":"03:41.870","Text":"if I write it out,"},{"Start":"03:41.870 ","End":"03:48.000","Text":"it\u0027s the product of all the numbers from 3n plus 3 down to 1,"},{"Start":"03:48.000 ","End":"03:52.725","Text":"so it\u0027s 3n plus 3, 3n plus 2,"},{"Start":"03:52.725 ","End":"03:59.910","Text":"3n plus 1, 3n and so on and so on,"},{"Start":"03:59.910 ","End":"04:04.010","Text":"right down to, let\u0027s make this a bit longer,"},{"Start":"04:04.010 ","End":"04:07.430","Text":"times 2 times 1 at the end."},{"Start":"04:07.430 ","End":"04:11.100","Text":"I\u0027ll extend this a bit here too."},{"Start":"04:12.570 ","End":"04:16.585","Text":"But notice that from this point onwards,"},{"Start":"04:16.585 ","End":"04:20.325","Text":"this thing is just 3n factorial,"},{"Start":"04:20.325 ","End":"04:25.540","Text":"which will then cancel with this in the numerator,"},{"Start":"04:25.540 ","End":"04:27.340","Text":"leaving just 1 here."},{"Start":"04:27.340 ","End":"04:31.060","Text":"All we have is 1 over 1 this, this, and this."},{"Start":"04:31.060 ","End":"04:36.400","Text":"Up to here it cancels."},{"Start":"04:36.410 ","End":"04:38.845","Text":"Now, the second bit,"},{"Start":"04:38.845 ","End":"04:42.170","Text":"the double asterisk part,"},{"Start":"04:42.170 ","End":"04:47.540","Text":"that is n plus 1 factorial,"},{"Start":"04:47.540 ","End":"04:53.990","Text":"which is n plus 1 times n times n minus 1 and down"},{"Start":"04:53.990 ","End":"05:01.800","Text":"to 2 times 1 over n factorial,"},{"Start":"05:01.800 ","End":"05:06.634","Text":"which is n, n minus 1,"},{"Start":"05:06.634 ","End":"05:10.520","Text":"and so on down to 2 times 1,"},{"Start":"05:10.520 ","End":"05:15.515","Text":"and this thing all cubed."},{"Start":"05:15.515 ","End":"05:18.529","Text":"But look, this part,"},{"Start":"05:18.529 ","End":"05:20.720","Text":"which is the n factorial part,"},{"Start":"05:20.720 ","End":"05:22.280","Text":"is the same as this."},{"Start":"05:22.280 ","End":"05:25.615","Text":"All we\u0027re left with is this cubed."},{"Start":"05:25.615 ","End":"05:29.805","Text":"Now let\u0027s plug these 2 back here."},{"Start":"05:29.805 ","End":"05:36.845","Text":"We have the limit as n goes to infinity,"},{"Start":"05:36.845 ","End":"05:41.420","Text":"dividing line on the numerator,"},{"Start":"05:41.420 ","End":"05:44.550","Text":"n plus 1 cubed."},{"Start":"05:47.720 ","End":"05:55.910","Text":"On the denominator, 3n plus 3,"},{"Start":"05:55.910 ","End":"06:02.490","Text":"3n plus 2, 3n plus 1."},{"Start":"06:02.630 ","End":"06:08.540","Text":"Now, I\u0027m not going to completely expand these,"},{"Start":"06:08.540 ","End":"06:12.050","Text":"that\u0027s too much work and it\u0027s not necessary,"},{"Start":"06:12.050 ","End":"06:19.010","Text":"because with polynomials, what counts is the dominant term, the highest coefficient."},{"Start":"06:19.010 ","End":"06:20.810","Text":"I know if I multiply this out,"},{"Start":"06:20.810 ","End":"06:27.365","Text":"I get n cubed plus something n squared and so on,"},{"Start":"06:27.365 ","End":"06:29.665","Text":"that\u0027s the highest power."},{"Start":"06:29.665 ","End":"06:33.450","Text":"Here, the highest power is 3n,"},{"Start":"06:33.450 ","End":"06:37.740","Text":"3n, 3n, which is 27n cubed."},{"Start":"06:37.740 ","End":"06:42.985","Text":"The rest of it is going to also be lower power n squared and so on."},{"Start":"06:42.985 ","End":"06:46.580","Text":"With limits of rational functions,"},{"Start":"06:46.580 ","End":"06:50.690","Text":"polynomial over polynomial, what counts is only the leading terms."},{"Start":"06:50.690 ","End":"06:59.665","Text":"This is actually equal to the limit as n goes to infinity of"},{"Start":"06:59.665 ","End":"07:06.125","Text":"just n cubed"},{"Start":"07:06.125 ","End":"07:11.670","Text":"over 27n cubed."},{"Start":"07:11.930 ","End":"07:14.650","Text":"Because as I said, that\u0027s what we do,"},{"Start":"07:14.650 ","End":"07:16.220","Text":"we just take the leading term."},{"Start":"07:16.220 ","End":"07:24.200","Text":"Now, this is now clear because n cubed cancels and this is just 1/27."},{"Start":"07:24.200 ","End":"07:26.330","Text":"It\u0027s a limit of a constant."},{"Start":"07:26.330 ","End":"07:32.800","Text":"Limit of a constant is the constant itself, so it\u0027s 1/27."},{"Start":"07:32.800 ","End":"07:36.940","Text":"Now this is what we called,"},{"Start":"07:37.010 ","End":"07:39.435","Text":"if you look back here,"},{"Start":"07:39.435 ","End":"07:43.990","Text":"this is the limit a,"},{"Start":"07:43.990 ","End":"07:50.105","Text":"and a came out in our case to be 1/27,"},{"Start":"07:50.105 ","End":"07:54.645","Text":"which falls under the case a less than 1."},{"Start":"07:54.645 ","End":"08:01.950","Text":"So the answer is that our original series converges and we\u0027re done."}],"ID":10547},{"Watched":false,"Name":"Exercise 8 Part b","Duration":"","ChapterTopicVideoID":10224,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this exercise we\u0027re given a series,"},{"Start":"00:04.440 ","End":"00:08.100","Text":"it\u0027s an infinite series because of the infinity and as"},{"Start":"00:08.100 ","End":"00:12.930","Text":"such it will converge or diverge and we have to decide which."},{"Start":"00:12.930 ","End":"00:16.160","Text":"Because it contains factorials and exponents,"},{"Start":"00:16.160 ","End":"00:21.570","Text":"the natural approach would be to try the ratio test,"},{"Start":"00:21.570 ","End":"00:23.805","Text":"and that\u0027s summarized in the box here."},{"Start":"00:23.805 ","End":"00:29.579","Text":"It relates to positive series and this certainly is a positive series."},{"Start":"00:29.579 ","End":"00:36.970","Text":"What we have to basically do is compute the limit of the ratio of consecutive terms,"},{"Start":"00:36.970 ","End":"00:44.190","Text":"so I\u0027ve copied it down here and we can get some more space."},{"Start":"00:45.110 ","End":"00:52.785","Text":"What I\u0027m going to do is this part is going to be our an."},{"Start":"00:52.785 ","End":"00:57.440","Text":"What I\u0027ll do is first of all without the summation,"},{"Start":"00:57.440 ","End":"01:02.755","Text":"let\u0027s just compute an plus 1 over an,"},{"Start":"01:02.755 ","End":"01:09.030","Text":"and see what that equals and then we can take the limit as n goes to infinity."},{"Start":"01:09.520 ","End":"01:16.805","Text":"an plus 1 is what I get if I replace n by n plus 1,"},{"Start":"01:16.805 ","End":"01:23.405","Text":"so it\u0027s n plus 1 factorial over"},{"Start":"01:23.405 ","End":"01:30.720","Text":"n plus 1^n plus 1."},{"Start":"01:30.720 ","End":"01:33.105","Text":"Now I have to divide by n."},{"Start":"01:33.105 ","End":"01:35.795","Text":"Now, remember when you divide by a fraction,"},{"Start":"01:35.795 ","End":"01:38.630","Text":"you\u0027ll multiply by the inverse fraction."},{"Start":"01:38.630 ","End":"01:42.520","Text":"Instead of dividing by this over this I\u0027m going to multiply by this over this,"},{"Start":"01:42.520 ","End":"01:49.495","Text":"so here I have n to the n over n factorial."},{"Start":"01:49.495 ","End":"01:52.930","Text":"Now I\u0027d like to simplify this."},{"Start":"01:53.810 ","End":"02:02.490","Text":"n plus 1 factorial is just n plus 1 times n factorial,"},{"Start":"02:02.490 ","End":"02:04.250","Text":"you\u0027ve seen this trick before."},{"Start":"02:04.250 ","End":"02:08.675","Text":"This is all the numbers from 1 to n plus 1 multiplied together,"},{"Start":"02:08.675 ","End":"02:12.665","Text":"and here it\u0027s just from 1 to n. This has got 1 extra factor,"},{"Start":"02:12.665 ","End":"02:14.450","Text":"an extra n plus 1,"},{"Start":"02:14.450 ","End":"02:24.525","Text":"so this is just n plus 1 added preceding the product for n factorial."},{"Start":"02:24.525 ","End":"02:29.040","Text":"That\u0027s that. Now, I\u0027m going to write this part under here,"},{"Start":"02:29.040 ","End":"02:32.760","Text":"so it will be clearer when I\u0027m canceling."},{"Start":"02:32.760 ","End":"02:39.185","Text":"Then also multiply by n^n,"},{"Start":"02:39.185 ","End":"02:48.500","Text":"and over here I\u0027ll write this part here but instead of ^n plus 1,"},{"Start":"02:48.500 ","End":"02:57.475","Text":"allow me to use the rules of exponents and say it\u0027s n plus 1 to the n times n plus 1^1."},{"Start":"02:57.475 ","End":"03:02.930","Text":"I didn\u0027t need to write the 1,"},{"Start":"03:02.930 ","End":"03:09.015","Text":"it\u0027s unnecessary, I\u0027ll erase that."},{"Start":"03:09.015 ","End":"03:13.840","Text":"Now, stuff will start to cancel."},{"Start":"03:14.660 ","End":"03:22.230","Text":"This n plus 1 will cancel with this n plus"},{"Start":"03:22.230 ","End":"03:32.170","Text":"1 and this n factorial will cancel with this n factorial."},{"Start":"03:32.170 ","End":"03:38.750","Text":"What we\u0027re left with is just this over this,"},{"Start":"03:38.750 ","End":"03:47.550","Text":"which I can write as n over n plus 1^n."},{"Start":"03:48.590 ","End":"03:53.240","Text":"Then we have to take the limit of this as n goes to infinity,"},{"Start":"03:53.240 ","End":"03:55.740","Text":"and that\u0027s not clear at all."},{"Start":"03:56.050 ","End":"04:00.630","Text":"But there\u0027s a bit of a trick we can use,"},{"Start":"04:00.630 ","End":"04:03.300","Text":"and I\u0027m going to do this at the side."},{"Start":"04:03.300 ","End":"04:08.525","Text":"I\u0027m just going to label this with an asterisk and do the asterisk over here."},{"Start":"04:08.525 ","End":"04:14.150","Text":"What we have is n over"},{"Start":"04:14.150 ","End":"04:21.655","Text":"n plus 1^n equals,"},{"Start":"04:21.655 ","End":"04:24.030","Text":"and this is not necessarily intuitive,"},{"Start":"04:24.030 ","End":"04:29.450","Text":"there\u0027s a trick and you\u0027ll see where I\u0027m heading very shortly."},{"Start":"04:29.450 ","End":"04:34.190","Text":"I\u0027m going to write it as the inverse fraction but in the denominator,"},{"Start":"04:34.190 ","End":"04:42.510","Text":"so I\u0027m going to write it as 1 over and then reverse the fraction n plus 1 over n^n,"},{"Start":"04:43.810 ","End":"04:48.600","Text":"and this is equal to 1 over,"},{"Start":"04:48.620 ","End":"04:51.354","Text":"n over n is 1,"},{"Start":"04:51.354 ","End":"04:59.900","Text":"1 over n is 1 over n. I\u0027m just going to divide both of these by n^n."},{"Start":"05:00.740 ","End":"05:04.960","Text":"Now, I\u0027m going to take the limit."},{"Start":"05:05.650 ","End":"05:16.355","Text":"The limit as n goes to infinity of an plus 1 over an"},{"Start":"05:16.355 ","End":"05:22.650","Text":"is the limit as n goes to"},{"Start":"05:22.650 ","End":"05:31.690","Text":"infinity of 1 over 1 plus 1 over n^n."},{"Start":"05:34.670 ","End":"05:40.290","Text":"Because the denominator has a limit,"},{"Start":"05:41.270 ","End":"05:43.875","Text":"my limit is just 1 over that."},{"Start":"05:43.875 ","End":"05:46.010","Text":"Let me say it in another way,"},{"Start":"05:46.010 ","End":"05:48.740","Text":"I can actually put the limit in the denominator."},{"Start":"05:48.740 ","End":"05:54.210","Text":"It\u0027s actually equal to 1 over the limit as"},{"Start":"05:54.210 ","End":"06:00.355","Text":"n goes to infinity of 1 plus 1 over n^n."},{"Start":"06:00.355 ","End":"06:03.440","Text":"Now this happens to be a famous limit,"},{"Start":"06:03.440 ","End":"06:11.885","Text":"and I hope you remember it when you learned about the number e and exponential stuff."},{"Start":"06:11.885 ","End":"06:20.070","Text":"This limit is e. Let me just scroll down a bit more."},{"Start":"06:20.070 ","End":"06:25.815","Text":"This is equal to 1 over e,"},{"Start":"06:25.815 ","End":"06:28.290","Text":"look it up in your notes."},{"Start":"06:28.290 ","End":"06:30.650","Text":"I hope you remember it even,"},{"Start":"06:30.650 ","End":"06:36.740","Text":"it\u0027s a famous limit and the answer is e. The total thing is 1 over e. Now,"},{"Start":"06:36.740 ","End":"06:39.275","Text":"I go back up and say,"},{"Start":"06:39.275 ","End":"06:41.315","Text":"okay, in our case,"},{"Start":"06:41.315 ","End":"06:47.500","Text":"a turned out to be 1 over e. 1 over e,"},{"Start":"06:47.500 ","End":"06:49.670","Text":"I don\u0027t need to know much about e,"},{"Start":"06:49.670 ","End":"06:52.040","Text":"but I do know that e is bigger than 1,"},{"Start":"06:52.040 ","End":"06:56.180","Text":"it\u0027s roughly 2.71828 and whatever."},{"Start":"06:56.180 ","End":"07:05.355","Text":"1 over e happens to be less than 1 because it\u0027s 1 over 2 point something."},{"Start":"07:05.355 ","End":"07:11.115","Text":"Therefore according to this test our series converges,"},{"Start":"07:11.115 ","End":"07:12.815","Text":"and that is the answer."},{"Start":"07:12.815 ","End":"07:17.910","Text":"The original series converges. We\u0027re done."}],"ID":10548},{"Watched":false,"Name":"Exercise 8 Part c","Duration":"","ChapterTopicVideoID":10225,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.430 ","End":"00:10.245","Text":"an infinite series and we have to decide does it converge or does it diverge?"},{"Start":"00:10.245 ","End":"00:13.635","Text":"It has exponents and it has factorials,"},{"Start":"00:13.635 ","End":"00:17.475","Text":"which is usually an indication that we should be using the ratio test,"},{"Start":"00:17.475 ","End":"00:22.125","Text":"and I have it handy here in the box summary of the ratio test,"},{"Start":"00:22.125 ","End":"00:25.800","Text":"which basically talks about positive series,"},{"Start":"00:25.800 ","End":"00:27.350","Text":"and this is a positive series."},{"Start":"00:27.350 ","End":"00:32.550","Text":"We just need to compute the limit of the ratio of successive terms."},{"Start":"00:32.550 ","End":"00:35.190","Text":"According to the result of that,"},{"Start":"00:35.190 ","End":"00:38.760","Text":"we can decide it converges or diverges."},{"Start":"00:38.760 ","End":"00:44.910","Text":"I copied the exercise down here so we could get some more space."},{"Start":"00:45.680 ","End":"00:51.915","Text":"Let\u0027s call this general term a_n,"},{"Start":"00:51.915 ","End":"00:55.605","Text":"so we have the sum of a_n."},{"Start":"00:55.605 ","End":"01:02.700","Text":"Let\u0027s start by computing this ratio first and then we\u0027ll take the limit,"},{"Start":"01:02.700 ","End":"01:08.250","Text":"so a_n plus 1 over a_n,"},{"Start":"01:08.250 ","End":"01:12.135","Text":"for a general n is going to equal,"},{"Start":"01:12.135 ","End":"01:20.195","Text":"a_n plus 1 is what I get when I put n plus 1 instead of n. Here I\u0027ll get n plus"},{"Start":"01:20.195 ","End":"01:27.800","Text":"4 factorial over n plus"},{"Start":"01:27.800 ","End":"01:36.330","Text":"1 factorial times 3"},{"Start":"01:36.330 ","End":"01:39.495","Text":"to the power of n plus 1,"},{"Start":"01:39.495 ","End":"01:44.260","Text":"and a_n, which I\u0027m going to divide by."},{"Start":"01:45.470 ","End":"01:52.680","Text":"What I\u0027m going do is just instead dividing by a_n,"},{"Start":"01:52.680 ","End":"01:55.820","Text":"I\u0027m going to multiply by the reciprocal of a_n."},{"Start":"01:55.820 ","End":"01:57.414","Text":"That\u0027s how we do division."},{"Start":"01:57.414 ","End":"01:58.990","Text":"We multiply by the opposite,"},{"Start":"01:58.990 ","End":"02:00.520","Text":"the upside down fraction."},{"Start":"02:00.520 ","End":"02:04.990","Text":"I\u0027ll put the n factorial 3 to the n here,"},{"Start":"02:04.990 ","End":"02:07.240","Text":"and on the bottom,"},{"Start":"02:07.240 ","End":"02:12.765","Text":"the n plus 3 factorial."},{"Start":"02:12.765 ","End":"02:17.150","Text":"Okay. Let\u0027s see, try and simplify this."},{"Start":"02:17.150 ","End":"02:24.580","Text":"N plus 4 factorial is the product of all the numbers from 1 up to n plus 4."},{"Start":"02:24.580 ","End":"02:30.230","Text":"If you think about it, that\u0027s the same as taking the product of"},{"Start":"02:30.230 ","End":"02:37.460","Text":"all the numbers from 1 to n plus 3 and then having an extra factor of n plus 4."},{"Start":"02:37.460 ","End":"02:39.920","Text":"This is tricky, it\u0027s been done before,"},{"Start":"02:39.920 ","End":"02:41.375","Text":"it should be familiar."},{"Start":"02:41.375 ","End":"02:43.860","Text":"That\u0027s this part."},{"Start":"02:45.710 ","End":"02:55.420","Text":"On the denominator, I\u0027m going to have n plus 1 factorial,"},{"Start":"02:55.420 ","End":"03:03.860","Text":"but I\u0027m also going to write it as n plus 1 times n factorial."},{"Start":"03:03.860 ","End":"03:06.570","Text":"All numbers from 1 to n plus 1 is like"},{"Start":"03:06.570 ","End":"03:09.820","Text":"all numbers from 1 to n and an extra n plus 1 factor."},{"Start":"03:09.820 ","End":"03:15.445","Text":"I\u0027m doing it this way because my intention is to get stuff to cancel."},{"Start":"03:15.445 ","End":"03:17.840","Text":"The n plus 3 factorial,"},{"Start":"03:17.840 ","End":"03:21.390","Text":"I can see already ahead that I\u0027m going to cancel with that and"},{"Start":"03:21.390 ","End":"03:29.690","Text":"the n factorial with n factorial is just a master of the power of 3."},{"Start":"03:30.200 ","End":"03:35.880","Text":"What I\u0027m going to do is write the 3 to the n plus 1 as 3 to"},{"Start":"03:35.880 ","End":"03:42.030","Text":"the n times 3 to the 1, just 3."},{"Start":"03:42.030 ","End":"03:50.340","Text":"All this is just by rewriting the first part and I still have n factorial 3 to the n,"},{"Start":"03:50.340 ","End":"03:55.665","Text":"n plus 3 factorial."},{"Start":"03:55.665 ","End":"04:01.170","Text":"Okay. Now I can start a lot of canceling,"},{"Start":"04:01.170 ","End":"04:05.470","Text":"3 to the n with 3 to the n,"},{"Start":"04:05.570 ","End":"04:12.525","Text":"n plus 3 factorial with n plus 3 factorial,"},{"Start":"04:12.525 ","End":"04:18.990","Text":"and this n factorial with this n factorial."},{"Start":"04:18.990 ","End":"04:22.020","Text":"Let\u0027s see what\u0027s left."},{"Start":"04:22.020 ","End":"04:30.075","Text":"What remains after all this cancellation is n plus"},{"Start":"04:30.075 ","End":"04:38.085","Text":"4 over 3 times n plus 1,"},{"Start":"04:38.085 ","End":"04:42.975","Text":"which I\u0027ll write as 3n plus 3."},{"Start":"04:42.975 ","End":"04:47.190","Text":"Okay. That\u0027s a_n plus 1 over a_n,"},{"Start":"04:47.190 ","End":"04:49.560","Text":"but we want the limit."},{"Start":"04:49.560 ","End":"04:56.410","Text":"The limit as n goes to infinity of a_n plus 1 over"},{"Start":"04:56.410 ","End":"05:03.860","Text":"a_n is the limit as n goes to infinity of this."},{"Start":"05:03.860 ","End":"05:11.760","Text":"Just copy it, n plus 4 over 3n plus 3."},{"Start":"05:12.350 ","End":"05:17.645","Text":"When we have a limit of a rational function polynomials,"},{"Start":"05:17.645 ","End":"05:23.615","Text":"we can just take the leading term, the highest coefficient."},{"Start":"05:23.615 ","End":"05:29.550","Text":"That\u0027s the limit of just n over 3n,"},{"Start":"05:29.550 ","End":"05:33.135","Text":"n goes to infinity."},{"Start":"05:33.135 ","End":"05:40.035","Text":"N over 3n, this cancels and this cancels,"},{"Start":"05:40.035 ","End":"05:41.955","Text":"and you cancel and leave a 1."},{"Start":"05:41.955 ","End":"05:48.390","Text":"Limit of a constant is just the constant itself,"},{"Start":"05:48.390 ","End":"05:50.715","Text":"so this is 1/3."},{"Start":"05:50.715 ","End":"05:56.270","Text":"Okay. Now, I can go back up and say"},{"Start":"05:56.270 ","End":"06:02.270","Text":"that we have this limit and this limit happens to be 1/3,"},{"Start":"06:02.270 ","End":"06:08.730","Text":"1/3 falls under the category of less than 1."},{"Start":"06:08.740 ","End":"06:12.680","Text":"According to the ratio test theorem,"},{"Start":"06:12.680 ","End":"06:18.300","Text":"the series converges, and we\u0027re done."}],"ID":10549},{"Watched":false,"Name":"Exercise 8 Part d","Duration":"","ChapterTopicVideoID":10226,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"Here we have an infinite series,"},{"Start":"00:02.565 ","End":"00:04.655","Text":"see it goes from 1 to infinity."},{"Start":"00:04.655 ","End":"00:07.800","Text":"As such, it will either converge or diverge,"},{"Start":"00:07.800 ","End":"00:11.370","Text":"and our task is to decide which of the two."},{"Start":"00:11.370 ","End":"00:15.930","Text":"Because I see exponents and factorials,"},{"Start":"00:15.930 ","End":"00:24.405","Text":"the most natural thing to try is the ratio test which is summarized in this box here."},{"Start":"00:24.405 ","End":"00:27.840","Text":"First of all, let\u0027s just make sure that it\u0027s a positive series,"},{"Start":"00:27.840 ","End":"00:30.615","Text":"for sure everything here is positive."},{"Start":"00:30.615 ","End":"00:37.170","Text":"Then we have to compute the following limit and see what we get and decide accordingly,"},{"Start":"00:37.570 ","End":"00:40.730","Text":"so some more space here."},{"Start":"00:40.730 ","End":"00:44.270","Text":"This is a_n. In other words,"},{"Start":"00:44.270 ","End":"00:46.040","Text":"the typical term here,"},{"Start":"00:46.040 ","End":"00:47.855","Text":"we\u0027ll call it a_n."},{"Start":"00:47.855 ","End":"00:51.550","Text":"What we have to do is figure out the limit."},{"Start":"00:51.550 ","End":"00:59.565","Text":"As n goes to infinity of a_n plus 1 over a_n."},{"Start":"00:59.565 ","End":"01:06.525","Text":"This equals the limit as n goes to infinity."},{"Start":"01:06.525 ","End":"01:16.500","Text":"Now, a_n plus 1 is what happens when we substitute n plus 1 instead of n. Here,"},{"Start":"01:16.500 ","End":"01:18.480","Text":"if n is replaced by n plus 1,"},{"Start":"01:18.480 ","End":"01:21.280","Text":"we get 2n plus 2."},{"Start":"01:22.610 ","End":"01:28.660","Text":"Here we get n plus 1 factorial."},{"Start":"01:28.730 ","End":"01:40.170","Text":"Here we have 2n plus 2 to the power of n plus 1."},{"Start":"01:40.170 ","End":"01:43.075","Text":"That\u0027s the a_n plus 1 part."},{"Start":"01:43.075 ","End":"01:47.085","Text":"Now we have to divide it by a_n, which is this."},{"Start":"01:47.085 ","End":"01:50.060","Text":"As you know, a division by a fraction is the same as"},{"Start":"01:50.060 ","End":"01:53.900","Text":"multiplying by the upside down fraction, the reciprocal."},{"Start":"01:53.900 ","End":"01:57.015","Text":"We have n factorial"},{"Start":"01:57.015 ","End":"02:05.220","Text":"times 2n^n over 2n factorial."},{"Start":"02:07.010 ","End":"02:13.580","Text":"Now we want to do a little bit of canceling."},{"Start":"02:14.150 ","End":"02:18.420","Text":"To simplify it, so what we\u0027ll say is this."},{"Start":"02:18.420 ","End":"02:24.540","Text":"This is equal to the limit 2n plus 2 factorial is"},{"Start":"02:24.540 ","End":"02:31.439","Text":"2n plus 2 times 2n plus 1 times 2n,"},{"Start":"02:31.439 ","End":"02:33.745","Text":"and so on down to 1."},{"Start":"02:33.745 ","End":"02:35.090","Text":"But if you think about it,"},{"Start":"02:35.090 ","End":"02:40.770","Text":"what\u0027s left here is precisely 2n factorial."},{"Start":"02:42.660 ","End":"02:49.420","Text":"Now, I\u0027m going to divide this by n"},{"Start":"02:49.420 ","End":"02:55.625","Text":"plus 1 factorial using the same idea is just n plus 1 times n times n minus 1,"},{"Start":"02:55.625 ","End":"02:58.940","Text":"and the remainder is just n factorial."},{"Start":"02:58.940 ","End":"03:04.320","Text":"This here, I can at least split"},{"Start":"03:04.320 ","End":"03:14.789","Text":"the exponent n plus 1 into a product and say it\u0027s 2n plus 2 to the power of n,"},{"Start":"03:14.789 ","End":"03:19.620","Text":"times 2n plus 2 to the power of 1."},{"Start":"03:19.620 ","End":"03:21.720","Text":"I don\u0027t write the 1."},{"Start":"03:21.720 ","End":"03:24.800","Text":"That\u0027s the first part and the second part,"},{"Start":"03:24.800 ","End":"03:30.560","Text":"I\u0027ll just leave it as is n factorial 2n to"},{"Start":"03:30.560 ","End":"03:37.575","Text":"the power of n over 2n factorial,"},{"Start":"03:37.575 ","End":"03:40.590","Text":"limit as n goes to infinity."},{"Start":"03:40.590 ","End":"03:43.305","Text":"Let\u0027s see what cancels."},{"Start":"03:43.305 ","End":"03:49.590","Text":"Let\u0027s see, 2n factorial with 2n factorial,"},{"Start":"03:49.590 ","End":"03:54.910","Text":"n factorial with n factorial."},{"Start":"03:55.130 ","End":"04:03.105","Text":"Also 2n plus 2 with 2n plus 2."},{"Start":"04:03.105 ","End":"04:05.880","Text":"Now we can rewrite this. Let\u0027s see."},{"Start":"04:05.880 ","End":"04:09.580","Text":"The 2n plus 1 over the n plus 1."},{"Start":"04:10.580 ","End":"04:14.830","Text":"I can write this part first,"},{"Start":"04:14.830 ","End":"04:20.255","Text":"and then the second part, this over this,"},{"Start":"04:20.255 ","End":"04:25.190","Text":"I can write as 2n over 2n plus 2,"},{"Start":"04:27.550 ","End":"04:33.290","Text":"all this to the power of n,"},{"Start":"04:33.290 ","End":"04:38.485","Text":"and then the limit as n goes to infinity."},{"Start":"04:38.485 ","End":"04:41.650","Text":"Now, one thing about limits,"},{"Start":"04:41.650 ","End":"04:43.475","Text":"when you have a product,"},{"Start":"04:43.475 ","End":"04:46.430","Text":"if each limit separately exist,"},{"Start":"04:46.430 ","End":"04:50.900","Text":"the limit of this part and the limit of the other part,"},{"Start":"04:50.900 ","End":"04:54.620","Text":"they both exist and they\u0027re finite,"},{"Start":"04:54.620 ","End":"04:59.840","Text":"then I can take the separate limits and then multiply them at the end."},{"Start":"04:59.840 ","End":"05:02.760","Text":"What I\u0027m saying is,"},{"Start":"05:03.340 ","End":"05:06.745","Text":"the first limit I\u0027ll call,"},{"Start":"05:06.745 ","End":"05:11.050","Text":"let\u0027s say asterisk for the first part,"},{"Start":"05:11.120 ","End":"05:14.835","Text":"let\u0027s say double asterisk for the second part."},{"Start":"05:14.835 ","End":"05:17.050","Text":"The limit of this and the limit of this,"},{"Start":"05:17.050 ","End":"05:21.320","Text":"let\u0027s do each of those separately at the side."},{"Start":"05:21.320 ","End":"05:24.410","Text":"First the asterisk part."},{"Start":"05:24.410 ","End":"05:30.680","Text":"What I have is the limit as n goes to infinity of 2n plus"},{"Start":"05:30.680 ","End":"05:40.310","Text":"1 over n plus 1."},{"Start":"05:40.310 ","End":"05:44.095","Text":"Because these are polynomials,"},{"Start":"05:44.095 ","End":"05:49.945","Text":"we can just take the leading term for each and it will be the same limit."},{"Start":"05:49.945 ","End":"05:57.320","Text":"It\u0027s the limit of 2n over n. That\u0027s how we deal with rational functions."},{"Start":"05:57.320 ","End":"06:00.410","Text":"In each polynomial, take the one with the highest coefficient,"},{"Start":"06:00.410 ","End":"06:02.525","Text":"that\u0027s the leading term, this and this."},{"Start":"06:02.525 ","End":"06:04.805","Text":"Still n goes to infinity."},{"Start":"06:04.805 ","End":"06:07.810","Text":"Now 2n over n is a constant 2."},{"Start":"06:07.810 ","End":"06:10.894","Text":"The limit of a constant is just that constant,"},{"Start":"06:10.894 ","End":"06:13.205","Text":"so this is just 2."},{"Start":"06:13.205 ","End":"06:18.690","Text":"Let\u0027s take the other one, the double asterisk."},{"Start":"06:19.850 ","End":"06:22.190","Text":"Let me just do it over here."},{"Start":"06:22.190 ","End":"06:30.725","Text":"Here we get the limit as n goes to infinity of this."},{"Start":"06:30.725 ","End":"06:33.425","Text":"Let\u0027s just divide top and bottom by 2,"},{"Start":"06:33.425 ","End":"06:42.770","Text":"so we get n over n plus 1 to the power of n. You\u0027ve probably seen this before."},{"Start":"06:42.770 ","End":"06:46.190","Text":"It\u0027s a fairly standard trick here."},{"Start":"06:46.190 ","End":"06:48.140","Text":"What we do is,"},{"Start":"06:48.140 ","End":"06:58.769","Text":"we say that this is 1 over the upside down."},{"Start":"06:59.230 ","End":"07:01.835","Text":"If I take 1 over this,"},{"Start":"07:01.835 ","End":"07:06.575","Text":"it\u0027s n plus 1 over n to the power of"},{"Start":"07:06.575 ","End":"07:13.490","Text":"n. You can actually put the limit in the denominator,"},{"Start":"07:13.490 ","End":"07:18.410","Text":"your limit as n goes to infinity."},{"Start":"07:18.410 ","End":"07:21.230","Text":"If this turns out to exist and not 0,"},{"Start":"07:21.230 ","End":"07:23.105","Text":"then it means that they are equal."},{"Start":"07:23.105 ","End":"07:25.340","Text":"Let\u0027s continue with this."},{"Start":"07:25.340 ","End":"07:29.550","Text":"This is equal to 1"},{"Start":"07:29.550 ","End":"07:36.240","Text":"over the limit as n goes to infinity."},{"Start":"07:36.240 ","End":"07:41.810","Text":"This fraction, the n plus 1 over n can be rewritten as 1 plus 1 over"},{"Start":"07:41.810 ","End":"07:47.290","Text":"n to the power of n. This limit is a well known limit."},{"Start":"07:47.290 ","End":"07:49.920","Text":"You must have seen it before."},{"Start":"07:49.920 ","End":"07:52.950","Text":"This limit is e, so this is 1/e."},{"Start":"07:52.950 ","End":"07:55.340","Text":"Because this limit exists and not 0,"},{"Start":"07:55.340 ","End":"07:58.710","Text":"then it means that this is also equal to this."},{"Start":"07:58.710 ","End":"08:02.390","Text":"We\u0027ve taken care of the asterisk and the double asterisk."},{"Start":"08:02.390 ","End":"08:05.590","Text":"Back to this main line."},{"Start":"08:05.590 ","End":"08:08.895","Text":"We now continue this equals."},{"Start":"08:08.895 ","End":"08:12.164","Text":"The first part is 2,"},{"Start":"08:12.164 ","End":"08:15.600","Text":"the second part is 1/e,"},{"Start":"08:15.600 ","End":"08:18.975","Text":"and so it\u0027s 2/e."},{"Start":"08:18.975 ","End":"08:23.490","Text":"Now, I don\u0027t know the exact value"},{"Start":"08:23.490 ","End":"08:27.445","Text":"of e. In fact all I need to know is that it\u0027s bigger than 2."},{"Start":"08:27.445 ","End":"08:29.535","Text":"E is somewhere between 2 and 3."},{"Start":"08:29.535 ","End":"08:33.550","Text":"It\u0027s 2.71828 whatever, but it\u0027s bigger than 2."},{"Start":"08:33.550 ","End":"08:38.090","Text":"What I\u0027m saying is that this thing is less than 1."},{"Start":"08:38.090 ","End":"08:41.890","Text":"E is 2 point something."},{"Start":"08:41.900 ","End":"08:46.070","Text":"If we go back up to look at the situation,"},{"Start":"08:46.070 ","End":"08:53.010","Text":"we found that this a is equal to 2/e."},{"Start":"08:53.010 ","End":"08:54.390","Text":"We know it\u0027s less than 1,"},{"Start":"08:54.390 ","End":"08:58.320","Text":"so we\u0027re in the case where it\u0027s less than 1."},{"Start":"08:58.320 ","End":"09:03.855","Text":"We conclude that the series a_n converges but that\u0027s our original series."},{"Start":"09:03.855 ","End":"09:09.100","Text":"It converges and we are done."}],"ID":10550},{"Watched":false,"Name":"Exercise 8 Part e","Duration":"","ChapterTopicVideoID":10227,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.675","Text":"In this exercise, we\u0027re given an infinite series,"},{"Start":"00:03.675 ","End":"00:11.325","Text":"n goes from 1 to infinity and we have to decide if this series converges or diverges."},{"Start":"00:11.325 ","End":"00:15.090","Text":"Because of the presence of exponents and factorials,"},{"Start":"00:15.090 ","End":"00:17.280","Text":"the natural thing to try was"},{"Start":"00:17.280 ","End":"00:21.795","Text":"the ratio test and I\u0027ve written the ratio test here in the box."},{"Start":"00:21.795 ","End":"00:24.120","Text":"It applies to positive series,"},{"Start":"00:24.120 ","End":"00:29.355","Text":"this is positive and we have to compute a certain limit,"},{"Start":"00:29.355 ","End":"00:38.270","Text":"which is the ratio of two consecutive terms like this and according to the limit,"},{"Start":"00:38.270 ","End":"00:42.590","Text":"we can decide which of the two cases we\u0027re in."},{"Start":"00:42.590 ","End":"00:43.850","Text":"But first of all,"},{"Start":"00:43.850 ","End":"00:46.370","Text":"let\u0027s compute this limit."},{"Start":"00:46.370 ","End":"00:54.430","Text":"I must say that a_n is the general term so we have the sum of a_n."},{"Start":"00:56.210 ","End":"00:58.970","Text":"Let me take it without the limit first,"},{"Start":"00:58.970 ","End":"01:04.020","Text":"let\u0027s just see what is a_n plus 1 over a_n."},{"Start":"01:04.020 ","End":"01:06.000","Text":"Well, if this is a_n,"},{"Start":"01:06.000 ","End":"01:10.220","Text":"a_n plus 1 is the same thing with n replaced by n plus 1 so I"},{"Start":"01:10.220 ","End":"01:17.385","Text":"get 3_n plus 1 and then 1 plus instead of n,"},{"Start":"01:17.385 ","End":"01:20.880","Text":"n plus 1 squared."},{"Start":"01:20.880 ","End":"01:27.690","Text":"On the denominator, I have n plus 1 factorial and"},{"Start":"01:27.690 ","End":"01:30.940","Text":"now I\u0027m going to divide by a_n dividing"},{"Start":"01:30.940 ","End":"01:34.820","Text":"by a fraction is like multiplying by the upside-down fraction."},{"Start":"01:34.820 ","End":"01:41.330","Text":"The n factorial goes in the numerator and this bit here goes in the denominator,"},{"Start":"01:41.330 ","End":"01:46.350","Text":"3_n times 1 plus n squared."},{"Start":"01:46.690 ","End":"01:53.060","Text":"Now we\u0027ll do a little bit of algebra and then rewrite the 3_n plus 1"},{"Start":"01:53.060 ","End":"01:58.595","Text":"as 3_n times 3"},{"Start":"01:58.595 ","End":"02:03.525","Text":"over the n plus 1 factorial."},{"Start":"02:03.525 ","End":"02:04.680","Text":"This is an old trick."},{"Start":"02:04.680 ","End":"02:09.820","Text":"We know that it\u0027s n plus 1 times n factorial."},{"Start":"02:09.820 ","End":"02:16.460","Text":"Product of all the numbers from 1 to n and if we add this onto the list of factors,"},{"Start":"02:16.460 ","End":"02:20.515","Text":"we get products of the numbers from 1 to n plus 1."},{"Start":"02:20.515 ","End":"02:24.050","Text":"You\u0027ve seen this before probably and this here,"},{"Start":"02:24.050 ","End":"02:26.950","Text":"I\u0027ll just expand and write it as, let\u0027s see."},{"Start":"02:26.950 ","End":"02:33.720","Text":"This is n squared plus 2n plus 1 but because of the 1 here,"},{"Start":"02:33.720 ","End":"02:35.685","Text":"so it\u0027s plus 2."},{"Start":"02:35.685 ","End":"02:43.560","Text":"Here, as before, n factorial over 3_n ,"},{"Start":"02:43.560 ","End":"02:45.760","Text":"1 plus n squared."},{"Start":"02:45.760 ","End":"02:50.700","Text":"I\u0027ll write it as n squared plus 1 in decreasing order just like this one."},{"Start":"02:51.370 ","End":"02:55.140","Text":"First of all, let\u0027s do some cancellation,"},{"Start":"02:55.510 ","End":"02:59.460","Text":"3_n goes with 3_n,"},{"Start":"02:59.560 ","End":"03:04.860","Text":"n factorial with n factorial."},{"Start":"03:09.080 ","End":"03:15.870","Text":"What we\u0027re left with is 3 times n squared plus 2n plus 2"},{"Start":"03:15.870 ","End":"03:23.770","Text":"over n plus 1 times n squared plus 1."},{"Start":"03:23.770 ","End":"03:27.680","Text":"It\u0027s a rational expression,"},{"Start":"03:27.680 ","End":"03:33.710","Text":"polynomial over polynomial and when we take the limit of such a thing,"},{"Start":"03:33.710 ","End":"03:38.105","Text":"then only the leading terms count."},{"Start":"03:38.105 ","End":"03:45.830","Text":"The leading term here is 3n squared and if you expand the denominator,"},{"Start":"03:45.830 ","End":"03:50.240","Text":"the leading term is the n cubed you get from multiplying this by this,"},{"Start":"03:50.240 ","End":"03:54.390","Text":"I don\u0027t need the rest of it over n cubed."},{"Start":"03:58.790 ","End":"04:02.160","Text":"Of course, I need the limit of this,"},{"Start":"04:02.160 ","End":"04:07.170","Text":"limit as n goes to infinity,"},{"Start":"04:07.170 ","End":"04:13.435","Text":"but this is just the limit of 3 over n,"},{"Start":"04:13.435 ","End":"04:18.125","Text":"I divide top and bottom by n squared and when n goes to infinity,"},{"Start":"04:18.125 ","End":"04:21.425","Text":"this thing is a 0,"},{"Start":"04:21.425 ","End":"04:24.590","Text":"so the limit is 0."},{"Start":"04:24.590 ","End":"04:27.170","Text":"Now I go back up here and say, okay,"},{"Start":"04:27.170 ","End":"04:30.410","Text":"we found that a, its limit is 0."},{"Start":"04:30.410 ","End":"04:33.515","Text":"0 obviously less than 1,"},{"Start":"04:33.515 ","End":"04:36.870","Text":"so this is the case we\u0027re in and in that case,"},{"Start":"04:36.870 ","End":"04:39.270","Text":"the series a_n converges,"},{"Start":"04:39.270 ","End":"04:41.610","Text":"a_n is our original series so, therefore,"},{"Start":"04:41.610 ","End":"04:47.650","Text":"we conclude that this converges and we\u0027re done."}],"ID":10551},{"Watched":false,"Name":"Exercise 8 Part f","Duration":"","ChapterTopicVideoID":10228,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:04.080","Text":"Here we have an infinite series as follows and"},{"Start":"00:04.080 ","End":"00:07.710","Text":"we have to decide if it converges or diverges."},{"Start":"00:07.710 ","End":"00:13.545","Text":"What we\u0027re going to use here is something called the root test."},{"Start":"00:13.545 ","End":"00:18.165","Text":"The root test is good when you have exponents,"},{"Start":"00:18.165 ","End":"00:20.820","Text":"especially something to the power of n,"},{"Start":"00:20.820 ","End":"00:25.604","Text":"and preferably without factorials being present."},{"Start":"00:25.604 ","End":"00:27.930","Text":"In this case, root test is recommended,"},{"Start":"00:27.930 ","End":"00:32.160","Text":"although you probably could do it with the ratio test also."},{"Start":"00:32.160 ","End":"00:33.840","Text":"Anyway, we\u0027ll use the root test,"},{"Start":"00:33.840 ","End":"00:41.630","Text":"which says that what we compute is the limit of the nth root of the general term a_n."},{"Start":"00:41.630 ","End":"00:43.565","Text":"According to the result,"},{"Start":"00:43.565 ","End":"00:45.020","Text":"less than 1 converges,"},{"Start":"00:45.020 ","End":"00:46.595","Text":"greater than 1, diverges,"},{"Start":"00:46.595 ","End":"00:49.290","Text":"equals 1 we don\u0027t know."},{"Start":"00:49.720 ","End":"00:52.309","Text":"I need some more space,"},{"Start":"00:52.309 ","End":"00:57.025","Text":"that\u0027s why I rewrote the series."},{"Start":"00:57.025 ","End":"01:00.660","Text":"In general, we have here that a_n"},{"Start":"01:00.660 ","End":"01:09.760","Text":"is n^1000e to the minus n. So let\u0027s first of all compute the nth root."},{"Start":"01:09.760 ","End":"01:12.220","Text":"Then we\u0027ll take the limit."},{"Start":"01:14.720 ","End":"01:16.970","Text":"If I take the nth root,"},{"Start":"01:16.970 ","End":"01:20.880","Text":"I can take the nth root of each piece separately."},{"Start":"01:21.440 ","End":"01:24.830","Text":"But also remember that in general,"},{"Start":"01:24.830 ","End":"01:27.020","Text":"if I have some number,"},{"Start":"01:27.020 ","End":"01:31.280","Text":"say a, and I take the nth root,"},{"Start":"01:31.280 ","End":"01:35.420","Text":"it\u0027s the same as taking it to the power of 1."},{"Start":"01:35.420 ","End":"01:38.250","Text":"I can say that it\u0027s n^1000,"},{"Start":"01:38.410 ","End":"01:45.125","Text":"to the power of 1,"},{"Start":"01:45.125 ","End":"01:52.910","Text":"and e to the minus n^1."},{"Start":"01:52.910 ","End":"02:02.095","Text":"Given this, I can go back to the nth root here."},{"Start":"02:02.095 ","End":"02:09.200","Text":"I have a reason I\u0027m going to go back to saying it\u0027s the nth root of n^1000."},{"Start":"02:09.200 ","End":"02:17.330","Text":"But here I\u0027ll use the exponent rules and multiply the exponents and get e to the minus 1."},{"Start":"02:17.330 ","End":"02:23.005","Text":"Now the reason I did that is because we have a result you should have seen before,"},{"Start":"02:23.005 ","End":"02:28.580","Text":"and that is that it\u0027s a well-known limit that as n goes to infinity,"},{"Start":"02:28.580 ","End":"02:35.254","Text":"if we take the nth root of n to the power of any positive k,"},{"Start":"02:35.254 ","End":"02:38.050","Text":"that this is equal to 1."},{"Start":"02:38.050 ","End":"02:39.770","Text":"I\u0027m going to use that result here."},{"Start":"02:39.770 ","End":"02:43.085","Text":"I\u0027m not going to prove this from scratch."},{"Start":"02:43.085 ","End":"02:51.170","Text":"The limit as n goes to infinity of the nth root of a_n,"},{"Start":"02:51.170 ","End":"02:53.600","Text":"this is a constant."},{"Start":"02:53.600 ","End":"02:58.160","Text":"This thing, using this formula, equals 1."},{"Start":"02:58.160 ","End":"03:04.255","Text":"It\u0027s 1 times that constant, other words, 1/e."},{"Start":"03:04.255 ","End":"03:06.975","Text":"You don\u0027t even have to know exactly what e is,"},{"Start":"03:06.975 ","End":"03:08.770","Text":"but we know that e is bigger than 1,"},{"Start":"03:08.770 ","End":"03:11.870","Text":"it\u0027s 2.71828 or something,"},{"Start":"03:11.870 ","End":"03:14.874","Text":"but it\u0027s less than 1 and that\u0027s what counts."},{"Start":"03:14.874 ","End":"03:17.850","Text":"We\u0027re in the less than 1 case."},{"Start":"03:17.850 ","End":"03:20.515","Text":"If we\u0027re in less than 1 case,"},{"Start":"03:20.515 ","End":"03:27.690","Text":"then we have our answer that the series converges, and we\u0027re done."}],"ID":10553},{"Watched":false,"Name":"Exercise 8 Part g","Duration":"","ChapterTopicVideoID":10220,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.625","Text":"Here we have an infinite series,"},{"Start":"00:02.625 ","End":"00:06.465","Text":"and we have to decide if it converges or diverges."},{"Start":"00:06.465 ","End":"00:09.090","Text":"I\u0027m going to do it with the ratio test,"},{"Start":"00:09.090 ","End":"00:12.645","Text":"which is summarized in this box."},{"Start":"00:12.645 ","End":"00:15.389","Text":"It applies to a positive series,"},{"Start":"00:15.389 ","End":"00:17.760","Text":"which is what we have here,"},{"Start":"00:17.760 ","End":"00:22.935","Text":"and n goes from 2,"},{"Start":"00:22.935 ","End":"00:25.920","Text":"because if I started if n equals 1,"},{"Start":"00:25.920 ","End":"00:27.810","Text":"natural log of 1 is 0."},{"Start":"00:27.810 ","End":"00:29.340","Text":"That\u0027s the reason we start from 2,"},{"Start":"00:29.340 ","End":"00:31.170","Text":"but we still go to infinity."},{"Start":"00:31.170 ","End":"00:34.540","Text":"Now, with the ratio test,"},{"Start":"00:34.540 ","End":"00:39.380","Text":"we need to figure out the ratio of 2 successive terms,"},{"Start":"00:39.380 ","End":"00:41.555","Text":"generally a_n plus 1 over a_n,"},{"Start":"00:41.555 ","End":"00:44.390","Text":"and see what the limit is and decide accordingly."},{"Start":"00:44.390 ","End":"00:50.190","Text":"Let\u0027s call this our a_n."},{"Start":"00:51.200 ","End":"00:53.450","Text":"Forget the limit for the moment."},{"Start":"00:53.450 ","End":"00:58.210","Text":"Let\u0027s just see what is a_n plus 1, over a_n,"},{"Start":"00:58.210 ","End":"01:01.485","Text":"computed simplify it then we\u0027ll take the limit."},{"Start":"01:01.485 ","End":"01:08.005","Text":"a_n plus 1 is what we have here for replace n by n plus 1."},{"Start":"01:08.005 ","End":"01:11.565","Text":"It\u0027s 4 to the power of n plus 1."},{"Start":"01:11.565 ","End":"01:18.470","Text":"Then I\u0027ll use square brackets because we have n plus 1 squared,"},{"Start":"01:18.470 ","End":"01:23.970","Text":"plus 4 n plus 1, plus 5."},{"Start":"01:23.970 ","End":"01:26.070","Text":"That\u0027s the numerator."},{"Start":"01:26.070 ","End":"01:28.380","Text":"On the denominator,"},{"Start":"01:28.380 ","End":"01:35.085","Text":"we have 3 to the power of n plus 1,"},{"Start":"01:35.085 ","End":"01:42.200","Text":"times natural log of n plus 1."},{"Start":"01:42.200 ","End":"01:45.995","Text":"Now, all this is just the a_n plus 1 part."},{"Start":"01:45.995 ","End":"01:49.495","Text":"I still have to divide by a_n."},{"Start":"01:49.495 ","End":"01:53.840","Text":"As usual, when we divide by a fraction,"},{"Start":"01:53.840 ","End":"01:56.585","Text":"we multiply by the inverse fraction."},{"Start":"01:56.585 ","End":"02:03.045","Text":"This is 3 to the n natural log of n over."},{"Start":"02:03.045 ","End":"02:04.920","Text":"I\u0027m just copying from here,"},{"Start":"02:04.920 ","End":"02:07.050","Text":"4 to the n,"},{"Start":"02:07.050 ","End":"02:13.335","Text":"n squared, plus 4n, plus 5."},{"Start":"02:13.335 ","End":"02:16.030","Text":"Now let\u0027s see."},{"Start":"02:16.030 ","End":"02:20.960","Text":"I can rewrite this a bit and then do some canceling."},{"Start":"02:20.960 ","End":"02:23.970","Text":"This is equal to."},{"Start":"02:24.560 ","End":"02:35.985","Text":"Now, I already can see that 4 to the n plus 1 over 4 to the n, is just 4."},{"Start":"02:35.985 ","End":"02:40.470","Text":"Because this is 4 to the n times 4 to the 1."},{"Start":"02:40.470 ","End":"02:46.095","Text":"This is just 4 an easy exponent question."},{"Start":"02:46.095 ","End":"02:47.600","Text":"Then this, over this,"},{"Start":"02:47.600 ","End":"02:50.605","Text":"or the other way round is over 3."},{"Start":"02:50.605 ","End":"02:53.930","Text":"Then, actually invited in separate bits."},{"Start":"02:53.930 ","End":"02:58.080","Text":"Then I can write natural log of n,"},{"Start":"02:58.080 ","End":"03:02.210","Text":"over natural log of n plus 1."},{"Start":"03:02.210 ","End":"03:04.610","Text":"That\u0027s this bit and this bit."},{"Start":"03:04.610 ","End":"03:08.695","Text":"Then we have a polynomial over a polynomial."},{"Start":"03:08.695 ","End":"03:15.615","Text":"Here, we just take the n squared plus 4n plus 5."},{"Start":"03:15.615 ","End":"03:18.185","Text":"Here, the result of expanding this,"},{"Start":"03:18.185 ","End":"03:22.670","Text":"I actually don\u0027t need to expand it I only need the first term."},{"Start":"03:22.670 ","End":"03:24.950","Text":"With polynomials, you only need the highest tenth,"},{"Start":"03:24.950 ","End":"03:26.960","Text":"so it\u0027s n squared plus something."},{"Start":"03:26.960 ","End":"03:31.655","Text":"But, we could figure out what this is."},{"Start":"03:31.655 ","End":"03:35.345","Text":"This is n squared plus 2n plus 1,"},{"Start":"03:35.345 ","End":"03:39.680","Text":"plus 4n, plus 1 plus 5."},{"Start":"03:39.680 ","End":"03:44.060","Text":"I think it comes out to be let\u0027s see this 2n and 4n is 6n,"},{"Start":"03:44.060 ","End":"03:46.490","Text":"and 1, and 4,"},{"Start":"03:46.490 ","End":"03:50.135","Text":"and 5 is 10."},{"Start":"03:50.135 ","End":"03:51.830","Text":"But like I said, it doesn\u0027t matter."},{"Start":"03:51.830 ","End":"03:53.750","Text":"We just need to know the leading term."},{"Start":"03:53.750 ","End":"03:55.295","Text":"What we\u0027re going to do now,"},{"Start":"03:55.295 ","End":"03:57.280","Text":"is take the limit,"},{"Start":"03:57.280 ","End":"04:04.245","Text":"as n goes to infinity of a_n plus 1 over a_n."},{"Start":"04:04.245 ","End":"04:08.500","Text":"This is equal to a constant comes out before the limit."},{"Start":"04:08.500 ","End":"04:15.604","Text":"It\u0027s 4 over 3 times the limit as n goes to infinity of this product."},{"Start":"04:15.604 ","End":"04:19.715","Text":"Now, the limit of a product is the product of the limits"},{"Start":"04:19.715 ","End":"04:25.095","Text":"provided both of them come out to be finite."},{"Start":"04:25.095 ","End":"04:28.580","Text":"The result, the ends justify the means."},{"Start":"04:28.580 ","End":"04:30.395","Text":"If the result comes out, okay,"},{"Start":"04:30.395 ","End":"04:31.940","Text":"it would have been okay,"},{"Start":"04:31.940 ","End":"04:35.470","Text":"so I\u0027ll take the limit of natural log of n,"},{"Start":"04:35.470 ","End":"04:39.820","Text":"over natural log of n plus 1,"},{"Start":"04:40.900 ","End":"04:47.635","Text":"times the limit as n goes to infinity."},{"Start":"04:47.635 ","End":"04:51.680","Text":"I just did a copy paste of this thing here."},{"Start":"04:51.680 ","End":"04:54.800","Text":"Now, we don\u0027t have a problem with the polynomial."},{"Start":"04:54.800 ","End":"05:02.285","Text":"With the polynomial, this bit here is the limit."},{"Start":"05:02.285 ","End":"05:04.355","Text":"We just take the leading terms."},{"Start":"05:04.355 ","End":"05:07.940","Text":"We can forget about the 6n plus 10,"},{"Start":"05:07.940 ","End":"05:09.110","Text":"and the 4n plus 5."},{"Start":"05:09.110 ","End":"05:11.825","Text":"It\u0027s n squared over n squared, which is 1."},{"Start":"05:11.825 ","End":"05:14.630","Text":"This limit is equal to 1."},{"Start":"05:14.630 ","End":"05:18.115","Text":"But the question is, what is this limit?"},{"Start":"05:18.115 ","End":"05:22.200","Text":"That\u0027s something that\u0027s not quite clear."},{"Start":"05:22.200 ","End":"05:26.420","Text":"What I\u0027m going to do is they\u0027ll do an exercise at the side."},{"Start":"05:26.420 ","End":"05:30.160","Text":"To compute this, I\u0027m going to use calculus."},{"Start":"05:30.160 ","End":"05:32.900","Text":"The method I suggest is replacing"},{"Start":"05:32.900 ","End":"05:38.090","Text":"the integer n by a general variable x going to infinity."},{"Start":"05:38.090 ","End":"05:46.400","Text":"I\u0027m going to discuss what is the limit as x goes to infinity of the natural log of x,"},{"Start":"05:46.400 ","End":"05:51.500","Text":"over the natural log of x plus 1."},{"Start":"05:51.500 ","End":"05:53.810","Text":"If I find this limit,"},{"Start":"05:53.810 ","End":"05:57.200","Text":"then that will be the same as this limit."},{"Start":"05:57.200 ","End":"06:03.935","Text":"Now this is 1 of those cases where we have infinity over infinity."},{"Start":"06:03.935 ","End":"06:10.680","Text":"We use L\u0027Hospital\u0027s rule to spell his name."},{"Start":"06:10.680 ","End":"06:12.690","Text":"Gets an honorable mention."},{"Start":"06:12.690 ","End":"06:17.955","Text":"L\u0027Hospital, who dealt with this limit."},{"Start":"06:17.955 ","End":"06:22.280","Text":"He basically said that if you have an infinity over infinity,"},{"Start":"06:22.280 ","End":"06:24.950","Text":"you can differentiate each piece and"},{"Start":"06:24.950 ","End":"06:28.130","Text":"the new limit will be the same value as the old limit."},{"Start":"06:28.130 ","End":"06:32.070","Text":"This is equal to the limit."},{"Start":"06:32.070 ","End":"06:37.954","Text":"Put the equal somewhere equals the limit."},{"Start":"06:37.954 ","End":"06:39.710","Text":"As x goes to infinity."},{"Start":"06:39.710 ","End":"06:43.400","Text":"The derivative of this is 1 over x,"},{"Start":"06:43.400 ","End":"06:49.500","Text":"and the derivative of the denominator is 1 over x plus 1."},{"Start":"06:50.480 ","End":"06:56.240","Text":"This is equal to the limit of,"},{"Start":"06:56.240 ","End":"07:01.090","Text":"if I just play around with the fractions x plus 1 over x,"},{"Start":"07:01.090 ","End":"07:10.335","Text":"it just comes out to be 1 plus 1 over x as x goes to infinity."},{"Start":"07:10.335 ","End":"07:12.180","Text":"Obviously as x goes to infinity,"},{"Start":"07:12.180 ","End":"07:14.085","Text":"1 over x goes to 0."},{"Start":"07:14.085 ","End":"07:17.160","Text":"This is just equal to 1."},{"Start":"07:17.160 ","End":"07:20.165","Text":"Now we\u0027ve got the missing piece of the puzzle."},{"Start":"07:20.165 ","End":"07:24.480","Text":"This question mark is actually 1."},{"Start":"07:26.360 ","End":"07:33.910","Text":"For this limit the product of 4/3 times 1 times 1."},{"Start":"07:34.300 ","End":"07:44.285","Text":"In other words, the limit here is equal to 4/3."},{"Start":"07:44.285 ","End":"07:49.110","Text":"Our a here, this limit is 4/3,"},{"Start":"07:49.110 ","End":"07:52.065","Text":"and we are now in case 2,"},{"Start":"07:52.065 ","End":"07:54.540","Text":"where a is bigger than 1,"},{"Start":"07:54.540 ","End":"07:56.445","Text":"obviously 4/3 is bigger than 1,"},{"Start":"07:56.445 ","End":"08:00.525","Text":"and that means that a_n diverges."},{"Start":"08:00.525 ","End":"08:02.285","Text":"That\u0027s our answer."},{"Start":"08:02.285 ","End":"08:07.290","Text":"The infinite series diverges and we\u0027re done."}],"ID":10554},{"Watched":false,"Name":"Exercise 8 Part h","Duration":"","ChapterTopicVideoID":10221,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Here, we have an infinite series,"},{"Start":"00:03.000 ","End":"00:05.445","Text":"it goes from 1 to infinity."},{"Start":"00:05.445 ","End":"00:08.565","Text":"It\u0027s a positive series."},{"Start":"00:08.565 ","End":"00:12.945","Text":"We have to decide if it converges or diverges."},{"Start":"00:12.945 ","End":"00:16.395","Text":"There might be other ways of doing it but I\u0027m going to go with"},{"Start":"00:16.395 ","End":"00:22.605","Text":"the ratio test which is summarized in this box here."},{"Start":"00:22.605 ","End":"00:26.710","Text":"I\u0027ll just get some more space here."},{"Start":"00:26.720 ","End":"00:31.440","Text":"What we have to do basically in"},{"Start":"00:31.440 ","End":"00:36.440","Text":"this ratio test is to figure out the limit of the ratio of successive terms,"},{"Start":"00:36.440 ","End":"00:38.705","Text":"and according to what we get,"},{"Start":"00:38.705 ","End":"00:42.445","Text":"decide as written here."},{"Start":"00:42.445 ","End":"00:45.090","Text":"Let\u0027s call this general term,"},{"Start":"00:45.090 ","End":"00:48.945","Text":"this bit here, this will be our a_n."},{"Start":"00:48.945 ","End":"00:51.030","Text":"Then we\u0027re going to compute this."},{"Start":"00:51.030 ","End":"00:56.670","Text":"First, we\u0027ll just see what is a_n plus 1 over a_n in general,"},{"Start":"00:56.670 ","End":"01:01.050","Text":"and then we\u0027ll take the limit and see what we get."},{"Start":"01:01.450 ","End":"01:07.400","Text":"a_n plus 1 is what we get here if we replace n by n plus 1."},{"Start":"01:07.400 ","End":"01:13.740","Text":"We get 4 n plus 1"},{"Start":"01:13.740 ","End":"01:20.280","Text":"squared plus 5 times n plus 1 plus 1."},{"Start":"01:20.280 ","End":"01:24.810","Text":"All this to the power of n plus"},{"Start":"01:24.810 ","End":"01:31.984","Text":"1 over 4 to the n plus 1,"},{"Start":"01:31.984 ","End":"01:37.760","Text":"and n plus 1 to the power of twice n plus 1,"},{"Start":"01:37.760 ","End":"01:40.295","Text":"I\u0027ll write it as 2n plus 2."},{"Start":"01:40.295 ","End":"01:42.880","Text":"That\u0027s just the a_n part."},{"Start":"01:42.880 ","End":"01:46.155","Text":"Now, we still have to divide by a_n."},{"Start":"01:46.155 ","End":"01:51.800","Text":"Dividing by a fraction is like multiplying by the inverse fraction."},{"Start":"01:51.800 ","End":"01:55.205","Text":"I\u0027ll put the 4 to the n,"},{"Start":"01:55.205 ","End":"01:58.085","Text":"n to the 2n on the top,"},{"Start":"01:58.085 ","End":"02:00.050","Text":"that was what was here, and this,"},{"Start":"02:00.050 ","End":"02:09.795","Text":"we\u0027ll put on the bottom here is 4n squared plus 5n plus 1."},{"Start":"02:09.795 ","End":"02:14.655","Text":"This to the power of n. It looks quite a mess."},{"Start":"02:14.655 ","End":"02:18.810","Text":"Hopefully, we can simplify it with a bit of algebra."},{"Start":"02:19.030 ","End":"02:25.490","Text":"Let\u0027s see. Now, I don\u0027t actually"},{"Start":"02:25.490 ","End":"02:30.980","Text":"have to multiply this out because only the leading term counts,"},{"Start":"02:30.980 ","End":"02:32.030","Text":"but I\u0027ll do it anyway."},{"Start":"02:32.030 ","End":"02:36.250","Text":"I did it on the side and I\u0027ll just quote the result."},{"Start":"02:36.250 ","End":"02:39.855","Text":"Well, perhaps, I\u0027ll give you a quick explanation, let\u0027 just see."},{"Start":"02:39.855 ","End":"02:43.035","Text":"From this part, we get n squared plus 2n plus 1."},{"Start":"02:43.035 ","End":"02:46.500","Text":"There\u0027s a 4n squared plus 8n,"},{"Start":"02:46.500 ","End":"02:51.300","Text":"but the 8n combines with the 5n, so it\u0027s 13n."},{"Start":"02:51.300 ","End":"02:56.190","Text":"The plus 1 is plus 4 goes with the plus 5 and the plus 1,"},{"Start":"02:56.190 ","End":"03:00.765","Text":"so we get actually plus 10."},{"Start":"03:00.765 ","End":"03:06.790","Text":"All this to the power of n plus 1."},{"Start":"03:08.870 ","End":"03:11.730","Text":"I\u0027ll put the dividing line now,"},{"Start":"03:11.730 ","End":"03:13.110","Text":"make it a long one,"},{"Start":"03:13.110 ","End":"03:14.685","Text":"a lot of stuff here."},{"Start":"03:14.685 ","End":"03:20.315","Text":"You know what? I\u0027ll take this part of the denominator first because it belongs."},{"Start":"03:20.315 ","End":"03:28.100","Text":"4n squared plus 5n plus"},{"Start":"03:28.100 ","End":"03:35.390","Text":"1 to the power of n. You know what?"},{"Start":"03:35.390 ","End":"03:39.220","Text":"I want to save a step because it\u0027s tedious."},{"Start":"03:39.220 ","End":"03:45.785","Text":"This n plus 1, I\u0027ll write as to the power of n and then to the power of 1."},{"Start":"03:45.785 ","End":"03:48.360","Text":"I\u0027ll erase this plus"},{"Start":"03:48.360 ","End":"03:53.920","Text":"1 and I\u0027ll just write another one of these factors to the power of 1,"},{"Start":"03:53.920 ","End":"03:57.745","Text":"4n squared plus 13n plus 10."},{"Start":"03:57.745 ","End":"03:59.230","Text":"That\u0027s the n, that\u0027s the 1."},{"Start":"03:59.230 ","End":"04:01.520","Text":"You don\u0027t see the 1, you don\u0027t need it."},{"Start":"04:01.520 ","End":"04:03.925","Text":"Now, let\u0027s see what else."},{"Start":"04:03.925 ","End":"04:09.050","Text":"I\u0027ll put the 4 to the power of n next."},{"Start":"04:11.630 ","End":"04:18.635","Text":"Down here, I\u0027ll take the 4 to the power of n plus 1,"},{"Start":"04:18.635 ","End":"04:22.410","Text":"a bit closer to this."},{"Start":"04:22.410 ","End":"04:28.270","Text":"Then here, we have n to the 2n."},{"Start":"04:30.950 ","End":"04:37.020","Text":"Here, I\u0027ll take n plus 1,"},{"Start":"04:37.020 ","End":"04:40.555","Text":"first to the power of 2n,"},{"Start":"04:40.555 ","End":"04:44.285","Text":"and the plus 2, I\u0027ll write separately,"},{"Start":"04:44.285 ","End":"04:48.270","Text":"n plus 1 squared."},{"Start":"04:49.390 ","End":"04:52.920","Text":"It\u0027s beginning to take shape."},{"Start":"04:53.050 ","End":"04:57.305","Text":"Now, let\u0027s start simplifying."},{"Start":"04:57.305 ","End":"05:01.520","Text":"Now, this over this,"},{"Start":"05:01.520 ","End":"05:09.270","Text":"the both of the power of n so I can write that bit as 4n squared"},{"Start":"05:09.270 ","End":"05:17.400","Text":"plus 13n plus 10 over 4n squared"},{"Start":"05:17.400 ","End":"05:20.535","Text":"plus 5n plus 1,"},{"Start":"05:20.535 ","End":"05:24.660","Text":"all this to the power of n, that\u0027s these here."},{"Start":"05:24.660 ","End":"05:31.515","Text":"Next, I\u0027ll take the 4 to the n over 4 to the n plus 1."},{"Start":"05:31.515 ","End":"05:37.000","Text":"That gives me a quarter."},{"Start":"05:37.940 ","End":"05:45.490","Text":"Then I\u0027ll take the n to the 2n over n plus 1 to the 2n,"},{"Start":"05:45.490 ","End":"05:53.100","Text":"and write it as n over n plus 1 to the power of 2n."},{"Start":"05:54.350 ","End":"05:56.415","Text":"Let\u0027s see."},{"Start":"05:56.415 ","End":"05:58.920","Text":"What else do we have?"},{"Start":"05:58.920 ","End":"06:03.315","Text":"What we have remaining is this,"},{"Start":"06:03.315 ","End":"06:11.640","Text":"4n squared plus 13n plus 10."},{"Start":"06:11.640 ","End":"06:17.050","Text":"Here, we have n plus 1 squared."},{"Start":"06:22.460 ","End":"06:25.490","Text":"Instead of putting the quarter here,"},{"Start":"06:25.490 ","End":"06:29.250","Text":"I think I\u0027d like to throw this 4 here."},{"Start":"06:29.360 ","End":"06:31.500","Text":"I\u0027ll write the 4 here,"},{"Start":"06:31.500 ","End":"06:33.960","Text":"but now, I have to erase this."},{"Start":"06:33.960 ","End":"06:36.445","Text":"There we are."},{"Start":"06:36.445 ","End":"06:39.595","Text":"Maybe I\u0027ll just squeeze it a bit closer."},{"Start":"06:39.595 ","End":"06:42.575","Text":"Just move it a bit."},{"Start":"06:42.575 ","End":"06:44.945","Text":"There we go."},{"Start":"06:44.945 ","End":"06:48.360","Text":"That\u0027s it."},{"Start":"06:50.330 ","End":"06:54.105","Text":"Let\u0027s get some more space."},{"Start":"06:54.105 ","End":"06:57.860","Text":"The tricky part is actually this first part."},{"Start":"06:57.860 ","End":"07:04.309","Text":"I had the idea that we could probably factorize and it might come out simpler."},{"Start":"07:04.309 ","End":"07:06.440","Text":"Now, I don\u0027t want to waste too much time with"},{"Start":"07:06.440 ","End":"07:12.060","Text":"the factorization so I\u0027ll just write it out for you."},{"Start":"07:20.360 ","End":"07:23.524","Text":"No, I\u0027ll do the denominator first."},{"Start":"07:23.524 ","End":"07:33.205","Text":"The denominator is 4n plus 1 times n plus 1."},{"Start":"07:33.205 ","End":"07:36.645","Text":"You probably know how to factor."},{"Start":"07:36.645 ","End":"07:39.135","Text":"There\u0027s many ways of doing it."},{"Start":"07:39.135 ","End":"07:43.325","Text":"All I\u0027m willing to do at this stage is just to check by multiplication."},{"Start":"07:43.325 ","End":"07:48.245","Text":"4n times n is 4n squared plus n plus 4n is 5n plus 1."},{"Start":"07:48.245 ","End":"07:49.865","Text":"You should know how to factor."},{"Start":"07:49.865 ","End":"07:54.029","Text":"The top comes out to be 4n plus"},{"Start":"07:54.029 ","End":"08:01.360","Text":"5, n plus 2."},{"Start":"08:02.750 ","End":"08:05.160","Text":"Let\u0027s just check that."},{"Start":"08:05.160 ","End":"08:10.560","Text":"4n times n is 4n squared plus 5n plus 8n is 13n,"},{"Start":"08:10.560 ","End":"08:11.970","Text":"5 times 2 is 10."},{"Start":"08:11.970 ","End":"08:17.720","Text":"This is to the power of n. What I can do is"},{"Start":"08:17.720 ","End":"08:23.585","Text":"write each piece to the power of n. I separated them a bit."},{"Start":"08:23.585 ","End":"08:32.130","Text":"Now, I can say this to the power of n and this to the power of n,"},{"Start":"08:32.130 ","End":"08:40.285","Text":"and then n over n plus 1,"},{"Start":"08:40.285 ","End":"08:44.965","Text":"and this time, it\u0027s to the power of 2n."},{"Start":"08:44.965 ","End":"08:48.545","Text":"Then the last bit is going to be the easiest,"},{"Start":"08:48.545 ","End":"08:57.550","Text":"it\u0027s 4n squared plus 13n plus 10 over,"},{"Start":"08:57.550 ","End":"08:59.370","Text":"let\u0027s open it up,"},{"Start":"08:59.370 ","End":"09:05.630","Text":"4n squared plus 8n plus 4."},{"Start":"09:05.630 ","End":"09:09.910","Text":"Now, we want to take the limit"},{"Start":"09:09.910 ","End":"09:20.380","Text":"of a_n plus 1 over a_n as n goes to infinity."},{"Start":"09:20.380 ","End":"09:24.700","Text":"But this is made up of a product of 4 separate things."},{"Start":"09:24.700 ","End":"09:27.715","Text":"I\u0027d like to do each one separately."},{"Start":"09:27.715 ","End":"09:35.560","Text":"The first 3 are similar and the last one is very straightforward."},{"Start":"09:35.560 ","End":"09:40.485","Text":"But let me just say that this is the limit."},{"Start":"09:40.485 ","End":"09:44.240","Text":"Well, tell you what,"},{"Start":"09:44.240 ","End":"09:47.960","Text":"I\u0027ll just number them."},{"Start":"09:47.960 ","End":"09:53.310","Text":"Let\u0027s say this is expression."},{"Start":"09:54.720 ","End":"09:59.725","Text":"I don\u0027t know, A, and this one B,"},{"Start":"09:59.725 ","End":"10:06.830","Text":"and this one C, and this one D just symbolically could have said 2 asterisks."},{"Start":"10:07.260 ","End":"10:12.550","Text":"What I\u0027m going to say is that, first of all,"},{"Start":"10:12.550 ","End":"10:22.165","Text":"compute the limit of A and then multiply it by the limit of B,"},{"Start":"10:22.165 ","End":"10:29.965","Text":"just symbolically, and then limit of the third part,"},{"Start":"10:29.965 ","End":"10:36.805","Text":"and then limit the of last part,"},{"Start":"10:36.805 ","End":"10:38.995","Text":"which happens to be the easiest."},{"Start":"10:38.995 ","End":"10:42.655","Text":"This is just symbolically and I didn\u0027t write the n to infinity in each one."},{"Start":"10:42.655 ","End":"10:45.280","Text":"This mean I\u0027m going to take the limit of each one separately,"},{"Start":"10:45.280 ","End":"10:47.035","Text":"multiply them all together,"},{"Start":"10:47.035 ","End":"10:49.660","Text":"and this is legitimate to do,"},{"Start":"10:49.660 ","End":"10:54.175","Text":"provided that all these limits happen to exist."},{"Start":"10:54.175 ","End":"10:56.890","Text":"I\u0027ll start with the easy one,"},{"Start":"10:56.890 ","End":"11:02.665","Text":"the last one, and scroll down."},{"Start":"11:02.665 ","End":"11:05.890","Text":"Let\u0027s take D first, this right,"},{"Start":"11:05.890 ","End":"11:16.359","Text":"that I\u0027m working on D. Limit as n goes to infinity of,"},{"Start":"11:16.359 ","End":"11:18.445","Text":"I\u0027ll just copy it here over,"},{"Start":"11:18.445 ","End":"11:19.750","Text":"and we know how to do this."},{"Start":"11:19.750 ","End":"11:26.470","Text":"With polynomials, what counts is the leading term 4n squared over 4n squared, which is 1."},{"Start":"11:26.470 ","End":"11:28.060","Text":"We\u0027ve seen this trick before,"},{"Start":"11:28.060 ","End":"11:29.635","Text":"so I\u0027m not going to dwell on it,"},{"Start":"11:29.635 ","End":"11:32.750","Text":"the answer to this is 1."},{"Start":"11:33.180 ","End":"11:39.235","Text":"Now part C, we\u0027ve seen something similar to it before,"},{"Start":"11:39.235 ","End":"11:41.020","Text":"maybe not with 2n, but with n,"},{"Start":"11:41.020 ","End":"11:44.500","Text":"so let\u0027s do part C. I\u0027m going to say is that"},{"Start":"11:44.500 ","End":"11:50.395","Text":"the limit as n goes to infinity of this thing is,"},{"Start":"11:50.395 ","End":"11:53.630","Text":"I can write it as,"},{"Start":"11:54.270 ","End":"11:58.420","Text":"if I reverse the fraction,"},{"Start":"11:58.420 ","End":"12:04.330","Text":"let me make it upside down and write n plus 1 over n,"},{"Start":"12:04.330 ","End":"12:09.715","Text":"and then I can take it to the power of minus 2_n."},{"Start":"12:09.715 ","End":"12:17.300","Text":"The minus 2n, I\u0027m going to write to the power of n,"},{"Start":"12:18.120 ","End":"12:22.300","Text":"to the power of minus 2."},{"Start":"12:22.300 ","End":"12:26.390","Text":"I hope that\u0027s clear, I just quickly mentioned again what I did."},{"Start":"12:26.390 ","End":"12:29.550","Text":"I inverted the fraction,"},{"Start":"12:29.550 ","End":"12:30.900","Text":"so that\u0027s 1 over,"},{"Start":"12:30.900 ","End":"12:34.185","Text":"which means that makes it to the power of minus 1,"},{"Start":"12:34.185 ","End":"12:37.125","Text":"so I have to the power of minus 2n,"},{"Start":"12:37.125 ","End":"12:38.929","Text":"and the minus 2n,"},{"Start":"12:38.929 ","End":"12:43.310","Text":"I split into n times minus 2."},{"Start":"12:45.210 ","End":"12:50.080","Text":"Now, when you have something to the power of something minus 2,"},{"Start":"12:50.080 ","End":"12:53.965","Text":"I could take the whole thing to the power of minus 2."},{"Start":"12:53.965 ","End":"12:58.120","Text":"It\u0027s the limit as n goes to infinity of this thing,"},{"Start":"12:58.120 ","End":"13:00.249","Text":"I\u0027ll write it in a moment,"},{"Start":"13:00.249 ","End":"13:03.430","Text":"to the power of minus 2."},{"Start":"13:03.430 ","End":"13:07.390","Text":"Now, this thing, instead of writing it as is,"},{"Start":"13:07.390 ","End":"13:13.945","Text":"I\u0027ll write it as 1 plus 1 over n to the power of n. I just took the fraction,"},{"Start":"13:13.945 ","End":"13:15.160","Text":"so then n over n is 1,"},{"Start":"13:15.160 ","End":"13:18.415","Text":"1 over n, I\u0027m sure you\u0027ve seen this trick before."},{"Start":"13:18.415 ","End":"13:22.570","Text":"This is a well-known limit and it equals e,"},{"Start":"13:22.570 ","End":"13:26.815","Text":"the number e, so altogether we have e to the minus 2,"},{"Start":"13:26.815 ","End":"13:27.895","Text":"or if you like,"},{"Start":"13:27.895 ","End":"13:30.475","Text":"1 over e squared,"},{"Start":"13:30.475 ","End":"13:39.265","Text":"that\u0027s part C. Now what about parts A and B, are equally difficult."},{"Start":"13:39.265 ","End":"13:42.580","Text":"Let me do part B first."},{"Start":"13:42.580 ","End":"13:44.920","Text":"Just see if I can scroll a bit,"},{"Start":"13:44.920 ","End":"13:47.635","Text":"but still keep it in sight."},{"Start":"13:47.635 ","End":"13:55.730","Text":"Part B, we want the limit as n goes to infinity."},{"Start":"14:15.510 ","End":"14:19.075","Text":"I\u0027ll just copy it first and then I\u0027ll show you the trick."},{"Start":"14:19.075 ","End":"14:27.750","Text":"It\u0027s n plus 2 over n plus 1 to the power of n,"},{"Start":"14:27.750 ","End":"14:36.025","Text":"and it looks quite a lot like n plus 1 over n, but not quite."},{"Start":"14:36.025 ","End":"14:46.660","Text":"The trick I\u0027m going to suggest is to make a substitution that m is going to be n plus 1."},{"Start":"14:46.660 ","End":"14:48.190","Text":"I will just write at the side,"},{"Start":"14:48.190 ","End":"14:53.260","Text":"I\u0027m going to let m equal n plus 1,"},{"Start":"14:53.260 ","End":"15:01.730","Text":"and then the other way around n is m minus 1."},{"Start":"15:01.890 ","End":"15:11.040","Text":"I also can see that n plus 2 will be m plus 1,"},{"Start":"15:11.040 ","End":"15:14.100","Text":"if I just add 1 to both sides here."},{"Start":"15:14.100 ","End":"15:19.695","Text":"The other thing is that when n goes to infinity,"},{"Start":"15:19.695 ","End":"15:23.815","Text":"so does m, m also goes to infinity."},{"Start":"15:23.815 ","End":"15:29.395","Text":"Because something plus 1 also goes,"},{"Start":"15:29.395 ","End":"15:33.730","Text":"m is just 1 more just gets to infinity,"},{"Start":"15:33.730 ","End":"15:37.075","Text":"one step ahead, it\u0027s still infinity."},{"Start":"15:37.075 ","End":"15:39.640","Text":"When n goes to infinity,"},{"Start":"15:39.640 ","End":"15:47.305","Text":"then m also goes to infinity."},{"Start":"15:47.305 ","End":"15:55.450","Text":"What we are left with here is the limit as m goes to infinity,"},{"Start":"15:55.450 ","End":"16:06.010","Text":"m plus 1 over m to the power of n,"},{"Start":"16:06.010 ","End":"16:08.679","Text":"which is m minus 1,"},{"Start":"16:08.679 ","End":"16:11.290","Text":"so instead of m minus 1,"},{"Start":"16:11.290 ","End":"16:16.165","Text":"I\u0027ll put m, and instead of the minus 1,"},{"Start":"16:16.165 ","End":"16:23.920","Text":"I will multiply by"},{"Start":"16:23.920 ","End":"16:28.225","Text":"m over m plus 1."},{"Start":"16:28.225 ","End":"16:35.150","Text":"This thing to the power of minus 1 its inverse."},{"Start":"16:36.930 ","End":"16:43.030","Text":"Now this bit we\u0027ve done up here,"},{"Start":"16:43.030 ","End":"16:47.719","Text":"this comes out to be e,"},{"Start":"16:49.230 ","End":"16:53.950","Text":"and this other part is 1,"},{"Start":"16:53.950 ","End":"17:01.840","Text":"so altogether this limit is e. Let\u0027s see,"},{"Start":"17:01.840 ","End":"17:07.495","Text":"why don\u0027t I just shade the ones we\u0027ve got already, I mean highlight."},{"Start":"17:07.495 ","End":"17:11.425","Text":"Part D came out to be 1,"},{"Start":"17:11.425 ","End":"17:19.270","Text":"part C came out to be 1 over e squared."},{"Start":"17:19.270 ","End":"17:22.569","Text":"That\u0027s D, that\u0027s C,"},{"Start":"17:22.569 ","End":"17:24.625","Text":"B we just did,"},{"Start":"17:24.625 ","End":"17:26.979","Text":"comes out to be e,"},{"Start":"17:26.979 ","End":"17:31.000","Text":"and we\u0027re just left with"},{"Start":"17:31.000 ","End":"17:38.780","Text":"A. I\u0027ll continue with A over here."},{"Start":"17:40.650 ","End":"17:45.625","Text":"I\u0027ll just divide this fraction top and bottom by 4,"},{"Start":"17:45.625 ","End":"17:58.640","Text":"and I\u0027ve got the limit as n goes to infinity of n"},{"Start":"17:59.130 ","End":"18:08.155","Text":"plus 5 over 4 divided by n plus"},{"Start":"18:08.155 ","End":"18:14.620","Text":"a 1/4 to the power of"},{"Start":"18:14.620 ","End":"18:22.675","Text":"n. I could use a very similar trick as in part B,"},{"Start":"18:22.675 ","End":"18:32.900","Text":"let\u0027s assume n is a continuous variable like x. I could let n plus a 1/4 equal m,"},{"Start":"18:33.390 ","End":"18:39.265","Text":"and everything would proceed pretty much like in step B,"},{"Start":"18:39.265 ","End":"18:42.110","Text":"and we would get the same answer."},{"Start":"18:42.480 ","End":"18:46.285","Text":"We\u0027re taking too long on this anyway."},{"Start":"18:46.285 ","End":"18:51.310","Text":"Just do the same strategy as in B,"},{"Start":"18:51.310 ","End":"18:57.430","Text":"pretty much like this and we\u0027ll get the answer also that the limit is equal"},{"Start":"18:57.430 ","End":"19:03.370","Text":"to e. I\u0027ll highlight"},{"Start":"19:03.370 ","End":"19:08.290","Text":"that the answer to part A is e. Now we have all the 4 parts,"},{"Start":"19:08.290 ","End":"19:12.080","Text":"and now we have to just multiply them together."},{"Start":"19:13.890 ","End":"19:20.540","Text":"Let\u0027s see now, we\u0027re running out of space there."},{"Start":"19:22.110 ","End":"19:28.720","Text":"The answer that we need is that the limit as n goes to"},{"Start":"19:28.720 ","End":"19:34.930","Text":"infinity of a_n plus 1 over a_n is equal to,"},{"Start":"19:34.930 ","End":"19:42.020","Text":"it\u0027s through the in order of A, B,"},{"Start":"19:42.020 ","End":"19:48.790","Text":"C, D. What we"},{"Start":"19:48.790 ","End":"19:57.580","Text":"get is 1."},{"Start":"19:57.580 ","End":"20:07.715","Text":"Unfortunately, this is problematic because in the theorem,"},{"Start":"20:07.715 ","End":"20:09.860","Text":"in the ratio test,"},{"Start":"20:09.860 ","End":"20:14.650","Text":"we actually don\u0027t know what happens when A equals 1,"},{"Start":"20:14.650 ","End":"20:17.520","Text":"and anything could happen."},{"Start":"20:18.680 ","End":"20:23.250","Text":"At this point I\u0027m just going to give up and say,"},{"Start":"20:23.500 ","End":"20:29.540","Text":"the ratio test has failed me and I really don\u0027t know."},{"Start":"20:29.540 ","End":"20:33.680","Text":"That\u0027s that."}],"ID":10555},{"Watched":false,"Name":"Exercise 8 Part i","Duration":"","ChapterTopicVideoID":10222,"CourseChapterTopicPlaylistID":52096,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.580 ","End":"00:06.420","Text":"this infinite series and we have to decide if it converges or"},{"Start":"00:06.420 ","End":"00:14.805","Text":"diverges and I\u0027ll already tell you that we\u0027re going to use what is called the root test."},{"Start":"00:14.805 ","End":"00:19.740","Text":"That\u0027s something that is usually recommended when you have exponents,"},{"Start":"00:19.740 ","End":"00:22.740","Text":"especially something to the power of n,"},{"Start":"00:22.740 ","End":"00:26.760","Text":"and when you don\u0027t have factorials."},{"Start":"00:26.760 ","End":"00:28.650","Text":"Then the root test is recommended,"},{"Start":"00:28.650 ","End":"00:32.835","Text":"although this could also be done with the ratio test,"},{"Start":"00:32.835 ","End":"00:34.935","Text":"they will both work."},{"Start":"00:34.935 ","End":"00:39.110","Text":"The root test basically says that when you have such an infinite series,"},{"Start":"00:39.110 ","End":"00:41.300","Text":"you take the general term a_n,"},{"Start":"00:41.300 ","End":"00:44.015","Text":"you take its nth root and find the limit,"},{"Start":"00:44.015 ","End":"00:46.625","Text":"and if that limit happens to be less than 1,"},{"Start":"00:46.625 ","End":"00:49.860","Text":"converges, greater than 1, diverges."},{"Start":"00:49.860 ","End":"00:52.580","Text":"If you get equal 1, you don\u0027t know."},{"Start":"00:52.580 ","End":"00:54.500","Text":"It works for positive series,"},{"Start":"00:54.500 ","End":"00:56.315","Text":"which is the case here."},{"Start":"00:56.315 ","End":"00:59.435","Text":"Let\u0027s get to this."},{"Start":"00:59.435 ","End":"01:02.820","Text":"I rewrote it down here."},{"Start":"01:03.250 ","End":"01:06.750","Text":"Our general term,"},{"Start":"01:08.330 ","End":"01:15.600","Text":"a_n is n squared over 2^n."},{"Start":"01:15.600 ","End":"01:16.890","Text":"Before I do the limit,"},{"Start":"01:16.890 ","End":"01:23.645","Text":"I would just like to say what is the nth root of a_n,"},{"Start":"01:23.645 ","End":"01:29.885","Text":"and I can take the nth root of the numerator separately."},{"Start":"01:29.885 ","End":"01:35.690","Text":"So I\u0027ve got the nth root of n squared,"},{"Start":"01:35.690 ","End":"01:39.470","Text":"and I can take the nth root of the denominator separately."},{"Start":"01:39.470 ","End":"01:48.630","Text":"Now, the nth root of 2^n is just going to be 2, in general."},{"Start":"01:48.760 ","End":"01:51.710","Text":"Take the nth power, then take the nth root,"},{"Start":"01:51.710 ","End":"01:53.930","Text":"you\u0027re back to the number itself."},{"Start":"01:53.930 ","End":"02:02.705","Text":"Or if you prefer, you can use exponent rule and say that the nth root of"},{"Start":"02:02.705 ","End":"02:12.020","Text":"2^n is 2^n to"},{"Start":"02:12.020 ","End":"02:15.200","Text":"the power of 1 over n. Because the nth root,"},{"Start":"02:15.200 ","End":"02:18.440","Text":"one way of defining it is by"},{"Start":"02:18.440 ","End":"02:24.195","Text":"fractional exponent and then we could use the rule of power over power."},{"Start":"02:24.195 ","End":"02:27.050","Text":"We multiply the exponents n times 1 over n,"},{"Start":"02:27.050 ","End":"02:29.960","Text":"which is 2^1, which is 2."},{"Start":"02:29.960 ","End":"02:31.645","Text":"But that\u0027s the long way around."},{"Start":"02:31.645 ","End":"02:33.620","Text":"So we\u0027re at this point."},{"Start":"02:33.620 ","End":"02:36.080","Text":"Now I\u0027m ready to take the limit."},{"Start":"02:36.080 ","End":"02:38.195","Text":"Well, almost ready."},{"Start":"02:38.195 ","End":"02:45.429","Text":"I also have to mention a result from Calculus 1,"},{"Start":"02:45.429 ","End":"02:47.420","Text":"which you should have encountered,"},{"Start":"02:47.420 ","End":"02:53.900","Text":"is that the limit as n goes to infinity of the nth root of n to the power of,"},{"Start":"02:53.900 ","End":"02:57.485","Text":"not just 2, but any positive integer k,"},{"Start":"02:57.485 ","End":"02:58.850","Text":"this is equal to 1."},{"Start":"02:58.850 ","End":"03:03.805","Text":"That\u0027s the result I\u0027m going to use and I\u0027m not going to prove it from scratch."},{"Start":"03:03.805 ","End":"03:12.240","Text":"What we get is that the limit of the nth root of a_n,"},{"Start":"03:12.240 ","End":"03:14.285","Text":"and here is our general term."},{"Start":"03:14.285 ","End":"03:16.535","Text":"The 2 is constant,"},{"Start":"03:16.535 ","End":"03:19.370","Text":"we have the limit of the numerator,"},{"Start":"03:19.370 ","End":"03:22.205","Text":"which is according to this formula,"},{"Start":"03:22.205 ","End":"03:24.980","Text":"equals 1 over 2."},{"Start":"03:24.980 ","End":"03:28.880","Text":"It comes out straight away because numerator fits this pattern,"},{"Start":"03:28.880 ","End":"03:31.700","Text":"denominator is a constant, there it is there."},{"Start":"03:31.700 ","End":"03:36.310","Text":"The important thing about 1/2 is that it\u0027s less than 1."},{"Start":"03:36.310 ","End":"03:39.825","Text":"Less than 1 means we\u0027re in this case,"},{"Start":"03:39.825 ","End":"03:46.170","Text":"and so our answer is that the series converges, and we\u0027re done."}],"ID":10556}],"Thumbnail":null,"ID":52096},{"Name":"The Alternating Series Test","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 9 Part a","Duration":"7m 8s","ChapterTopicVideoID":10231,"CourseChapterTopicPlaylistID":52097,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10231.jpeg","UploadDate":"2017-08-22T04:52:46.1430000","DurationForVideoObject":"PT7M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, we\u0027re given an infinite series,"},{"Start":"00:03.750 ","End":"00:08.325","Text":"this 1 and we have to decide if it converges or diverges."},{"Start":"00:08.325 ","End":"00:13.829","Text":"This minus 1 to the n. It means that we alternate signs"},{"Start":"00:13.829 ","End":"00:19.785","Text":"and that gave me the idea that we should use the alternating series test,"},{"Start":"00:19.785 ","End":"00:22.679","Text":"which is due to Leibnitz."},{"Start":"00:22.679 ","End":"00:31.695","Text":"That if you have something of this form with minus 1 to the n and the a_n part,"},{"Start":"00:31.695 ","End":"00:35.100","Text":"which is the part without the minus 1 to the n. If it"},{"Start":"00:35.100 ","End":"00:39.195","Text":"decreases ultimately to 0 at the limit,"},{"Start":"00:39.195 ","End":"00:41.755","Text":"then we know the series is convergent."},{"Start":"00:41.755 ","End":"00:45.260","Text":"What we\u0027d have to do then we would just be to show that this"},{"Start":"00:45.260 ","End":"00:51.060","Text":"decreases and its limit is 0 and I\u0027ll just to get some more space here."},{"Start":"00:53.420 ","End":"00:57.560","Text":"The easier part is to show that the limit is 0"},{"Start":"00:57.560 ","End":"01:02.970","Text":"and what is commonly done, often with series,"},{"Start":"01:02.970 ","End":"01:05.090","Text":"is to replace the variable n,"},{"Start":"01:05.090 ","End":"01:06.530","Text":"which is 1, 2, 3, 4, 5,"},{"Start":"01:06.530 ","End":"01:09.800","Text":"etc with a continuous variable x,"},{"Start":"01:09.800 ","End":"01:12.895","Text":"and then use calculus methods."},{"Start":"01:12.895 ","End":"01:18.760","Text":"Let me just say that in our case our a_n is 1"},{"Start":"01:18.760 ","End":"01:25.250","Text":"over the square root of 4n plus 1."},{"Start":"01:25.250 ","End":"01:29.815","Text":"Instead of looking at a_n we look at a function of x,"},{"Start":"01:29.815 ","End":"01:36.925","Text":"which is 1 over the square root of 4x plus 1."},{"Start":"01:36.925 ","End":"01:41.410","Text":"We did a similar thing in the integral test for a series."},{"Start":"01:41.410 ","End":"01:44.770","Text":"Now I\u0027m going to show 2 things instead of with a_n"},{"Start":"01:44.770 ","End":"01:48.365","Text":"I\u0027m going to use f of x. I\u0027m going to show that the limit,"},{"Start":"01:48.365 ","End":"01:51.165","Text":"as x goes to infinity,"},{"Start":"01:51.165 ","End":"01:59.095","Text":"of f of x is 0 and that will prove that the limit as n goes to infinity is also 0."},{"Start":"01:59.095 ","End":"02:00.670","Text":"Let\u0027s see what this is."},{"Start":"02:00.670 ","End":"02:07.395","Text":"This is the limit as x goes to infinity,"},{"Start":"02:07.395 ","End":"02:16.095","Text":"1 over the square root of 4x plus 1."},{"Start":"02:16.095 ","End":"02:23.390","Text":"Now that\u0027s fairly easy to see that this is equal to if x is infinity,"},{"Start":"02:23.390 ","End":"02:27.590","Text":"I\u0027ll write it symbolically as 1 over"},{"Start":"02:27.590 ","End":"02:34.725","Text":"the square root of 4 times infinity plus 1."},{"Start":"02:34.725 ","End":"02:40.925","Text":"Basically what this means is that 4 times infinity plus 1 is infinity."},{"Start":"02:40.925 ","End":"02:43.580","Text":"I have 1 over the square root of infinity,"},{"Start":"02:43.580 ","End":"02:47.280","Text":"and the square root of infinity is also infinity."},{"Start":"02:47.960 ","End":"02:55.130","Text":"We have 1 over infinity and this is equal to 0."},{"Start":"02:55.130 ","End":"02:57.215","Text":"This is all symbolic."},{"Start":"02:57.215 ","End":"03:00.365","Text":"What we really mean is as x goes to infinity,"},{"Start":"03:00.365 ","End":"03:04.820","Text":"4x plus 1 goes to infinity and if something goes to infinity,"},{"Start":"03:04.820 ","End":"03:07.010","Text":"the square root of it goes to infinity,"},{"Start":"03:07.010 ","End":"03:08.420","Text":"and if something goes to infinity,"},{"Start":"03:08.420 ","End":"03:10.465","Text":"1 over it goes to 0."},{"Start":"03:10.465 ","End":"03:18.180","Text":"That\u0027s implies that this is 0,"},{"Start":"03:18.180 ","End":"03:26.014","Text":"so that a_n also goes to 0 as n goes to infinity,"},{"Start":"03:26.014 ","End":"03:28.715","Text":"which not written here, but that\u0027s what is meant."},{"Start":"03:28.715 ","End":"03:31.910","Text":"That was this part that,"},{"Start":"03:31.910 ","End":"03:35.220","Text":"we still have to do the decreasing part."},{"Start":"03:37.040 ","End":"03:42.500","Text":"It\u0027s possible to do this without calculus using inequalities,"},{"Start":"03:42.500 ","End":"03:47.045","Text":"but I\u0027ll do it using calculus using the same function of x."},{"Start":"03:47.045 ","End":"03:55.160","Text":"What I have to show is that the derivative is negative and then it will be decreasing."},{"Start":"03:55.160 ","End":"04:04.090","Text":"Let\u0027s see, I\u0027ll just start here and let\u0027s see what f prime of x is equal to in general."},{"Start":"04:04.090 ","End":"04:15.200","Text":"You look at it as a quotient or you can look at it as something to the minus 1."},{"Start":"04:15.480 ","End":"04:18.385","Text":"I\u0027ll look at it as a quotient."},{"Start":"04:18.385 ","End":"04:24.850","Text":"What we have is the derivative is the denominator"},{"Start":"04:24.850 ","End":"04:33.410","Text":"squared down below and then we have the derivative of the numerator which is 0,"},{"Start":"04:33.410 ","End":"04:42.730","Text":"times the denominator, square root of 4x plus 1 minus the numerator as is."},{"Start":"04:42.730 ","End":"04:47.070","Text":"I just extend this,"},{"Start":"04:47.070 ","End":"04:57.525","Text":"derivative of the denominator is 1 over twice the square root of 4x plus 1."},{"Start":"04:57.525 ","End":"05:01.565","Text":"But because 4x plus 1 as a function of x,"},{"Start":"05:01.565 ","End":"05:11.420","Text":"the anti-derivative is 4 we put that here and after simplification,"},{"Start":"05:11.420 ","End":"05:14.075","Text":"what we get is just minus,"},{"Start":"05:14.075 ","End":"05:16.940","Text":"now 4/2 is 2,"},{"Start":"05:16.940 ","End":"05:20.725","Text":"so we get minus 2 over"},{"Start":"05:20.725 ","End":"05:28.730","Text":"4x plus 1 times the square root of 4 x plus 1."},{"Start":"05:28.730 ","End":"05:33.165","Text":"Could have written 4x to the power of 1.5 or 3/2."},{"Start":"05:33.165 ","End":"05:36.655","Text":"In any event, x, of course,"},{"Start":"05:36.655 ","End":"05:44.155","Text":"is positive because we\u0027re only taking n from fact x is even bigger than 1 if you like."},{"Start":"05:44.155 ","End":"05:47.210","Text":"This thing is negative."},{"Start":"05:47.960 ","End":"05:52.750","Text":"Here this thing, I\u0027ll use another color."},{"Start":"05:52.750 ","End":"05:58.450","Text":"This is negative, square root is always"},{"Start":"05:58.450 ","End":"06:05.200","Text":"positive 4x plus 1 is positive because x is bigger than 0 or 1,"},{"Start":"06:05.200 ","End":"06:09.040","Text":"this is even bigger than 4, certainly positive,"},{"Start":"06:09.040 ","End":"06:14.360","Text":"and the square root of something positive is always positive if it\u0027s defined."},{"Start":"06:14.360 ","End":"06:20.150","Text":"We have negative over positive positive so altogether,"},{"Start":"06:20.150 ","End":"06:23.850","Text":"what we get is negative."},{"Start":"06:24.800 ","End":"06:32.135","Text":"What we can say is that f prime of x is less than 0,"},{"Start":"06:32.135 ","End":"06:35.075","Text":"so f is decreasing."},{"Start":"06:35.075 ","End":"06:37.730","Text":"If f is decreasing,"},{"Start":"06:37.730 ","End":"06:41.510","Text":"it implies also that a_n is a decreasing series."},{"Start":"06:41.510 ","End":"06:45.660","Text":"I\u0027ll illustrate this briefly with a down arrow, that means decreasing."},{"Start":"06:45.950 ","End":"06:50.390","Text":"Now we\u0027ve proven the tens to 0 part."},{"Start":"06:50.390 ","End":"06:53.045","Text":"We\u0027ve proved the decreasing part,"},{"Start":"06:53.045 ","End":"06:59.450","Text":"and we\u0027ve satisfied the conditions of Leibnitz\u0027s test and that means that"},{"Start":"06:59.450 ","End":"07:08.140","Text":"the original series is convergent and that answers the question. We\u0027re done."}],"ID":10557},{"Watched":false,"Name":"Exercise 9 Part b","Duration":"7m 32s","ChapterTopicVideoID":10229,"CourseChapterTopicPlaylistID":52097,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10229.jpeg","UploadDate":"2017-08-22T04:52:17.5270000","DurationForVideoObject":"PT7M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.430","Text":"In this exercise, we have to find out whether this series,"},{"Start":"00:05.430 ","End":"00:08.580","Text":"the infinite series converges or diverges,"},{"Start":"00:08.580 ","End":"00:13.860","Text":"and the minus 1 to the n here means it has alternating signs,"},{"Start":"00:13.860 ","End":"00:15.975","Text":"minus, plus, minus, plus et cetera."},{"Start":"00:15.975 ","End":"00:21.525","Text":"We\u0027re going to try the alternating series test,"},{"Start":"00:21.525 ","End":"00:24.000","Text":"and this form of it."},{"Start":"00:24.000 ","End":"00:28.665","Text":"1 is for minus plus-minus and 1 is for plus-minus- plus."},{"Start":"00:28.665 ","End":"00:30.360","Text":"Same thing pretty much."},{"Start":"00:30.360 ","End":"00:33.435","Text":"We have to show 2 things before we can do it."},{"Start":"00:33.435 ","End":"00:36.930","Text":"First of all, we have to identify what is a_n."},{"Start":"00:36.930 ","End":"00:42.850","Text":"I\u0027m going to let a_n be the part without the sign, just this."},{"Start":"00:48.800 ","End":"00:51.260","Text":"In this exercise, we\u0027re given"},{"Start":"00:51.260 ","End":"00:56.060","Text":"an infinite series and we have to decide if it converges or diverges."},{"Start":"00:56.060 ","End":"00:59.640","Text":"I see that it has alternating signs from the minus 1 to the n,"},{"Start":"00:59.640 ","End":"01:07.540","Text":"so we\u0027re going to try the alternating series test due to the mathematician Leibniz,"},{"Start":"01:07.640 ","End":"01:10.550","Text":"which basically says that if you had something like"},{"Start":"01:10.550 ","End":"01:13.835","Text":"this and if you just call this part a_n."},{"Start":"01:13.835 ","End":"01:17.280","Text":"I\u0027ll continue down here, more room."},{"Start":"01:17.280 ","End":"01:19.455","Text":"We\u0027ll let this part be a_n,"},{"Start":"01:19.455 ","End":"01:28.770","Text":"just the natural log of n over n. What we have to show is 2 things;"},{"Start":"01:28.770 ","End":"01:31.350","Text":"that a_n is decreasing,"},{"Start":"01:31.350 ","End":"01:34.085","Text":"so I\u0027ll just symbolically write it like this,"},{"Start":"01:34.085 ","End":"01:36.590","Text":"and that a_n tends to 0,"},{"Start":"01:36.590 ","End":"01:39.485","Text":"it means as n goes to infinity."},{"Start":"01:39.485 ","End":"01:42.910","Text":"These are the 2 things that we have to show,"},{"Start":"01:42.910 ","End":"01:48.550","Text":"and then we\u0027ll be able to deduce that our series is convergent."},{"Start":"01:48.740 ","End":"01:57.350","Text":"A common thing to do is to switch from a discrete function of n,"},{"Start":"01:57.350 ","End":"02:01.730","Text":"which is a_n, and go to a continuous function f of x."},{"Start":"02:01.730 ","End":"02:03.670","Text":"It\u0027s very widely used this trick."},{"Start":"02:03.670 ","End":"02:12.755","Text":"Instead I look at a function of not n but of x to equal natural log of x over x."},{"Start":"02:12.755 ","End":"02:18.680","Text":"Then I show that f of x is a decreasing function,"},{"Start":"02:18.680 ","End":"02:22.280","Text":"which will automatically imply that at the whole numbers it\u0027s decreasing,"},{"Start":"02:22.280 ","End":"02:26.680","Text":"and I\u0027ll also show that f of x goes to 0 as x goes to infinity."},{"Start":"02:26.680 ","End":"02:30.405","Text":"Let\u0027s first of all show the limit part,"},{"Start":"02:30.405 ","End":"02:41.730","Text":"and the limit as x goes to infinity of f of x,"},{"Start":"02:41.980 ","End":"02:49.490","Text":"which is natural log of x over x is equal to."},{"Start":"02:49.490 ","End":"02:54.240","Text":"Here we have a situation of infinity over infinity."},{"Start":"02:54.520 ","End":"03:02.435","Text":"We are going to use the L\u0027Hopital trick."},{"Start":"03:02.435 ","End":"03:04.850","Text":"I\u0027ll just write his name."},{"Start":"03:04.850 ","End":"03:08.570","Text":"French mathematician L\u0027Hopital, just to remind you,"},{"Start":"03:08.570 ","End":"03:13.309","Text":"he said that if we have a limit that\u0027s 0 over 0 or infinity over infinity,"},{"Start":"03:13.309 ","End":"03:18.350","Text":"what you can do is differentiate numerator and denominator,"},{"Start":"03:18.350 ","End":"03:19.790","Text":"get a different limit,"},{"Start":"03:19.790 ","End":"03:21.695","Text":"but it will come out the same answer."},{"Start":"03:21.695 ","End":"03:24.590","Text":"This is equal to the limit."},{"Start":"03:24.590 ","End":"03:32.930","Text":"Now, write L for L\u0027Hopital and it\u0027s the infinity over infinity case."},{"Start":"03:32.930 ","End":"03:34.895","Text":"We\u0027ll see what we\u0027re doing here,"},{"Start":"03:34.895 ","End":"03:37.955","Text":"and then we get the limit as x goes to infinity."},{"Start":"03:37.955 ","End":"03:41.235","Text":"Derivative of this is 1 over x,"},{"Start":"03:41.235 ","End":"03:44.740","Text":"derivative of this is 1."},{"Start":"03:44.750 ","End":"03:49.010","Text":"The limit is just 1 over x as x goes to infinity,"},{"Start":"03:49.010 ","End":"03:50.870","Text":"which is 0."},{"Start":"03:50.870 ","End":"03:55.545","Text":"This part, check."},{"Start":"03:55.545 ","End":"03:57.455","Text":"Let\u0027s see about the decreasing."},{"Start":"03:57.455 ","End":"04:00.740","Text":"The decreasing is commonly done by showing"},{"Start":"04:00.740 ","End":"04:04.490","Text":"that the derivative is negative and then the function\u0027s decreasing."},{"Start":"04:04.490 ","End":"04:10.470","Text":"At least we have to show it\u0027s negative from 1 onwards because n only starts at 1,"},{"Start":"04:10.470 ","End":"04:12.420","Text":"so we can start x at 1."},{"Start":"04:12.420 ","End":"04:15.005","Text":"Let\u0027s see, the derivative,"},{"Start":"04:15.005 ","End":"04:17.250","Text":"and I\u0027ll do that over here,"},{"Start":"04:17.840 ","End":"04:22.170","Text":"f prime of x equals."},{"Start":"04:22.170 ","End":"04:28.585","Text":"I\u0027m going to use the quotient formula and you should have it memorized by now."},{"Start":"04:28.585 ","End":"04:32.405","Text":"The easiest part is the denominator is squared,"},{"Start":"04:32.405 ","End":"04:36.920","Text":"and then it\u0027s the derivative of the numerator times"},{"Start":"04:36.920 ","End":"04:43.175","Text":"the denominator less the other way round numerator as is,"},{"Start":"04:43.175 ","End":"04:49.620","Text":"and the derivative of the denominator, which is 1."},{"Start":"04:49.970 ","End":"04:59.685","Text":"This is equal to 1 minus natural log of x over x squared."},{"Start":"04:59.685 ","End":"05:02.479","Text":"Immediately obvious that it\u0027s negative,"},{"Start":"05:02.479 ","End":"05:05.195","Text":"the denominator is positive, that\u0027s for sure."},{"Start":"05:05.195 ","End":"05:07.940","Text":"We\u0027re taking x is bigger than 1."},{"Start":"05:07.940 ","End":"05:17.870","Text":"Now, it turns out that it\u0027s not true that for all x that this is negative,"},{"Start":"05:17.870 ","End":"05:21.139","Text":"but let\u0027s see if I could translate this condition,"},{"Start":"05:21.139 ","End":"05:24.860","Text":"to say that 1 minus natural log of x is negative,"},{"Start":"05:24.860 ","End":"05:26.210","Text":"which is what I want,"},{"Start":"05:26.210 ","End":"05:32.019","Text":"is to say that 1 is less than natural log of x."},{"Start":"05:33.680 ","End":"05:38.150","Text":"This is true because natural log is an increasing function."},{"Start":"05:38.150 ","End":"05:40.430","Text":"This is natural log of e,"},{"Start":"05:40.430 ","End":"05:44.930","Text":"so it\u0027s to say that natural log of e is less than"},{"Start":"05:44.930 ","End":"05:50.659","Text":"natural log of x because I took e to the power of 1.This is true,"},{"Start":"05:50.659 ","End":"05:56.820","Text":"provided that e is less than x,"},{"Start":"05:56.820 ","End":"06:06.215","Text":"so x is bigger than e. It\u0027s not true that it\u0027s decreasing everywhere."},{"Start":"06:06.215 ","End":"06:09.800","Text":"If I\u0027m talking about n, a whole number,"},{"Start":"06:09.800 ","End":"06:16.320","Text":"I can say that it\u0027s decreasing from n equals 3 onwards,"},{"Start":"06:16.320 ","End":"06:17.840","Text":"and then if n is bigger or equal to 3,"},{"Start":"06:17.840 ","End":"06:19.535","Text":"It\u0027s certainly bigger than e,"},{"Start":"06:19.535 ","End":"06:22.835","Text":"and then the a_n series is decreasing."},{"Start":"06:22.835 ","End":"06:28.085","Text":"Actually the formulation should be"},{"Start":"06:28.085 ","End":"06:34.140","Text":"almost always decreasing or decreasing except for maybe a few at the beginning."},{"Start":"06:36.590 ","End":"06:40.625","Text":"Mostly, I\u0027ll just put the word mostly."},{"Start":"06:40.625 ","End":"06:45.710","Text":"Almost always decreasing, that\u0027s good enough."},{"Start":"06:45.710 ","End":"06:50.870","Text":"If I tried n equals 1 and n equals 2,"},{"Start":"06:50.870 ","End":"06:58.215","Text":"it doesn\u0027t decrease because when n is 1, then it\u0027s 0."},{"Start":"06:58.215 ","End":"07:03.195","Text":"But it decreases from 3 onwards. We\u0027re okay."},{"Start":"07:03.195 ","End":"07:06.915","Text":"When this is true, this is less than 0,"},{"Start":"07:06.915 ","End":"07:11.850","Text":"and so we proved yes."},{"Start":"07:11.850 ","End":"07:14.460","Text":"Check here too."},{"Start":"07:14.460 ","End":"07:19.969","Text":"If you want to be pedantic from n bigger or equal to 3 is decreasing,"},{"Start":"07:19.969 ","End":"07:21.700","Text":"which is good enough."},{"Start":"07:21.700 ","End":"07:24.495","Text":"It satisfied the conditions,"},{"Start":"07:24.495 ","End":"07:33.430","Text":"and therefore the conclusion is that the series is convergent. Done."}],"ID":10558},{"Watched":false,"Name":"Exercise 9 Part c","Duration":"5m 39s","ChapterTopicVideoID":10230,"CourseChapterTopicPlaylistID":52097,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/10230.jpeg","UploadDate":"2017-08-22T04:52:30.0170000","DurationForVideoObject":"PT5M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we\u0027re given an infinite series and"},{"Start":"00:04.050 ","End":"00:09.105","Text":"our task is to determine if this series converges or diverges."},{"Start":"00:09.105 ","End":"00:13.920","Text":"You can see right away it\u0027s an alternating series because of the minus 1^n,"},{"Start":"00:13.920 ","End":"00:17.160","Text":"and that brings to mind the Leibniz test"},{"Start":"00:17.160 ","End":"00:20.700","Text":"or the alternating series test which relates to this."},{"Start":"00:20.700 ","End":"00:23.290","Text":"Let\u0027s just get some more room here."},{"Start":"00:23.290 ","End":"00:28.885","Text":"That basically if the bit besides the minus"},{"Start":"00:28.885 ","End":"00:35.210","Text":"1^n happens to be decreasing and tends to 0 as n goes to infinity,"},{"Start":"00:35.210 ","End":"00:38.195","Text":"then we can guarantee that the series is convergent."},{"Start":"00:38.195 ","End":"00:45.330","Text":"In our case, the a_n is n plus 2 over n"},{"Start":"00:45.330 ","End":"00:52.640","Text":"squared plus n. We have to show other side of those 2 things."},{"Start":"00:52.640 ","End":"00:56.180","Text":"Let\u0027s first of all do this part, the limit part."},{"Start":"00:56.180 ","End":"01:02.184","Text":"The limit as n goes to infinity of a_n,"},{"Start":"01:02.184 ","End":"01:09.290","Text":"which is n plus 2 over n squared plus n is equal to,"},{"Start":"01:09.290 ","End":"01:12.710","Text":"remember with polynomials, all you have to do is take"},{"Start":"01:12.710 ","End":"01:17.800","Text":"the dominant term and you can throw the rest out and you get the same limit."},{"Start":"01:17.800 ","End":"01:19.430","Text":"It\u0027s n goes to infinity,"},{"Start":"01:19.430 ","End":"01:21.965","Text":"just n over n squared."},{"Start":"01:21.965 ","End":"01:25.805","Text":"Now, n over n squared is 1 over n,"},{"Start":"01:25.805 ","End":"01:27.590","Text":"and as n goes to infinity,"},{"Start":"01:27.590 ","End":"01:30.680","Text":"this is equal to 0. That\u0027s good."},{"Start":"01:30.680 ","End":"01:33.335","Text":"We can put a check mark on this."},{"Start":"01:33.335 ","End":"01:36.035","Text":"Now, how about the decreasing?"},{"Start":"01:36.035 ","End":"01:38.225","Text":"A common trick is to,"},{"Start":"01:38.225 ","End":"01:39.890","Text":"instead of taking a_n,"},{"Start":"01:39.890 ","End":"01:43.195","Text":"to think of this like as a function of n."},{"Start":"01:43.195 ","End":"01:48.785","Text":"Then to look at the function instead of an integer n of a whole number,"},{"Start":"01:48.785 ","End":"01:53.720","Text":"we can look at f of x in general with a variable that\u0027s"},{"Start":"01:53.720 ","End":"02:00.790","Text":"continuous and call it x plus 2 over x squared plus x."},{"Start":"02:01.360 ","End":"02:12.525","Text":"In our case, x is bigger than 1 because here it goes from 1 to infinity."},{"Start":"02:12.525 ","End":"02:17.930","Text":"It\u0027s easy to check for decreasing with functions because we have the derivative test."},{"Start":"02:17.930 ","End":"02:21.675","Text":"If the derivative is negative, then we\u0027re decreasing."},{"Start":"02:21.675 ","End":"02:23.850","Text":"Let\u0027s try and get the derivative,"},{"Start":"02:23.850 ","End":"02:26.445","Text":"and see if it\u0027s always negative."},{"Start":"02:26.445 ","End":"02:28.635","Text":"F prime of x is,"},{"Start":"02:28.635 ","End":"02:33.255","Text":"and using the quotient rule,"},{"Start":"02:33.255 ","End":"02:36.190","Text":"the quotient rule I start with the denominator sometimes."},{"Start":"02:36.190 ","End":"02:38.410","Text":"I know it\u0027s the denominator squared,"},{"Start":"02:38.410 ","End":"02:40.370","Text":"and it\u0027s the derivative of this,"},{"Start":"02:40.370 ","End":"02:48.445","Text":"which is 1 times this x squared plus x minus the other way around,"},{"Start":"02:48.445 ","End":"02:50.825","Text":"this one as is,"},{"Start":"02:50.825 ","End":"02:52.865","Text":"and the derivative of this,"},{"Start":"02:52.865 ","End":"02:57.000","Text":"which is 2x plus 1."},{"Start":"02:57.730 ","End":"03:00.790","Text":"Let\u0027s see what we get."},{"Start":"03:00.790 ","End":"03:06.330","Text":"On the denominator, I\u0027m not even going to bother evaluating it,"},{"Start":"03:06.330 ","End":"03:09.080","Text":"I\u0027m just going to say that it\u0027s positive,"},{"Start":"03:09.080 ","End":"03:11.630","Text":"it\u0027s a plus because it\u0027s something squared."},{"Start":"03:11.630 ","End":"03:14.150","Text":"I don\u0027t really care about the actual value,"},{"Start":"03:14.150 ","End":"03:19.080","Text":"just plus or minus, so not bother."},{"Start":"03:19.080 ","End":"03:21.410","Text":"Here, let\u0027s see what we get."},{"Start":"03:21.410 ","End":"03:23.150","Text":"If we expand the brackets,"},{"Start":"03:23.150 ","End":"03:28.890","Text":"we\u0027ve got x squared plus x minus- let\u0027s see."},{"Start":"03:28.890 ","End":"03:30.315","Text":"If I multiply this out,"},{"Start":"03:30.315 ","End":"03:41.085","Text":"x times 2x is 2x squared plus 4x plus x is plus 5x."},{"Start":"03:41.085 ","End":"03:46.780","Text":"Then plus 2 plus 1 is plus 2."},{"Start":"03:47.840 ","End":"03:51.780","Text":"After I cancel everything, let\u0027s see."},{"Start":"03:51.780 ","End":"03:54.975","Text":"We\u0027ve got x squared minus 2x squared,"},{"Start":"03:54.975 ","End":"03:58.410","Text":"that\u0027s minus x squared"},{"Start":"03:58.410 ","End":"04:07.960","Text":"plus x minus 5x is minus 4x."},{"Start":"04:08.960 ","End":"04:13.665","Text":"We also have a minus 2,"},{"Start":"04:13.665 ","End":"04:15.665","Text":"so you know what I\u0027m going to do?"},{"Start":"04:15.665 ","End":"04:17.480","Text":"They\u0027re all minuses."},{"Start":"04:17.480 ","End":"04:23.775","Text":"Why don\u0027t I just leave 1 minus and make them all plus there. How about that?"},{"Start":"04:23.775 ","End":"04:28.705","Text":"Then again, over something positive,"},{"Start":"04:28.705 ","End":"04:31.460","Text":"it\u0027s this thing squared, if you want to copy it,"},{"Start":"04:31.460 ","End":"04:33.830","Text":"copy it, but I\u0027m just saying it, it\u0027s positive."},{"Start":"04:33.830 ","End":"04:36.440","Text":"Now, this thing is positive for"},{"Start":"04:36.440 ","End":"04:40.700","Text":"all positive x because everything\u0027s with pluses and this is positive."},{"Start":"04:40.700 ","End":"04:46.340","Text":"What I have basically is, I\u0027ll write it symbolically."},{"Start":"04:46.340 ","End":"04:48.785","Text":"I have a minus, I have a plus,"},{"Start":"04:48.785 ","End":"04:52.070","Text":"and I have a plus and minus plus plus,"},{"Start":"04:52.070 ","End":"04:53.540","Text":"it doesn\u0027t matter I multiply or divide,"},{"Start":"04:53.540 ","End":"04:55.730","Text":"gives me a minus."},{"Start":"04:55.730 ","End":"05:03.270","Text":"In fact, f prime of x is indeed less than 0."},{"Start":"05:03.270 ","End":"05:05.590","Text":"Minus means negative."},{"Start":"05:06.220 ","End":"05:10.130","Text":"If f prime of x is negative,"},{"Start":"05:10.130 ","End":"05:13.290","Text":"then f of x is decreasing."},{"Start":"05:13.730 ","End":"05:17.175","Text":"The continuous version is decreasing,"},{"Start":"05:17.175 ","End":"05:20.095","Text":"a_n is also decreasing."},{"Start":"05:20.095 ","End":"05:24.740","Text":"That basically concludes it because we\u0027ve shown that a_n tends to 0."},{"Start":"05:24.740 ","End":"05:26.900","Text":"We\u0027ve shown that a_n is decreasing."},{"Start":"05:26.900 ","End":"05:30.260","Text":"The conclusion is that the series is convergent."},{"Start":"05:30.260 ","End":"05:40.030","Text":"We got to check mark here to decreasing tends to 0, so convergent. Done."}],"ID":10559}],"Thumbnail":null,"ID":52097},{"Name":"Absolute and Conditional Convergence of Series","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 10 Part a","Duration":"","ChapterTopicVideoID":10233,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"In this exercise, we have an infinite series and we have to decide,"},{"Start":"00:05.010 ","End":"00:09.360","Text":"if it\u0027s divergent or if it converges."},{"Start":"00:09.360 ","End":"00:12.610","Text":"Absolutely or conditionally."},{"Start":"00:12.710 ","End":"00:17.505","Text":"Here we\u0027ve repeated the definition in case you need them."},{"Start":"00:17.505 ","End":"00:23.230","Text":"What is absolutely convergent and what is conditionally convergent."},{"Start":"00:23.240 ","End":"00:26.505","Text":"There\u0027s also a theorem,"},{"Start":"00:26.505 ","End":"00:30.000","Text":"call it a fact that is something absolutely converges,"},{"Start":"00:30.000 ","End":"00:33.360","Text":"then it also converges and we could use that maybe."},{"Start":"00:33.360 ","End":"00:36.165","Text":"Anyway, in the first 1,"},{"Start":"00:36.165 ","End":"00:43.865","Text":"we see that we have an alternating series so it looks like it\u0027s going to"},{"Start":"00:43.865 ","End":"00:52.320","Text":"converge using the alternating series theorem due to Leibnitz."},{"Start":"00:53.180 ","End":"00:55.580","Text":"With an alternating series,"},{"Start":"00:55.580 ","End":"01:01.880","Text":"all I have to show is that if I take this as my a_n without the minus 1 part."},{"Start":"01:01.880 ","End":"01:07.320","Text":"If I take a_ n to be 1 over square root of n plus 1."},{"Start":"01:07.320 ","End":"01:09.389","Text":"If I can show 2 things,"},{"Start":"01:09.389 ","End":"01:17.435","Text":"that a_n is decreasing and that a_n goes to 0 as n goes to infinity,"},{"Start":"01:17.435 ","End":"01:21.405","Text":"then this series will be convergent."},{"Start":"01:21.405 ","End":"01:25.890","Text":"Afterwards we can decide if it\u0027s absolute or conditional."},{"Start":"01:26.080 ","End":"01:29.120","Text":"How do we do this?"},{"Start":"01:29.120 ","End":"01:34.445","Text":"Well, each part is fairly straightforward."},{"Start":"01:34.445 ","End":"01:38.760","Text":"The a_n goes to 0."},{"Start":"01:39.880 ","End":"01:42.305","Text":"I\u0027ll say it like this."},{"Start":"01:42.305 ","End":"01:43.970","Text":"If n goes to infinity,"},{"Start":"01:43.970 ","End":"01:45.695","Text":"so does n plus 1."},{"Start":"01:45.695 ","End":"01:47.360","Text":"If something goes to infinity,"},{"Start":"01:47.360 ","End":"01:49.060","Text":"so does the square root,"},{"Start":"01:49.060 ","End":"01:50.630","Text":"and if something goes to infinity,"},{"Start":"01:50.630 ","End":"01:52.220","Text":"1 over it goes to 0."},{"Start":"01:52.220 ","End":"02:01.160","Text":"Symbolically, I would write this limit is 1 over the square root of infinity plus 1,"},{"Start":"02:01.160 ","End":"02:05.450","Text":"and we sometimes it\u0027s symbolically we treat infinity like a number."},{"Start":"02:05.450 ","End":"02:07.310","Text":"Infinity plus 1 is infinity,"},{"Start":"02:07.310 ","End":"02:12.555","Text":"square root of infinity is infinity and 1 over infinity is 0."},{"Start":"02:12.555 ","End":"02:18.090","Text":"This part of going to 0 check."},{"Start":"02:18.090 ","End":"02:20.190","Text":"As for the decreasing,"},{"Start":"02:20.190 ","End":"02:24.755","Text":"we could do it with calculus and make this a function and differentiate it,"},{"Start":"02:24.755 ","End":"02:26.935","Text":"but it\u0027s actually simpler."},{"Start":"02:26.935 ","End":"02:29.220","Text":"To show that it\u0027s decreasing,"},{"Start":"02:29.220 ","End":"02:37.380","Text":"we have to show in general that a_n plus 1 is less than a_n."},{"Start":"02:37.380 ","End":"02:39.080","Text":"At least, we have to show this."},{"Start":"02:39.080 ","End":"02:40.865","Text":"I don\u0027t know that this is so."},{"Start":"02:40.865 ","End":"02:42.485","Text":"What am I saying here?"},{"Start":"02:42.485 ","End":"02:49.395","Text":"I\u0027m saying that 1 over the square root of n plus 1 plus 1,"},{"Start":"02:49.395 ","End":"02:51.390","Text":"which is n plus 2,"},{"Start":"02:51.390 ","End":"02:59.780","Text":"is less than the square root of n plus 1 on the denominator."},{"Start":"02:59.780 ","End":"03:02.695","Text":"This is what I have to show."},{"Start":"03:02.695 ","End":"03:05.070","Text":"This implies this."},{"Start":"03:05.070 ","End":"03:07.955","Text":"Now, in order to show this,"},{"Start":"03:07.955 ","End":"03:16.760","Text":"if I showed that the square root of n plus 2 is bigger than the square root of n plus 1."},{"Start":"03:16.760 ","End":"03:18.650","Text":"If this were true,"},{"Start":"03:18.650 ","End":"03:21.140","Text":"then this would be true because it\u0027s 1 over,"},{"Start":"03:21.140 ","End":"03:23.915","Text":"I\u0027m going backwards logically."},{"Start":"03:23.915 ","End":"03:27.440","Text":"I certainly know that n plus 2 is bigger than n plus 1."},{"Start":"03:27.440 ","End":"03:30.395","Text":"I think at this point we can agree and stop going back."},{"Start":"03:30.395 ","End":"03:33.200","Text":"I don\u0027t want to go back to 2 is bigger than 1."},{"Start":"03:33.200 ","End":"03:34.940","Text":"This is clear enough."},{"Start":"03:34.940 ","End":"03:38.060","Text":"If this is clear, then with positive numbers,"},{"Start":"03:38.060 ","End":"03:41.735","Text":"the square root preserves the direction."},{"Start":"03:41.735 ","End":"03:46.400","Text":"Square root of something bigger is also bigger and then 1 over reverses"},{"Start":"03:46.400 ","End":"03:50.420","Text":"direction and we can see that it is decreasing."},{"Start":"03:50.420 ","End":"03:53.310","Text":"I can put a check mark here."},{"Start":"03:54.050 ","End":"03:58.755","Text":"If this part is decreasing, it goes to 0,"},{"Start":"03:58.755 ","End":"04:02.950","Text":"this thing is convergent,"},{"Start":"04:02.990 ","End":"04:06.695","Text":"so the original series is convergent."},{"Start":"04:06.695 ","End":"04:12.215","Text":"Now, how do we know fits absolutely or conditionally?"},{"Start":"04:12.215 ","End":"04:14.555","Text":"We look at another series,"},{"Start":"04:14.555 ","End":"04:23.180","Text":"which is to take the absolute value of a_n and if this is convergent,"},{"Start":"04:23.180 ","End":"04:27.320","Text":"then we\u0027re absolutely convergent and otherwise it\u0027s conditional."},{"Start":"04:27.320 ","End":"04:30.230","Text":"Let\u0027s just show what I\u0027m saying."},{"Start":"04:30.230 ","End":"04:34.220","Text":"I\u0027m saying that we now look at the series 1 to"},{"Start":"04:34.220 ","End":"04:39.660","Text":"infinity of the absolute value of all this."},{"Start":"04:42.190 ","End":"04:44.570","Text":"I\u0027ve been a bit imprecise."},{"Start":"04:44.570 ","End":"04:47.810","Text":"I can\u0027t use the same a_n to the same letter twice."},{"Start":"04:47.810 ","End":"04:50.980","Text":"Let\u0027s call this thing b_n,"},{"Start":"04:50.980 ","End":"04:55.215","Text":"so we know that the sum of b_n is convergent,"},{"Start":"04:55.215 ","End":"04:59.895","Text":"and a_n is the series without b minus 1 to the n."},{"Start":"04:59.895 ","End":"05:05.375","Text":"What I have to show is that the sum of the absolute value."},{"Start":"05:05.375 ","End":"05:07.270","Text":"Everywhere here you see a,"},{"Start":"05:07.270 ","End":"05:08.740","Text":"it could be b."},{"Start":"05:08.740 ","End":"05:14.040","Text":"In this case I\u0027m going to take my a_n here as the b_n,"},{"Start":"05:14.040 ","End":"05:20.545","Text":"so a_n is going to be replaced by b_n in this here and then we\u0027re going to relate to b_n."},{"Start":"05:20.545 ","End":"05:23.785","Text":"The sum of absolute value of b_n is the sum"},{"Start":"05:23.785 ","End":"05:29.655","Text":"of absolute value just throws away the sign the plus or minus."},{"Start":"05:29.655 ","End":"05:36.055","Text":"My question is, what about 1 over the square root of n plus 1?"},{"Start":"05:36.055 ","End":"05:38.530","Text":"Is this convergent or not?"},{"Start":"05:38.530 ","End":"05:40.404","Text":"This is the absolute value,"},{"Start":"05:40.404 ","End":"05:42.920","Text":"the sum of the absolute value."},{"Start":"05:43.020 ","End":"05:45.160","Text":"How would we know this?"},{"Start":"05:45.160 ","End":"05:48.365","Text":"We have to make an educated guess first of all,"},{"Start":"05:48.365 ","End":"05:50.710","Text":"the n plus 1 is a polynomial,"},{"Start":"05:50.710 ","End":"05:53.850","Text":"the dominant term is"},{"Start":"05:53.850 ","End":"06:02.560","Text":"the n I use the squiggly say behaves like the sum of 1 over square root of n,"},{"Start":"06:02.560 ","End":"06:04.615","Text":"which is n to the 1/2."},{"Start":"06:04.615 ","End":"06:07.075","Text":"Now that\u0027s a p series."},{"Start":"06:07.075 ","End":"06:12.775","Text":"Remember when we have the sum of 1 over n to the power of p,"},{"Start":"06:12.775 ","End":"06:18.035","Text":"the convergence condition is that p should be bigger than 1."},{"Start":"06:18.035 ","End":"06:22.810","Text":"But here we have a 1/2 which is not bigger than 1."},{"Start":"06:23.000 ","End":"06:28.035","Text":"Because of that, this series diverges."},{"Start":"06:28.035 ","End":"06:30.960","Text":"I\u0027ll write that diverges."},{"Start":"06:30.960 ","End":"06:33.830","Text":"We\u0027re expecting this 1 to diverge."},{"Start":"06:33.830 ","End":"06:36.350","Text":"Now that this was the informal part,"},{"Start":"06:36.350 ","End":"06:41.470","Text":"now we have to just make it more precise and use 1 of the tests and"},{"Start":"06:41.470 ","End":"06:48.075","Text":"what I\u0027ll do is I\u0027ll use the comparison test."},{"Start":"06:48.075 ","End":"06:56.090","Text":"What I\u0027m saying is that I can actually say that the sum from"},{"Start":"06:56.090 ","End":"07:03.875","Text":"n equals 1 to infinity of 1 over the square root of n plus 1 is divergent."},{"Start":"07:03.875 ","End":"07:09.320","Text":"I want to use the comparison test on another series that I know is divergent."},{"Start":"07:09.320 ","End":"07:13.159","Text":"But for that I want to make it bigger or equal to something."},{"Start":"07:13.159 ","End":"07:20.985","Text":"I actually want to make the denominator bigger,"},{"Start":"07:20.985 ","End":"07:24.515","Text":"putting square root of n makes the denominator smaller."},{"Start":"07:24.515 ","End":"07:29.690","Text":"But suppose I said that 1 over the square root of n plus"},{"Start":"07:29.690 ","End":"07:37.045","Text":"1 is bigger or equal to 1 over the square root of 2n."},{"Start":"07:37.045 ","End":"07:42.080","Text":"Certainly n plus 1 is less than or equal to 2n,"},{"Start":"07:42.080 ","End":"07:47.425","Text":"so the square root is less than or equal to and the reciprocal reverses the direction."},{"Start":"07:47.425 ","End":"07:52.640","Text":"Since this is this and since the series,"},{"Start":"07:52.640 ","End":"07:56.390","Text":"the sum of 1 over square root of 2n,"},{"Start":"07:56.390 ","End":"08:02.300","Text":"this obviously diverges, I say obviously,"},{"Start":"08:02.300 ","End":"08:04.040","Text":"it may not be obvious."},{"Start":"08:04.040 ","End":"08:08.660","Text":"But at the side what I really can do is say it\u0027s 1 over the square root of"},{"Start":"08:08.660 ","End":"08:12.890","Text":"2 times the sum of 1 over the square root of"},{"Start":"08:12.890 ","End":"08:18.035","Text":"n. We\u0027ve already discussed that n to the power of a 1/2 diverges."},{"Start":"08:18.035 ","End":"08:23.410","Text":"Because this is bigger or equal to this and because this diverges,"},{"Start":"08:23.410 ","End":"08:28.730","Text":"it implies from the comparison test that this also"},{"Start":"08:28.730 ","End":"08:36.420","Text":"diverges or is divergent."},{"Start":"08:36.460 ","End":"08:40.770","Text":"Just being consistent with what I wrote up here."},{"Start":"08:40.780 ","End":"08:45.125","Text":"The series is convergent,"},{"Start":"08:45.125 ","End":"08:49.485","Text":"but the series with the absolute values is divergent and"},{"Start":"08:49.485 ","End":"08:54.740","Text":"so we are conditionally convergent and that\u0027s the answer,"},{"Start":"08:54.740 ","End":"09:04.720","Text":"conditionally convergent, and that\u0027s all there is to it and we are done."}],"ID":10560},{"Watched":false,"Name":"Exercise 10 Part b","Duration":"","ChapterTopicVideoID":10234,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.800","Text":"In this exercise, we have an infinite series and we have to decide if it\u0027s 1 of 3 things,"},{"Start":"00:07.800 ","End":"00:11.970","Text":"absolutely convergent, conditionally convergent or divergent."},{"Start":"00:11.970 ","End":"00:15.460","Text":"Every series is going to be 1 of those 3."},{"Start":"00:16.610 ","End":"00:22.860","Text":"We see the minus 1 to the power of n plus 1,"},{"Start":"00:22.860 ","End":"00:26.700","Text":"and that reminds us of an alternating series."},{"Start":"00:26.700 ","End":"00:32.565","Text":"I would like to first see if the original series converges or not"},{"Start":"00:32.565 ","End":"00:42.150","Text":"using what we call the Leibniz alternating series test."},{"Start":"00:42.150 ","End":"00:46.590","Text":"What we\u0027ll do, and I\u0027ll just get a bit more room here."},{"Start":"00:46.970 ","End":"00:49.410","Text":"In the Leibniz test,"},{"Start":"00:49.410 ","End":"00:51.545","Text":"if this is my series,"},{"Start":"00:51.545 ","End":"00:56.285","Text":"let\u0027s say that the original series is a_n,"},{"Start":"00:56.285 ","End":"01:03.930","Text":"then I just consider the part without the minus 1 to the n and call it b_n."},{"Start":"01:05.350 ","End":"01:08.960","Text":"If I can show 2 things,"},{"Start":"01:08.960 ","End":"01:13.755","Text":"that b_n is a decreasing series"},{"Start":"01:13.755 ","End":"01:19.650","Text":"and b_n tends to 0 as n goes to infinity,"},{"Start":"01:19.650 ","End":"01:26.385","Text":"then I can conclude that the original series a_n is convergent."},{"Start":"01:26.385 ","End":"01:35.120","Text":"Now, each of these is not very hard because there are several ways of doing it,"},{"Start":"01:35.120 ","End":"01:38.160","Text":"I can even do it without calculus,"},{"Start":"01:38.160 ","End":"01:40.530","Text":"just with inequalities, and I\u0027ll do that,"},{"Start":"01:40.530 ","End":"01:43.885","Text":"although it can be done with more sophisticated techniques."},{"Start":"01:43.885 ","End":"01:46.710","Text":"The first thing to show that,"},{"Start":"01:46.710 ","End":"01:48.870","Text":"let\u0027s say we\u0027ll take the easier 1 first,"},{"Start":"01:48.870 ","End":"01:50.950","Text":"that b_n tends to 0,"},{"Start":"01:50.950 ","End":"01:59.810","Text":"I\u0027ll show that the limit as n goes to infinity of 1 over n squared plus 1."},{"Start":"01:59.810 ","End":"02:02.215","Text":"This is equal to,"},{"Start":"02:02.215 ","End":"02:04.635","Text":"you could just say,"},{"Start":"02:04.635 ","End":"02:06.030","Text":"n goes to infinity,"},{"Start":"02:06.030 ","End":"02:07.710","Text":"so n squared goes to infinity,"},{"Start":"02:07.710 ","End":"02:10.540","Text":"so n squared plus 1 goes to infinity,"},{"Start":"02:10.540 ","End":"02:11.990","Text":"and if it\u0027s in the denominator,"},{"Start":"02:11.990 ","End":"02:13.190","Text":"it goes to 0."},{"Start":"02:13.190 ","End":"02:18.710","Text":"Or we can just write this as a symbolically shorthand way for saying what I just said."},{"Start":"02:18.710 ","End":"02:22.520","Text":"Which is that it\u0027s 1 over infinity squared plus 1,"},{"Start":"02:22.520 ","End":"02:26.490","Text":"and infinity squared is infinity."},{"Start":"02:27.800 ","End":"02:30.675","Text":"This is 1 over infinity,"},{"Start":"02:30.675 ","End":"02:32.240","Text":"and we can add the plus 1,"},{"Start":"02:32.240 ","End":"02:35.500","Text":"which is 1 over infinity, which is 0."},{"Start":"02:35.500 ","End":"02:39.045","Text":"So this part, I\u0027m going to 0 check."},{"Start":"02:39.045 ","End":"02:42.690","Text":"Now, how do we know that b_n is decreasing?"},{"Start":"02:42.690 ","End":"02:44.639","Text":"To show that it\u0027s decreasing,"},{"Start":"02:44.639 ","End":"02:47.570","Text":"I have to generally show that b_n plus"},{"Start":"02:47.570 ","End":"02:53.150","Text":"1 is less than or at any rate is less than or equal to b_n."},{"Start":"02:53.150 ","End":"02:59.205","Text":"Each successive term is no bigger than the original term."},{"Start":"02:59.205 ","End":"03:06.120","Text":"What this says is I have to show that 1 over n plus 1"},{"Start":"03:06.120 ","End":"03:14.655","Text":"squared plus 1 is less than or equal to 1 over n squared plus 1."},{"Start":"03:14.655 ","End":"03:19.690","Text":"But this is true because,"},{"Start":"03:20.360 ","End":"03:23.910","Text":"I can show something it\u0027s true that it implies this."},{"Start":"03:23.910 ","End":"03:32.740","Text":"If we just see it that the denominator here is bigger or equal to,"},{"Start":"03:32.740 ","End":"03:35.395","Text":"in fact actually bigger than, but doesn\u0027t matter,"},{"Start":"03:35.395 ","End":"03:42.250","Text":"than n squared plus 1,"},{"Start":"03:42.250 ","End":"03:43.975","Text":"then that will be okay."},{"Start":"03:43.975 ","End":"03:46.930","Text":"Now, of course this is bigger or equal to this."},{"Start":"03:46.930 ","End":"03:49.360","Text":"I could do it in several stages."},{"Start":"03:49.360 ","End":"03:55.990","Text":"I could say this is because n plus 1 squared is bigger or equal to n squared,"},{"Start":"03:55.990 ","End":"03:57.640","Text":"and why is this true?"},{"Start":"03:57.640 ","End":"04:02.350","Text":"This is true because n plus 1 is"},{"Start":"04:02.350 ","End":"04:07.790","Text":"bigger or equal to n. If this is bigger or equal to this,"},{"Start":"04:07.790 ","End":"04:10.700","Text":"when I square it and it\u0027s positive numbers,"},{"Start":"04:10.700 ","End":"04:12.485","Text":"then this is still bigger equal to this."},{"Start":"04:12.485 ","End":"04:15.560","Text":"Adding 1 still keeps the inequality."},{"Start":"04:15.560 ","End":"04:18.395","Text":"With positive numbers, and in general,"},{"Start":"04:18.395 ","End":"04:20.415","Text":"when you take the reciprocal,"},{"Start":"04:20.415 ","End":"04:23.735","Text":"it reverses the sign of the inequality,"},{"Start":"04:23.735 ","End":"04:25.110","Text":"and that leads us this."},{"Start":"04:25.110 ","End":"04:28.880","Text":"We start with something true and end up with what we wanted to prove,"},{"Start":"04:28.880 ","End":"04:33.170","Text":"so I can put a check mark here also."},{"Start":"04:33.170 ","End":"04:40.060","Text":"I prove the 2 conditions for the Leibniz alternating series test."},{"Start":"04:40.700 ","End":"04:45.290","Text":"This means that the original series,"},{"Start":"04:45.290 ","End":"04:50.795","Text":"the sum of a_n converges."},{"Start":"04:50.795 ","End":"04:53.480","Text":"Now we have these,"},{"Start":"04:53.480 ","End":"04:58.205","Text":"we\u0027ve ruled out the third possibility."},{"Start":"04:58.205 ","End":"05:05.745","Text":"In the exercise, we were asked to show whether it\u0027s convergent or divergent."},{"Start":"05:05.745 ","End":"05:07.560","Text":"We ruled out divergent."},{"Start":"05:07.560 ","End":"05:11.105","Text":"Now what we have to decide if it\u0027s absolutely or conditionally,"},{"Start":"05:11.105 ","End":"05:17.745","Text":"and the way we do that is to look at the definition."},{"Start":"05:17.745 ","End":"05:21.095","Text":"We want to know whether the series with absolute value."},{"Start":"05:21.095 ","End":"05:26.860","Text":"Now, the absolute value of"},{"Start":"05:27.230 ","End":"05:35.300","Text":"a_n is just the same thing without the minus 1 to the power of n plus 1."},{"Start":"05:35.300 ","End":"05:36.560","Text":"The sign doesn\u0027t matter."},{"Start":"05:36.560 ","End":"05:38.120","Text":"We take it to be positive."},{"Start":"05:38.120 ","End":"05:40.555","Text":"It\u0027s actually equal to b_n."},{"Start":"05:40.555 ","End":"05:43.095","Text":"But I\u0027ll write it again,"},{"Start":"05:43.095 ","End":"05:47.375","Text":"it\u0027s not significant, there just happens to equal b_n."},{"Start":"05:47.375 ","End":"05:50.060","Text":"What I have to ask is,"},{"Start":"05:50.060 ","End":"05:59.170","Text":"does the sum of 1 over n squared plus 1 converge?"},{"Start":"05:59.170 ","End":"06:01.280","Text":"Meanwhile, I\u0027m asking with a question mark,"},{"Start":"06:01.280 ","End":"06:05.400","Text":"I don\u0027t know, n goes from 1 to infinity."},{"Start":"06:05.540 ","End":"06:09.975","Text":"I can think of more than 1 way of showing this."},{"Start":"06:09.975 ","End":"06:19.865","Text":"For 1 thing I could use the limit comparison test or just the plain comparison test."},{"Start":"06:19.865 ","End":"06:22.715","Text":"I\u0027ll go with the limit comparison test."},{"Start":"06:22.715 ","End":"06:31.810","Text":"The comparison series I\u0027ll use will be just 1 over n squared,"},{"Start":"06:33.770 ","End":"06:38.680","Text":"and I know that this 1 converges."},{"Start":"06:40.060 ","End":"06:42.650","Text":"Why does this converge?"},{"Start":"06:42.650 ","End":"06:46.220","Text":"I\u0027ll remind you, because we\u0027ve seen it before,"},{"Start":"06:46.220 ","End":"06:53.195","Text":"but also there\u0027s a general theorem about p series that the sum of 1 over"},{"Start":"06:53.195 ","End":"07:01.010","Text":"n to the power of p converges if and only if p is bigger than 1."},{"Start":"07:01.010 ","End":"07:03.845","Text":"Certainly in our case, if p is 2,"},{"Start":"07:03.845 ","End":"07:07.010","Text":"2 is bigger than 1, so this converges."},{"Start":"07:07.010 ","End":"07:08.390","Text":"I\u0027m going to take this as"},{"Start":"07:08.390 ","End":"07:20.040","Text":"my comparison series in"},{"Start":"07:20.040 ","End":"07:22.950","Text":"order to check if this series converges."},{"Start":"07:22.950 ","End":"07:33.635","Text":"The comparison test says that we take the ratios of the comparison over the original."},{"Start":"07:33.635 ","End":"07:37.435","Text":"Now, in the limit comparison test,"},{"Start":"07:37.435 ","End":"07:42.715","Text":"we take the ratio of corresponding terms, I\u0027ll show you."},{"Start":"07:42.715 ","End":"07:44.965","Text":"It doesn\u0027t really matter which way round."},{"Start":"07:44.965 ","End":"07:47.995","Text":"Let\u0027s say I\u0027ll take this 1 over this 1,"},{"Start":"07:47.995 ","End":"07:55.200","Text":"so I take the limit as n goes to infinity of 1 over n"},{"Start":"07:55.200 ","End":"08:04.340","Text":"squared divided by 1 over n squared plus 1."},{"Start":"08:04.490 ","End":"08:08.230","Text":"In 1 variation of the limit comparison test,"},{"Start":"08:08.230 ","End":"08:13.295","Text":"we just want to look for a finite result that\u0027s not 0 and not infinity."},{"Start":"08:13.295 ","End":"08:18.680","Text":"If that\u0027s the case, if we get something bigger than 0 but not infinity,"},{"Start":"08:18.680 ","End":"08:21.500","Text":"then they both converge or both diverge."},{"Start":"08:21.500 ","End":"08:24.800","Text":"Their fates are tied together, so to speak."},{"Start":"08:24.800 ","End":"08:29.600","Text":"Let\u0027s see what is the limit of this ratio this is equal"},{"Start":"08:29.600 ","End":"08:34.535","Text":"to the limit as n goes to infinity."},{"Start":"08:34.535 ","End":"08:43.095","Text":"Doing some fractions, we get basically n squared plus 1 over n squared,"},{"Start":"08:43.095 ","End":"08:46.745","Text":"and this is the limit."},{"Start":"08:46.745 ","End":"08:50.390","Text":"We can just take the leading terms or we can actually take the trouble of"},{"Start":"08:50.390 ","End":"08:54.980","Text":"dividing it out and saying it\u0027s 1 plus 1 over n squared,"},{"Start":"08:54.980 ","End":"08:59.015","Text":"and this is equal to,"},{"Start":"08:59.015 ","End":"09:01.720","Text":"because n goes to infinity,"},{"Start":"09:01.720 ","End":"09:06.425","Text":"it\u0027s basically 1 minus 1 over infinity squared."},{"Start":"09:06.425 ","End":"09:09.140","Text":"It\u0027s just 1. I mean,"},{"Start":"09:09.140 ","End":"09:14.585","Text":"I could write 1 minus 1 over infinity squared is 1 over infinity is 0 and so on."},{"Start":"09:14.585 ","End":"09:20.190","Text":"Now, the fact that 1 is between 0 and infinity, not inclusive,"},{"Start":"09:20.190 ","End":"09:25.130","Text":"it means that we can apply the limit comparison test and say that if 1 converges,"},{"Start":"09:25.130 ","End":"09:27.155","Text":"the other converges and vice versa."},{"Start":"09:27.155 ","End":"09:29.405","Text":"Since we know this converges,"},{"Start":"09:29.405 ","End":"09:35.560","Text":"we conclude that this 1 converges also."},{"Start":"09:35.780 ","End":"09:42.855","Text":"The answer is that this sum,"},{"Start":"09:42.855 ","End":"09:44.790","Text":"which is the sum,"},{"Start":"09:44.790 ","End":"09:47.680","Text":"let me see where I can write it."},{"Start":"09:47.680 ","End":"09:49.640","Text":"I\u0027ll go to this corner here."},{"Start":"09:49.640 ","End":"09:56.540","Text":"This series was the sum from n goes from 1 to infinity of the absolute value of a_n,"},{"Start":"09:56.540 ","End":"09:58.760","Text":"where a_n was the original series,"},{"Start":"09:58.760 ","End":"10:02.010","Text":"that is this here."},{"Start":"10:02.360 ","End":"10:06.460","Text":"Now we\u0027ve shown that this converges"},{"Start":"10:18.380 ","End":"10:21.110","Text":"and if this converges,"},{"Start":"10:21.110 ","End":"10:24.095","Text":"then we know that the original series,"},{"Start":"10:24.095 ","End":"10:30.095","Text":"which was just a_n without the absolute value, converges absolutely."},{"Start":"10:30.095 ","End":"10:31.955","Text":"I\u0027ll write that down,"},{"Start":"10:31.955 ","End":"10:36.060","Text":"and it turns out that we actually, in retrospect,"},{"Start":"10:36.060 ","End":"10:42.755","Text":"did an unnecessary step of showing that the sum of a_n converges."},{"Start":"10:42.755 ","End":"10:45.350","Text":"Because as it says here,"},{"Start":"10:45.350 ","End":"10:49.500","Text":"if it\u0027s absolutely convergent, it\u0027s also convergent."},{"Start":"10:49.500 ","End":"10:58.070","Text":"The first part with the Leibniz alternating series was not needed."},{"Start":"10:58.070 ","End":"11:01.500","Text":"But if we knew in advance what to expect,"},{"Start":"11:01.500 ","End":"11:07.890","Text":"we could have gone and tried this absolute value right away."},{"Start":"11:08.890 ","End":"11:13.800","Text":"That\u0027s it for this 1, it converges absolutely."}],"ID":10561},{"Watched":false,"Name":"Exercise 10 Part c","Duration":"","ChapterTopicVideoID":10235,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this exercise, we\u0027re given the following infinite series and we have to"},{"Start":"00:04.350 ","End":"00:09.070","Text":"decide which of the 3 cases applies,"},{"Start":"00:09.070 ","End":"00:11.100","Text":"whether it\u0027s absolutely convergent,"},{"Start":"00:11.100 ","End":"00:14.040","Text":"conditionally convergent, or divergent."},{"Start":"00:14.040 ","End":"00:19.815","Text":"Now, I believe that it\u0027s not going to be conditionally convergent."},{"Start":"00:19.815 ","End":"00:25.840","Text":"The conditionally means that it converges but not in absolute value,"},{"Start":"00:25.840 ","End":"00:29.985","Text":"which means that if I throw out this minus 1^n plus 1,"},{"Start":"00:29.985 ","End":"00:32.744","Text":"that would make a difference."},{"Start":"00:32.744 ","End":"00:37.440","Text":"Now, if this was a decreasing series or sequence,"},{"Start":"00:37.440 ","End":"00:40.620","Text":"if the cosine n over n squared was decreasing,"},{"Start":"00:40.620 ","End":"00:47.025","Text":"I would use the Leibniz\u0027s alternating series test,"},{"Start":"00:47.025 ","End":"00:49.395","Text":"but I don\u0027t know that it\u0027s decreasing."},{"Start":"00:49.395 ","End":"00:52.580","Text":"Certainly, the denominator is increasing, but the numerator,"},{"Start":"00:52.580 ","End":"00:56.160","Text":"the cosine of a number jumps all over the place from"},{"Start":"00:56.160 ","End":"01:00.925","Text":"minus 1-1 and anything in between and it\u0027s hard to say."},{"Start":"01:00.925 ","End":"01:05.270","Text":"I would like to go right ahead and try and"},{"Start":"01:05.270 ","End":"01:09.845","Text":"see if we can get absolute convergence from the beginning."},{"Start":"01:09.845 ","End":"01:15.575","Text":"If we fail, then it\u0027s probably going to be divergent and then we\u0027ll deal with that."},{"Start":"01:15.575 ","End":"01:20.150","Text":"But I\u0027m mentally thinking that cosine"},{"Start":"01:20.150 ","End":"01:24.835","Text":"n in absolute value is less than 1 and I know that 1 over n squared is convergent,"},{"Start":"01:24.835 ","End":"01:27.350","Text":"so that\u0027s what motivates me to think that it\u0027s"},{"Start":"01:27.350 ","End":"01:31.200","Text":"probably going to be absolutely convergent."},{"Start":"01:32.060 ","End":"01:38.080","Text":"What I\u0027m going to do, let me get some more space here,"},{"Start":"01:38.080 ","End":"01:41.885","Text":"I\u0027m going to shoot for absolute convergence."},{"Start":"01:41.885 ","End":"01:46.880","Text":"That means that the series in absolute value is convergent."},{"Start":"01:46.880 ","End":"01:49.580","Text":"Let me just copy it. What was it?"},{"Start":"01:49.580 ","End":"01:57.870","Text":"It was the sum of n goes from 1 to infinity minus 1 to,"},{"Start":"01:57.870 ","End":"02:00.165","Text":"I forget if it was n or n plus 1."},{"Start":"02:00.165 ","End":"02:03.490","Text":"Let\u0027s just check, it was n plus 1,"},{"Start":"02:05.630 ","End":"02:12.855","Text":"times cosine n over n squared."},{"Start":"02:12.855 ","End":"02:19.275","Text":"This is going to be my term a_n."},{"Start":"02:19.275 ","End":"02:23.930","Text":"What I\u0027m going to do is take a look at the absolute value of this."},{"Start":"02:23.930 ","End":"02:30.200","Text":"The absolute value of this would be the series n goes from 1 to infinity."},{"Start":"02:30.200 ","End":"02:33.800","Text":"The absolute value of plus or minus 1 is just 1."},{"Start":"02:33.800 ","End":"02:37.265","Text":"What I get is the absolute value of cosine"},{"Start":"02:37.265 ","End":"02:41.870","Text":"n. We need the absolute value because cosine can be positive or negative."},{"Start":"02:41.870 ","End":"02:43.550","Text":"But in the denominator,"},{"Start":"02:43.550 ","End":"02:45.170","Text":"n squared is always positive,"},{"Start":"02:45.170 ","End":"02:47.730","Text":"so we don\u0027t need it there."},{"Start":"02:48.590 ","End":"02:51.335","Text":"This looks to me convergent."},{"Start":"02:51.335 ","End":"02:56.540","Text":"I\u0027m going to compare it somehow using 1 of the comparison tests to"},{"Start":"02:56.540 ","End":"03:03.310","Text":"the sum from 1 to infinity of 1 over n squared."},{"Start":"03:03.310 ","End":"03:07.655","Text":"It looks to me like the regular comparison test will do,"},{"Start":"03:07.655 ","End":"03:11.210","Text":"which means that, first of all,"},{"Start":"03:11.210 ","End":"03:13.220","Text":"I show that this is convergent."},{"Start":"03:13.220 ","End":"03:14.960","Text":"We\u0027re actually familiar with this,"},{"Start":"03:14.960 ","End":"03:16.070","Text":"and we know it\u0027s convergent,"},{"Start":"03:16.070 ","End":"03:18.065","Text":"but I\u0027ll go over that in a moment."},{"Start":"03:18.065 ","End":"03:21.260","Text":"Then we\u0027ll say that this series is less than or equal to"},{"Start":"03:21.260 ","End":"03:25.335","Text":"this series and they\u0027re both non-negative series."},{"Start":"03:25.335 ","End":"03:33.110","Text":"What I\u0027m going to show is that cosine n over n squared is"},{"Start":"03:33.110 ","End":"03:36.050","Text":"less than or equal to 1 over n"},{"Start":"03:36.050 ","End":"03:42.065","Text":"squared for all n or for all n from a certain place onwards."},{"Start":"03:42.065 ","End":"03:46.129","Text":"If I show this then by the comparison test,"},{"Start":"03:46.129 ","End":"03:49.340","Text":"and let\u0027s just throw in the 0 less than or equal to,"},{"Start":"03:49.340 ","End":"03:50.900","Text":"which is often written in,"},{"Start":"03:50.900 ","End":"03:53.465","Text":"that if this converges,"},{"Start":"03:53.465 ","End":"03:55.985","Text":"then this converges also."},{"Start":"03:55.985 ","End":"03:59.950","Text":"Now, we know that this series converges because, a,"},{"Start":"03:59.950 ","End":"04:03.060","Text":"we\u0027ve seen it so many times before,"},{"Start":"04:03.060 ","End":"04:07.660","Text":"and b, I usually give the same reasoning that it\u0027s a p-series."},{"Start":"04:07.660 ","End":"04:12.865","Text":"In general, when we have the sum of 1 over n^p,"},{"Start":"04:12.865 ","End":"04:18.670","Text":"the precise condition for convergence of this is that p should be bigger than 1."},{"Start":"04:18.670 ","End":"04:20.620","Text":"In our case, p is 2,"},{"Start":"04:20.620 ","End":"04:23.495","Text":"which is certainly bigger than 1."},{"Start":"04:23.495 ","End":"04:25.995","Text":"As for this inequality,"},{"Start":"04:25.995 ","End":"04:33.410","Text":"that\u0027s fairly clear because this holds because we know that cosine of anything,"},{"Start":"04:35.390 ","End":"04:38.550","Text":"in particular, cosine of n,"},{"Start":"04:38.550 ","End":"04:41.215","Text":"is always between minus 1 and 1."},{"Start":"04:41.215 ","End":"04:43.935","Text":"If something is between minus 1 and 1,"},{"Start":"04:43.935 ","End":"04:48.700","Text":"then its absolute value has to be less than or equal to 1."},{"Start":"04:48.700 ","End":"04:50.620","Text":"This is less than or equal to 1."},{"Start":"04:50.620 ","End":"04:54.265","Text":"It\u0027s the same positive denominator."},{"Start":"04:54.265 ","End":"04:56.770","Text":"This is less than or equal to this."},{"Start":"04:56.770 ","End":"05:01.710","Text":"This validates the comparison test."},{"Start":"05:01.710 ","End":"05:06.750","Text":"Because this converges, well,"},{"Start":"05:06.750 ","End":"05:10.490","Text":"the Sigma of it converges, this 1 converges."},{"Start":"05:10.490 ","End":"05:13.330","Text":"I don\u0027t need to write it twice."},{"Start":"05:13.330 ","End":"05:20.750","Text":"Because this converges and because we\u0027ve satisfied the comparison test,"},{"Start":"05:20.750 ","End":"05:23.900","Text":"then this 1 also converges."},{"Start":"05:23.900 ","End":"05:28.910","Text":"If this converges, it\u0027s the absolute value of the original series, this,"},{"Start":"05:28.910 ","End":"05:34.725","Text":"so this 1 converges absolutely."},{"Start":"05:34.725 ","End":"05:40.205","Text":"That\u0027s the definition of absolute convergence,"},{"Start":"05:40.205 ","End":"05:44.600","Text":"is that the series with absolute values is convergent."},{"Start":"05:44.600 ","End":"05:47.330","Text":"This is our answer to this question,"},{"Start":"05:47.330 ","End":"05:50.880","Text":"absolutely convergent. We\u0027re done."}],"ID":10562},{"Watched":false,"Name":"Exercise 10 Part d","Duration":"","ChapterTopicVideoID":10236,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.735","Text":"In this exercise, we have an infinite series"},{"Start":"00:03.735 ","End":"00:07.380","Text":"and we have to decide what convergence it has,"},{"Start":"00:07.380 ","End":"00:11.025","Text":"either absolute, or conditional,"},{"Start":"00:11.025 ","End":"00:15.375","Text":"or doesn\u0027t convergent at all, divergent."},{"Start":"00:15.375 ","End":"00:20.070","Text":"First of all, it looks like there\u0027s some nasty trigonometric expression here,"},{"Start":"00:20.070 ","End":"00:22.530","Text":"cosine n Pi, and what is that?"},{"Start":"00:22.530 ","End":"00:25.290","Text":"But if we think about it,"},{"Start":"00:25.290 ","End":"00:28.620","Text":"and let me scroll down,"},{"Start":"00:28.620 ","End":"00:30.855","Text":"get a bit more space here."},{"Start":"00:30.855 ","End":"00:35.370","Text":"If I think about what cosine n Pi is,"},{"Start":"00:35.370 ","End":"00:38.520","Text":"I claim it can be written quite simply."},{"Start":"00:38.520 ","End":"00:40.635","Text":"Let\u0027s take, for example,"},{"Start":"00:40.635 ","End":"00:44.340","Text":"where if I just take n as being 1,"},{"Start":"00:44.340 ","End":"00:48.645","Text":"2, 3, 4, and so on,"},{"Start":"00:48.645 ","End":"00:54.765","Text":"then here, cosine n Pi will give me cosine of Pi,"},{"Start":"00:54.765 ","End":"01:00.465","Text":"cosine 2Pi, cosine 3Pi,"},{"Start":"01:00.465 ","End":"01:04.830","Text":"cosine 4Pi, and so on."},{"Start":"01:04.830 ","End":"01:06.405","Text":"I can put commas then,"},{"Start":"01:06.405 ","End":"01:10.005","Text":"if I want, and so on."},{"Start":"01:10.005 ","End":"01:14.825","Text":"Now, the trigonometric function cosine has a period of 2Pi,"},{"Start":"01:14.825 ","End":"01:17.060","Text":"2Pi is a whole circle."},{"Start":"01:17.060 ","End":"01:20.150","Text":"Every time I increase by 2Pi,"},{"Start":"01:20.150 ","End":"01:22.415","Text":"I get to the same point."},{"Start":"01:22.415 ","End":"01:27.225","Text":"Now, cosine of Pi,"},{"Start":"01:27.225 ","End":"01:31.935","Text":"it\u0027s well-known, you can look it up on the calculator, is minus 1."},{"Start":"01:31.935 ","End":"01:35.140","Text":"Cosine 2Pi is the same as cosine of 0,"},{"Start":"01:35.140 ","End":"01:37.295","Text":"cosine 0 is 1."},{"Start":"01:37.295 ","End":"01:40.365","Text":"Since we\u0027re going in periods of 2Pi,"},{"Start":"01:40.365 ","End":"01:42.030","Text":"from here to here, it\u0027s 2Pi."},{"Start":"01:42.030 ","End":"01:43.350","Text":"This will also be minus 1,"},{"Start":"01:43.350 ","End":"01:44.895","Text":"this will also be 1."},{"Start":"01:44.895 ","End":"01:47.350","Text":"What we\u0027re going to get is minus 1,"},{"Start":"01:47.350 ","End":"01:50.500","Text":"1, minus 1, 1, and so on."},{"Start":"01:50.500 ","End":"01:52.775","Text":"It\u0027s going to be plus or minus 1."},{"Start":"01:52.775 ","End":"02:01.980","Text":"So that this has got to be either minus 1^n or minus 1^n plus 1."},{"Start":"02:01.980 ","End":"02:03.210","Text":"It doesn\u0027t really matter,"},{"Start":"02:03.210 ","End":"02:06.630","Text":"but let\u0027s just for precision sake, see which it is."},{"Start":"02:06.630 ","End":"02:13.105","Text":"If n is 1, this is negative and this is negative also."},{"Start":"02:13.105 ","End":"02:17.765","Text":"Yeah, so cosine n Pi is just minus 1^n,"},{"Start":"02:17.765 ","End":"02:25.680","Text":"which means that our original series can be rewritten as the sum"},{"Start":"02:25.680 ","End":"02:35.875","Text":"from 1 to infinity of minus 1^n times 1."},{"Start":"02:35.875 ","End":"02:42.110","Text":"Now, we\u0027ve probably seen this in some form before."},{"Start":"02:42.110 ","End":"02:44.180","Text":"This looks like a classical case of"},{"Start":"02:44.180 ","End":"02:48.670","Text":"conditional convergence because if I didn\u0027t have the minus 1^n,"},{"Start":"02:48.670 ","End":"02:51.680","Text":"if I put it always is 1, I get the 1,"},{"Start":"02:51.680 ","End":"02:55.450","Text":"which is the harmonic series and which is divergent."},{"Start":"02:55.450 ","End":"03:00.665","Text":"Usually, the first example of an alternating series is this."},{"Start":"03:00.665 ","End":"03:03.470","Text":"This is going to be conditionally convergent,"},{"Start":"03:03.470 ","End":"03:07.615","Text":"but let me just go over it again as to why."},{"Start":"03:07.615 ","End":"03:16.760","Text":"Basically, I\u0027ll show that it\u0027s convergent but not absolutely."},{"Start":"03:16.760 ","End":"03:21.050","Text":"Now, why it\u0027s convergent is because if I take minus 1^n,"},{"Start":"03:21.050 ","End":"03:22.940","Text":"and then times something else,"},{"Start":"03:22.940 ","End":"03:29.010","Text":"let\u0027s say that we\u0027ll call this whole thing a_n,"},{"Start":"03:29.010 ","End":"03:31.970","Text":"but this bit without the minus 1^n,"},{"Start":"03:31.970 ","End":"03:33.670","Text":"We\u0027ll call that b_n."},{"Start":"03:33.670 ","End":"03:36.560","Text":"According to the Leibnitz test,"},{"Start":"03:36.560 ","End":"03:40.235","Text":"if b_n has 2 things,"},{"Start":"03:40.235 ","End":"03:44.690","Text":"if it tends to 0 and it\u0027s decreasing,"},{"Start":"03:44.690 ","End":"03:50.090","Text":"then we can conclude that a_n is convergent."},{"Start":"03:50.090 ","End":"03:53.810","Text":"We haven\u0027t proven these 2 things yet,"},{"Start":"03:53.810 ","End":"03:57.730","Text":"but then it will mean that the original series is convergent."},{"Start":"03:57.730 ","End":"04:01.820","Text":"After we show that, I\u0027ll show that in absolute value, it isn\u0027t."},{"Start":"04:01.820 ","End":"04:05.490","Text":"Let\u0027s see, why does b_n go to 0?"},{"Start":"04:16.310 ","End":"04:24.810","Text":"Here\u0027s how I\u0027ll say it, minus 1^n times 1,"},{"Start":"04:24.810 ","End":"04:28.150","Text":"this part is bounded."},{"Start":"04:28.180 ","End":"04:33.080","Text":"It\u0027s bounded because it\u0027s always sandwiched between plus and minus 1,"},{"Start":"04:33.080 ","End":"04:36.320","Text":"because it\u0027s only equal to plus or minus 1, but anyway,"},{"Start":"04:36.320 ","End":"04:40.060","Text":"it\u0027s sandwiched, it\u0027s bounded within a certain range."},{"Start":"04:40.060 ","End":"04:44.450","Text":"This 1 goes to 0."},{"Start":"04:44.450 ","End":"04:51.110","Text":"As a theorem that something bounded times something that goes to 0 also goes to 0."},{"Start":"04:51.110 ","End":"04:53.195","Text":"You could use the sandwich theorem."},{"Start":"04:53.195 ","End":"04:55.100","Text":"This whole thing has got to be between"},{"Start":"04:55.100 ","End":"04:58.230","Text":"the upper bound times 0 and the lower bound times 0,"},{"Start":"04:58.230 ","End":"05:00.500","Text":"and either way it\u0027s between 0 and 0,"},{"Start":"05:00.500 ","End":"05:02.335","Text":"so it\u0027s forced to be 0."},{"Start":"05:02.335 ","End":"05:05.570","Text":"This thing indeed goes to 0."},{"Start":"05:06.530 ","End":"05:09.800","Text":"I can put a checkmark on this."},{"Start":"05:09.800 ","End":"05:12.230","Text":"Now as for the decreasing,"},{"Start":"05:12.230 ","End":"05:15.140","Text":"certainly because the series n,"},{"Start":"05:15.140 ","End":"05:17.135","Text":"n is always increasing,"},{"Start":"05:17.135 ","End":"05:18.650","Text":"1, 2, 3, 4, 5."},{"Start":"05:18.650 ","End":"05:20.750","Text":"If I take 1, 1/2,"},{"Start":"05:20.750 ","End":"05:23.775","Text":"1/3, 1/4, 1/5, 1/6 and so on."},{"Start":"05:23.775 ","End":"05:29.010","Text":"Obviously, decreasing. As 1 over something increasing is decreasing."},{"Start":"05:29.060 ","End":"05:32.970","Text":"I can put a checkmark here too,"},{"Start":"05:32.970 ","End":"05:35.790","Text":"b_n goes to 0, b_n is decreasing."},{"Start":"05:35.790 ","End":"05:38.070","Text":"The b_n is the 1 part."},{"Start":"05:38.070 ","End":"05:40.875","Text":"We conclude that an is convergent."},{"Start":"05:40.875 ","End":"05:43.370","Text":"Because of these 2, I can say yes."},{"Start":"05:43.370 ","End":"05:46.820","Text":"Now, conditionally convergent means that it\u0027s"},{"Start":"05:46.820 ","End":"05:51.320","Text":"convergent but the absolute value series is divergent."},{"Start":"05:51.320 ","End":"05:53.225","Text":"Let\u0027s just check that part."},{"Start":"05:53.225 ","End":"05:56.040","Text":"The absolute value"},{"Start":"06:06.260 ","End":"06:08.955","Text":"series"},{"Start":"06:08.955 ","End":"06:13.315","Text":"is the series n goes from 1 to infinity."},{"Start":"06:13.315 ","End":"06:19.790","Text":"The absolute value of this is just without the plus or minus 1."},{"Start":"06:19.940 ","End":"06:23.820","Text":"This is the harmonic series,"},{"Start":"06:23.820 ","End":"06:27.160","Text":"and it\u0027s known to be divergent."},{"Start":"06:28.940 ","End":"06:33.070","Text":"If you want a different reason other than remembering,"},{"Start":"06:33.070 ","End":"06:35.410","Text":"we could use the p series."},{"Start":"06:35.410 ","End":"06:40.375","Text":"For example, we can say that it\u0027s p series,"},{"Start":"06:40.375 ","End":"06:46.395","Text":"1 over n^p, where p equals 1."},{"Start":"06:46.395 ","End":"06:50.500","Text":"The convergence for this is precisely p greater than 1."},{"Start":"06:50.500 ","End":"06:53.210","Text":"In our case, p equals 1."},{"Start":"06:53.210 ","End":"06:57.420","Text":"It doesn\u0027t satisfy because 1 is not greater than 1."},{"Start":"06:57.420 ","End":"07:01.800","Text":"It doesn\u0027t converge, so it diverges."},{"Start":"07:01.800 ","End":"07:05.140","Text":"This 1 is divergent,"},{"Start":"07:05.710 ","End":"07:12.340","Text":"and this is the sum of absolute value of a_n."},{"Start":"07:12.340 ","End":"07:16.640","Text":"We\u0027ve met the condition that this is divergent and this is convergent,"},{"Start":"07:16.640 ","End":"07:23.010","Text":"and so we end up with the conditionally convergent and we\u0027re done."}],"ID":10563},{"Watched":false,"Name":"Exercise 10 Part e","Duration":"","ChapterTopicVideoID":10237,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we\u0027re given this infinite series and we have to"},{"Start":"00:04.320 ","End":"00:08.520","Text":"decide whether it\u0027s convergent absolutely or conditionally,"},{"Start":"00:08.520 ","End":"00:11.550","Text":"or maybe it doesn\u0027t converge at all."},{"Start":"00:11.550 ","End":"00:15.630","Text":"I remember seeing this exercise in the section on"},{"Start":"00:15.630 ","End":"00:20.070","Text":"the alternating series test due to Leibniz."},{"Start":"00:20.070 ","End":"00:22.340","Text":"But just in case you missed it,"},{"Start":"00:22.340 ","End":"00:24.320","Text":"I\u0027ll go over it again briefly."},{"Start":"00:24.320 ","End":"00:27.660","Text":"Just get some more space here."},{"Start":"00:30.430 ","End":"00:36.235","Text":"The whole thing could be called a_n,"},{"Start":"00:36.235 ","End":"00:39.980","Text":"and this part here could be called b_n."},{"Start":"00:39.980 ","End":"00:44.390","Text":"We showed 2 things that b_n is decreasing"},{"Start":"00:44.390 ","End":"00:49.375","Text":"and that b_n tends to zero as n goes to infinity."},{"Start":"00:49.375 ","End":"00:55.890","Text":"Then we used the Leibniz alternating"},{"Start":"00:55.890 ","End":"01:02.620","Text":"series test to conclude that a_n is in fact convergent."},{"Start":"01:02.620 ","End":"01:05.660","Text":"As a matter of fact, it turned out that this"},{"Start":"01:05.660 ","End":"01:08.300","Text":"was not always decreasing,"},{"Start":"01:08.300 ","End":"01:12.110","Text":"but only for n bigger or equal to 3"},{"Start":"01:12.110 ","End":"01:13.760","Text":"could we guarantee it."},{"Start":"01:13.760 ","End":"01:15.050","Text":"But that doesn\u0027t matter because"},{"Start":"01:15.050 ","End":"01:16.550","Text":"a finite number of terms never"},{"Start":"01:16.550 ","End":"01:19.825","Text":"affects convergence or divergence."},{"Start":"01:19.825 ","End":"01:22.260","Text":"Essentially we did this by looking"},{"Start":"01:22.260 ","End":"01:27.540","Text":"at the continuous function of f of x,"},{"Start":"01:27.540 ","End":"01:31.300","Text":"which was the natural log of x over x"},{"Start":"01:31.300 ","End":"01:35.785","Text":"and we prove both of them, the infinity part."},{"Start":"01:35.785 ","End":"01:37.670","Text":"We let x go to infinity."},{"Start":"01:37.670 ","End":"01:39.755","Text":"We got infinity over infinity."},{"Start":"01:39.755 ","End":"01:43.340","Text":"We use L\u0027Hopital\u0027s rule and we differentiate the top"},{"Start":"01:43.340 ","End":"01:45.305","Text":"and bottom and we got zero."},{"Start":"01:45.305 ","End":"01:47.960","Text":"Then the other case we differentiated"},{"Start":"01:47.960 ","End":"01:50.015","Text":"and found that the derivative,"},{"Start":"01:50.015 ","End":"01:54.890","Text":"at least for x bigger or equal to 3 was negative"},{"Start":"01:54.890 ","End":"01:56.255","Text":"and so it\u0027s decreasing."},{"Start":"01:56.255 ","End":"02:00.545","Text":"That\u0027s a quick summary of why this series converges."},{"Start":"02:00.545 ","End":"02:04.290","Text":"We\u0027ve ruled 1 of the 3 cases out."},{"Start":"02:05.450 ","End":"02:08.240","Text":"In the original series,"},{"Start":"02:08.240 ","End":"02:10.340","Text":"we\u0027ve already ruled out divergent."},{"Start":"02:10.340 ","End":"02:14.885","Text":"It\u0027s either conditionally or absolutely convergent."},{"Start":"02:14.885 ","End":"02:18.590","Text":"We\u0027ll check if it\u0027s absolutely convergent."},{"Start":"02:18.590 ","End":"02:20.780","Text":"Absolute convergence means that we now look"},{"Start":"02:20.780 ","End":"02:24.290","Text":"at the series of the absolute value of this."},{"Start":"02:24.290 ","End":"02:30.515","Text":"Now since these are positive or at least non-negative,"},{"Start":"02:30.515 ","End":"02:32.900","Text":"when n is zero."},{"Start":"02:32.900 ","End":"02:36.350","Text":"But the absolute value we just throw out the plus or minus,"},{"Start":"02:36.350 ","End":"02:38.000","Text":"which is the minus 1^n."},{"Start":"02:38.000 ","End":"02:43.200","Text":"We are asking about the series natural log of n over n"},{"Start":"02:43.200 ","End":"02:45.375","Text":"from 1 to infinity."},{"Start":"02:45.375 ","End":"02:51.570","Text":"Which is the absolute value of a_n in general."},{"Start":"02:51.570 ","End":"02:54.490","Text":"Is this convergent or not?"},{"Start":"02:54.490 ","End":"02:59.240","Text":"I would guess looking at it, that it\u0027s divergent."},{"Start":"02:59.240 ","End":"03:00.635","Text":"Why would I say that?"},{"Start":"03:00.635 ","End":"03:02.990","Text":"Because I\u0027m thinking of the harmonic series,"},{"Start":"03:02.990 ","End":"03:07.790","Text":"1 over n. Now 1 over n is divergent and"},{"Start":"03:07.790 ","End":"03:11.660","Text":"the natural log of n is mostly bigger than 1"},{"Start":"03:11.660 ","End":"03:14.435","Text":"or maybe when n equals 1 or 2,"},{"Start":"03:14.435 ","End":"03:15.770","Text":"it\u0027s less than 1,"},{"Start":"03:15.770 ","End":"03:18.019","Text":"that certainly from 3 onwards,"},{"Start":"03:18.019 ","End":"03:20.015","Text":"it\u0027s bigger than 1."},{"Start":"03:20.015 ","End":"03:22.430","Text":"So we\u0027re bigger than a divergent series"},{"Start":"03:22.430 ","End":"03:23.990","Text":"and so with divergent."},{"Start":"03:23.990 ","End":"03:28.320","Text":"Let me just say, do a little bit more formally."},{"Start":"03:28.320 ","End":"03:30.405","Text":"Why I expect this to be divergent,"},{"Start":"03:30.405 ","End":"03:34.250","Text":"I do a comparison and put a squiggly sign,"},{"Start":"03:34.250 ","End":"03:40.400","Text":"meaning compare this to the sum n equals 1 to infinity of 1"},{"Start":"03:40.400 ","End":"03:49.585","Text":"over n. This is absolute value of a_n and this might be cn."},{"Start":"03:49.585 ","End":"03:51.545","Text":"What I have to show,"},{"Start":"03:51.545 ","End":"03:56.810","Text":"is I have to show that the natural log of n over n is"},{"Start":"03:56.810 ","End":"04:02.675","Text":"bigger or equal to 1 over n. It doesn\u0027t have to be for all n,"},{"Start":"04:02.675 ","End":"04:05.120","Text":"but it can be for almost all n,"},{"Start":"04:05.120 ","End":"04:06.950","Text":"meaning from a certain n onwards."},{"Start":"04:06.950 ","End":"04:10.490","Text":"I claim that this is true for n bigger or"},{"Start":"04:10.490 ","End":"04:15.064","Text":"equal to 3 because when n is bigger or equal to 3,"},{"Start":"04:15.064 ","End":"04:24.485","Text":"the natural log of n is bigger or equal to the natural log of 3."},{"Start":"04:24.485 ","End":"04:28.010","Text":"The natural log of 3 is certainly bigger or equal"},{"Start":"04:28.010 ","End":"04:31.280","Text":"to the natural log of e, certainly,"},{"Start":"04:31.280 ","End":"04:33.135","Text":"but because e is less than 3,"},{"Start":"04:33.135 ","End":"04:37.919","Text":"e is 2.718 or something."},{"Start":"04:38.350 ","End":"04:42.125","Text":"If this is bigger or equal to natural log of e,"},{"Start":"04:42.125 ","End":"04:44.435","Text":"and this is equal to 1."},{"Start":"04:44.435 ","End":"04:48.320","Text":"Natural log of n is bigger or equal to 1"},{"Start":"04:48.320 ","End":"04:51.325","Text":"for most n and same denominator."},{"Start":"04:51.325 ","End":"04:53.294","Text":"The numerator counts."},{"Start":"04:53.294 ","End":"04:55.454","Text":"Because this is true,"},{"Start":"04:55.454 ","End":"04:59.310","Text":"and because this 1 is divergent,"},{"Start":"04:59.310 ","End":"05:01.310","Text":"by the comparison test,"},{"Start":"05:01.310 ","End":"05:03.710","Text":"if we\u0027re bigger or equal to divergent,"},{"Start":"05:03.710 ","End":"05:09.620","Text":"it implies that this is also divergent."},{"Start":"05:09.620 ","End":"05:16.295","Text":"We\u0027re now in a situation where the original series is convergent."},{"Start":"05:16.295 ","End":"05:19.280","Text":"I just briefly showed you why here."},{"Start":"05:19.280 ","End":"05:22.760","Text":"We did it before. We also showed"},{"Start":"05:22.760 ","End":"05:25.100","Text":"that this series is divergent."},{"Start":"05:25.100 ","End":"05:29.840","Text":"When this happens, then we are conditionally convergent."},{"Start":"05:29.840 ","End":"05:31.500","Text":"That\u0027s it."}],"ID":10564},{"Watched":false,"Name":"Exercise 10 Part f","Duration":"","ChapterTopicVideoID":10238,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have this infinite series and we"},{"Start":"00:03.360 ","End":"00:06.089","Text":"have to decide if it converges absolutely,"},{"Start":"00:06.089 ","End":"00:10.240","Text":"or conditionally, or perhaps it doesn\u0027t converge at all."},{"Start":"00:11.270 ","End":"00:15.359","Text":"We have to have a guess of what to try first,"},{"Start":"00:15.359 ","End":"00:21.450","Text":"and I have a feeling that this one is actually going to be absolutely convergent,"},{"Start":"00:21.450 ","End":"00:23.220","Text":"so let\u0027s try that. If that doesn\u0027t work,"},{"Start":"00:23.220 ","End":"00:26.020","Text":"we can always try the others."},{"Start":"00:26.270 ","End":"00:30.250","Text":"Let me just get some more space here."},{"Start":"00:32.150 ","End":"00:35.780","Text":"I\u0027m going to try for absolute convergence,"},{"Start":"00:35.780 ","End":"00:38.764","Text":"which means that if I take the absolute value,"},{"Start":"00:38.764 ","End":"00:40.669","Text":"so I won\u0027t take this series,"},{"Start":"00:40.669 ","End":"00:43.265","Text":"I\u0027ll take the series of absolute values,"},{"Start":"00:43.265 ","End":"00:46.220","Text":"which is the sum from 1 to infinity,"},{"Start":"00:46.220 ","End":"00:48.560","Text":"the same thing except without the minus,"},{"Start":"00:48.560 ","End":"00:54.530","Text":"because the denominator is positive, and the absolute value of minus 1^n is just 1,"},{"Start":"00:54.530 ","End":"01:02.180","Text":"so I have 1 over natural log of n to the power of n. Now,"},{"Start":"01:02.180 ","End":"01:12.590","Text":"we\u0027ve learned various tests and the power of n leads me to want to try the root test."},{"Start":"01:12.590 ","End":"01:17.360","Text":"Remember in the root test,"},{"Start":"01:17.360 ","End":"01:22.520","Text":"what we do is we take the limit as n goes to infinity of"},{"Start":"01:22.520 ","End":"01:28.010","Text":"the nth root of the general term,"},{"Start":"01:28.010 ","End":"01:38.265","Text":"which is 1 over natural log of n to the power of n. If this comes out bigger than 1,"},{"Start":"01:38.265 ","End":"01:40.710","Text":"diverges, less than 1,"},{"Start":"01:40.710 ","End":"01:43.610","Text":"converges, equal to 1, we don\u0027t know."},{"Start":"01:43.610 ","End":"01:47.185","Text":"Anyway, let\u0027s see what this is equal to."},{"Start":"01:47.185 ","End":"01:50.640","Text":"The nth root and the power of n cancel,"},{"Start":"01:50.640 ","End":"01:55.100","Text":"so we just have the limit as n goes to infinity of 1 over"},{"Start":"01:55.100 ","End":"02:00.619","Text":"the natural log of n. Because when n goes to infinity,"},{"Start":"02:00.619 ","End":"02:02.980","Text":"so does the natural log,"},{"Start":"02:02.980 ","End":"02:06.710","Text":"this is just equal to 1 over infinity,"},{"Start":"02:06.710 ","End":"02:11.120","Text":"which is 0, and 0 is less than 1."},{"Start":"02:11.120 ","End":"02:16.300","Text":"By the root test, this thing converges."},{"Start":"02:16.300 ","End":"02:20.480","Text":"If the series of absolute values converges,"},{"Start":"02:20.480 ","End":"02:24.950","Text":"then this one converges, absolutely."},{"Start":"02:24.950 ","End":"02:30.470","Text":"So this has absolute rather than it\u0027s absolutely convergent."},{"Start":"02:30.470 ","End":"02:34.800","Text":"That answers the question and we are done."}],"ID":10565},{"Watched":false,"Name":"Exercise 10 Part g","Duration":"","ChapterTopicVideoID":10232,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.595","Text":"In this exercise, we\u0027re given"},{"Start":"00:02.595 ","End":"00:06.780","Text":"an infinite series and we have to decide if it\u0027s absolutely convergent,"},{"Start":"00:06.780 ","End":"00:09.690","Text":"conditionally convergent or divergent."},{"Start":"00:09.690 ","End":"00:11.730","Text":"But before we start it,"},{"Start":"00:11.730 ","End":"00:18.370","Text":"let\u0027s do some informal mathematics just to see what we expect to get."},{"Start":"00:22.610 ","End":"00:25.710","Text":"If I just take the leading power,"},{"Start":"00:25.710 ","End":"00:28.270","Text":"it\u0027s n squared plus n is like n squared,"},{"Start":"00:28.270 ","End":"00:31.410","Text":"so the denominator is roughly like 1"},{"Start":"00:31.410 ","End":"00:40.210","Text":"over n and that diverges but because I have the alternating plus minus from here,"},{"Start":"00:40.510 ","End":"00:45.380","Text":"this thing generally goes to infinity in the denominator."},{"Start":"00:45.380 ","End":"00:47.855","Text":"In other words, the general term goes to 0."},{"Start":"00:47.855 ","End":"00:51.320","Text":"It looks like a classical case of conditional convergence"},{"Start":"00:51.320 ","End":"00:55.770","Text":"and I would use the Leibniz alternating series test."},{"Start":"00:56.300 ","End":"00:58.975","Text":"Why don\u0027t we do that?"},{"Start":"00:58.975 ","End":"01:04.624","Text":"Let\u0027s show that this converges according to the alternating series and then afterwards,"},{"Start":"01:04.624 ","End":"01:10.450","Text":"I\u0027ll show that it diverges using a comparison test with the harmonic series."},{"Start":"01:10.450 ","End":"01:13.830","Text":"Let me just get some more space here."},{"Start":"01:14.540 ","End":"01:17.760","Text":"First of all, to show convergence,"},{"Start":"01:17.760 ","End":"01:23.340","Text":"let\u0027s say that this is equal to"},{"Start":"01:23.340 ","End":"01:30.970","Text":"minus 1^n plus 1 times bn,"},{"Start":"01:30.970 ","End":"01:35.310","Text":"this part is the minus 1^n and bn is"},{"Start":"01:35.310 ","End":"01:43.700","Text":"the same thing and the words bn is just 1 over square root of n,"},{"Start":"01:43.700 ","End":"01:47.990","Text":"n plus 1 without the alternating sign."},{"Start":"01:47.990 ","End":"01:54.140","Text":"Now if I show that 2 things for the alternating series test,"},{"Start":"01:54.140 ","End":"02:03.380","Text":"if I show that bn turns to 0 and bn is a decreasing sequence,"},{"Start":"02:03.380 ","End":"02:09.620","Text":"then that will be enough to show that the sum of an,"},{"Start":"02:09.620 ","End":"02:15.005","Text":"this is my an, is convergent."},{"Start":"02:15.005 ","End":"02:20.730","Text":"Let\u0027s see. Probably the easiest way to do"},{"Start":"02:20.730 ","End":"02:26.540","Text":"it is to use calculus."},{"Start":"02:26.540 ","End":"02:28.340","Text":"I\u0027m just trying to think. On the other hand,"},{"Start":"02:28.340 ","End":"02:31.260","Text":"we could use inequalities."},{"Start":"02:31.940 ","End":"02:36.875","Text":"I\u0027ll tell you what, I\u0027ll do it this time without the calculus."},{"Start":"02:36.875 ","End":"02:44.070","Text":"What I can say is that bn let\u0027s try to prove the tenths to 0 part."},{"Start":"02:44.070 ","End":"02:50.720","Text":"So bn is this and if I take n outside the brackets here,"},{"Start":"02:50.720 ","End":"02:57.410","Text":"I\u0027ll say that this is equal to 1 over n. Now if I pull n outside,"},{"Start":"02:57.410 ","End":"03:00.170","Text":"not the brackets, I meant the square root sign,"},{"Start":"03:00.170 ","End":"03:05.450","Text":"I have to take n squared out so I\u0027ll take n out of here,"},{"Start":"03:05.450 ","End":"03:06.770","Text":"another n out of here,"},{"Start":"03:06.770 ","End":"03:13.555","Text":"and I\u0027m left with 1 plus 1 over n. Now,"},{"Start":"03:13.555 ","End":"03:22.075","Text":"this thing, n goes to infinity so 1 over n goes to 0."},{"Start":"03:22.075 ","End":"03:23.960","Text":"Under the square root sign,"},{"Start":"03:23.960 ","End":"03:28.100","Text":"it goes to 1 because this also goes to 0."},{"Start":"03:28.100 ","End":"03:34.660","Text":"Basically, what I\u0027m getting if I write it symbolically is 1"},{"Start":"03:34.660 ","End":"03:42.650","Text":"over infinity times the square root of 1 plus 1 over infinity."},{"Start":"03:44.210 ","End":"03:47.350","Text":"Since this bit 1 over infinity is 0,"},{"Start":"03:47.350 ","End":"03:49.105","Text":"square root of 1 is 1."},{"Start":"03:49.105 ","End":"03:50.845","Text":"It\u0027s 1 over infinity,"},{"Start":"03:50.845 ","End":"03:58.074","Text":"which is equal to 0 so we\u0027re okay with the bn going to 0."},{"Start":"03:58.074 ","End":"04:01.250","Text":"As for bn decreasing,"},{"Start":"04:01.350 ","End":"04:07.900","Text":"what I would like to say is that bn plus 1 is going to be less."},{"Start":"04:07.900 ","End":"04:11.290","Text":"That\u0027s enough for me to show that it\u0027s less than or equal to bn for"},{"Start":"04:11.290 ","End":"04:15.970","Text":"any n because bn is n times n plus"},{"Start":"04:15.970 ","End":"04:21.020","Text":"1 and bn plus 1 is when I replace n by"},{"Start":"04:21.020 ","End":"04:27.810","Text":"n plus 1 is n plus 1 times n plus 2."},{"Start":"04:28.790 ","End":"04:34.760","Text":"Now, 1 over the square root,"},{"Start":"04:36.320 ","End":"04:40.305","Text":"I\u0027m going to build up to bn is less than or equal to."},{"Start":"04:40.305 ","End":"04:42.735","Text":"Let\u0027s start off with this and say,"},{"Start":"04:42.735 ","End":"04:47.420","Text":"surely this is bigger or equal to this,"},{"Start":"04:47.420 ","End":"04:49.190","Text":"actually it\u0027s strictly bigger than but it doesn\u0027t"},{"Start":"04:49.190 ","End":"04:51.455","Text":"matter because this is bigger than this,"},{"Start":"04:51.455 ","End":"04:53.790","Text":"this is bigger than this,"},{"Start":"04:53.790 ","End":"04:55.560","Text":"and it\u0027s positive numbers."},{"Start":"04:55.560 ","End":"05:01.460","Text":"Now if I take the square root of this, n plus 1,"},{"Start":"05:01.460 ","End":"05:07.850","Text":"n plus 2, it will still be bigger or equal to the square root of n,"},{"Start":"05:07.850 ","End":"05:11.764","Text":"n plus 1 because we\u0027re dealing with positive numbers."},{"Start":"05:11.764 ","End":"05:16.460","Text":"But if I take a reciprocal 1 over the square root,"},{"Start":"05:16.460 ","End":"05:20.690","Text":"it will be less than or equal to 1 over the square root because the"},{"Start":"05:20.690 ","End":"05:25.820","Text":"1 over reverses the direction so here we have n plus 1,"},{"Start":"05:25.820 ","End":"05:30.800","Text":"n plus 2 and here n, n plus 1."},{"Start":"05:30.800 ","End":"05:34.920","Text":"This thing here is"},{"Start":"05:34.920 ","End":"05:43.720","Text":"exactly our bn and this is exactly our bn plus 1."},{"Start":"05:43.910 ","End":"05:48.815","Text":"With just inequalities, we can show that each term is"},{"Start":"05:48.815 ","End":"05:53.810","Text":"less than or equal to the previous term so it\u0027s decreasing so check on here also,"},{"Start":"05:53.810 ","End":"06:02.070","Text":"which means that this thing is convergent by the alternating series test."},{"Start":"06:04.010 ","End":"06:10.370","Text":"How about showing that this one is divergent,"},{"Start":"06:10.370 ","End":"06:12.170","Text":"which is what we suspected."},{"Start":"06:12.170 ","End":"06:15.455","Text":"What I\u0027m going to do is compare."},{"Start":"06:15.455 ","End":"06:18.440","Text":"Well, first of all, say what this series is."},{"Start":"06:18.440 ","End":"06:23.789","Text":"The other series we have to show is the series of absolute values."},{"Start":"06:24.670 ","End":"06:28.760","Text":"The absolute value just means throw out the plus or minus 1,"},{"Start":"06:28.760 ","End":"06:30.325","Text":"we don\u0027t need it."},{"Start":"06:30.325 ","End":"06:36.095","Text":"We have 1 over the square root of n,"},{"Start":"06:36.095 ","End":"06:43.830","Text":"n plus 1, and this is this part."},{"Start":"06:43.830 ","End":"06:48.345","Text":"This is the Sigma of absolute value of an,"},{"Start":"06:48.345 ","End":"06:50.245","Text":"that\u0027s what this is here."},{"Start":"06:50.245 ","End":"06:52.010","Text":"Now I\u0027m going to compare it,"},{"Start":"06:52.010 ","End":"06:53.750","Text":"use some comparison tests."},{"Start":"06:53.750 ","End":"06:58.850","Text":"I\u0027m not sure if the limit comparison test or regular comparison test but like I said,"},{"Start":"06:58.850 ","End":"07:04.445","Text":"it\u0027s very similar to the harmonic series which is 1"},{"Start":"07:04.445 ","End":"07:13.580","Text":"over n. I think I\u0027ll go with the limit comparison test,"},{"Start":"07:13.580 ","End":"07:16.940","Text":"which takes the ratio of these and takes them to the limit."},{"Start":"07:16.940 ","End":"07:20.670","Text":"Let\u0027s see what is the limit of the ratio,"},{"Start":"07:20.670 ","End":"07:24.110","Text":"it doesn\u0027t really matter which way around you divide."},{"Start":"07:24.110 ","End":"07:27.949","Text":"The idea is that if we get the ratio"},{"Start":"07:27.949 ","End":"07:32.690","Text":"being somewhere between 0 and infinity and it\u0027s finite but not 0,"},{"Start":"07:32.690 ","End":"07:34.850","Text":"then either both converge or both diverge,"},{"Start":"07:34.850 ","End":"07:36.350","Text":"they have the same fate."},{"Start":"07:36.350 ","End":"07:40.020","Text":"Let\u0027s take, for example,"},{"Start":"07:40.940 ","End":"07:49.455","Text":"this divided by this so we get dividing by 1 over n puts n in the numerator,"},{"Start":"07:49.455 ","End":"07:55.120","Text":"so n over the square root of n, n plus 1."},{"Start":"07:55.490 ","End":"07:58.425","Text":"This divided by this, the limit,"},{"Start":"07:58.425 ","End":"08:03.200","Text":"just look at the limit comparison test and we have to see what this is."},{"Start":"08:03.200 ","End":"08:07.850","Text":"Well, this is the limit as n goes to infinity,"},{"Start":"08:07.850 ","End":"08:09.260","Text":"divide top and bottom by"},{"Start":"08:09.260 ","End":"08:18.755","Text":"n. We\u0027ve done this calculation already over here,"},{"Start":"08:18.755 ","End":"08:27.000","Text":"that this is equal to 1 over n times the square root of"},{"Start":"08:27.000 ","End":"08:35.130","Text":"1 plus 1 over n. Sorry,"},{"Start":"08:35.130 ","End":"08:36.315","Text":"it\u0027s not 1 over,"},{"Start":"08:36.315 ","End":"08:42.125","Text":"it\u0027s n over because we forgot that here we had the 1 here we have the n, sorry."},{"Start":"08:42.125 ","End":"08:45.325","Text":"This is equal to,"},{"Start":"08:45.325 ","End":"08:48.600","Text":"now n over n is just 1 so this is"},{"Start":"08:48.600 ","End":"08:57.525","Text":"just the limit of 1 over the square root of 1 plus 1 over n. Now it\u0027s clear,"},{"Start":"08:57.525 ","End":"08:59.530","Text":"1 over n goes to 0,"},{"Start":"08:59.530 ","End":"09:03.170","Text":"so it\u0027s 1 over the square root of 1 and this is 1."},{"Start":"09:03.170 ","End":"09:06.775","Text":"Now, 1 is between 0 and infinity,"},{"Start":"09:06.775 ","End":"09:10.150","Text":"which is good for the limit comparison test."},{"Start":"09:10.150 ","End":"09:14.320","Text":"That means that both of these convergeable diverge but we know that this"},{"Start":"09:14.320 ","End":"09:21.190","Text":"diverges this one because we\u0027ve seen it so many times before it\u0027s the harmonic series."},{"Start":"09:21.190 ","End":"09:24.280","Text":"You could look at it as a p series with p equals"},{"Start":"09:24.280 ","End":"09:28.290","Text":"1 and it only converges when p is bigger than 1."},{"Start":"09:28.290 ","End":"09:33.480","Text":"So this diverges, hence, this diverges too."},{"Start":"09:33.680 ","End":"09:44.900","Text":"This one also diverges from the limit comparison test and so"},{"Start":"09:44.900 ","End":"09:47.810","Text":"we fulfill the condition that the original series is"},{"Start":"09:47.810 ","End":"09:50.645","Text":"convergent but the absolute value series"},{"Start":"09:50.645 ","End":"09:57.480","Text":"is divergent so we are conditionally convergent and we\u0027re done."}],"ID":10566},{"Watched":false,"Name":"Exercise 11 Part a","Duration":"","ChapterTopicVideoID":10240,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.920","Text":"This exercise is a proof question, approve or disapprove."},{"Start":"00:07.460 ","End":"00:13.080","Text":"The claim is that if we have 2 series, a_n and b_n,"},{"Start":"00:13.080 ","End":"00:15.105","Text":"and this series converges,"},{"Start":"00:15.105 ","End":"00:17.415","Text":"and this series diverges,"},{"Start":"00:17.415 ","End":"00:19.860","Text":"then the sum diverges."},{"Start":"00:19.860 ","End":"00:24.980","Text":"In other words, is convergent plus divergent equal to divergent?"},{"Start":"00:24.980 ","End":"00:28.499","Text":"Turns out, the answer is yes."},{"Start":"00:28.700 ","End":"00:30.900","Text":"I\u0027ll just say first of all,"},{"Start":"00:30.900 ","End":"00:32.220","Text":"the answer is yes,"},{"Start":"00:32.220 ","End":"00:34.140","Text":"and now the proof."},{"Start":"00:34.140 ","End":"00:37.560","Text":"Proof, there\u0027s many ways of proving things."},{"Start":"00:37.560 ","End":"00:40.760","Text":"In this case, we\u0027re going to use proof by contradiction,"},{"Start":"00:40.760 ","End":"00:46.435","Text":"and you may or may not have heard of proof by contradiction, I hope you have."},{"Start":"00:46.435 ","End":"00:52.010","Text":"Proof by contradiction usually starts off with something like,"},{"Start":"00:52.010 ","End":"00:56.285","Text":"suppose not, or suppose that this is not true,"},{"Start":"00:56.285 ","End":"00:58.820","Text":"and then by a series of arguments,"},{"Start":"00:58.820 ","End":"01:01.340","Text":"we reach a contradiction at the end."},{"Start":"01:01.340 ","End":"01:04.250","Text":"Because we reach a contradiction,"},{"Start":"01:04.250 ","End":"01:07.219","Text":"the supposition that this is false,"},{"Start":"01:07.219 ","End":"01:10.325","Text":"gives a contradiction, so it must be true."},{"Start":"01:10.325 ","End":"01:12.260","Text":"Let\u0027s suppose not,"},{"Start":"01:12.260 ","End":"01:14.650","Text":"so what does not so mean?"},{"Start":"01:14.650 ","End":"01:18.260","Text":"It means that we can find an example,"},{"Start":"01:20.080 ","End":"01:26.030","Text":"so we can find specific examples"},{"Start":"01:26.030 ","End":"01:33.450","Text":"of a_n, which converges,"},{"Start":"01:34.550 ","End":"01:42.520","Text":"or just to say it\u0027s convergent, and b_n divergent."},{"Start":"01:46.880 ","End":"01:49.625","Text":"But, here\u0027s the contradiction of the statement,"},{"Start":"01:49.625 ","End":"01:52.355","Text":"an example, but this is convergent,"},{"Start":"01:52.355 ","End":"01:59.615","Text":"but the sum of a_n plus b_n is convergent."},{"Start":"01:59.615 ","End":"02:04.310","Text":"All of these are n goes from something to infinity."},{"Start":"02:04.310 ","End":"02:10.550","Text":"Sometimes we just put the n, and not say from 1 to"},{"Start":"02:10.550 ","End":"02:13.370","Text":"infinity or from where, at least to show that the variable is"},{"Start":"02:13.370 ","End":"02:16.790","Text":"n. Now, I\u0027m going to show you how we reach a contradiction."},{"Start":"02:16.790 ","End":"02:18.800","Text":"Let\u0027s take this series,"},{"Start":"02:18.800 ","End":"02:25.175","Text":"the sum of a_n plus b_n, it\u0027s convergent."},{"Start":"02:25.175 ","End":"02:33.915","Text":"Now, also the sum of a_n is convergent."},{"Start":"02:33.915 ","End":"02:38.895","Text":"Now, I\u0027ll just write that this is convergent,"},{"Start":"02:38.895 ","End":"02:41.730","Text":"and I\u0027ll write it also here, convergent,"},{"Start":"02:41.730 ","End":"02:48.740","Text":"and we know that the sum or difference of convergent series is also convergent."},{"Start":"02:48.740 ","End":"02:51.620","Text":"In other words, what I\u0027m saying is if this is convergent,"},{"Start":"02:51.620 ","End":"02:52.940","Text":"and this is convergent,"},{"Start":"02:52.940 ","End":"02:55.535","Text":"then this difference is convergent."},{"Start":"02:55.535 ","End":"02:59.135","Text":"I mean that we can actually take the difference element-wise."},{"Start":"02:59.135 ","End":"03:01.355","Text":"If we have 2 convergent series,"},{"Start":"03:01.355 ","End":"03:05.020","Text":"then the series obtained by adding or subtracting,"},{"Start":"03:05.020 ","End":"03:09.300","Text":"so we got a_n plus b_n minus a_n,"},{"Start":"03:09.300 ","End":"03:11.210","Text":"going to put extra brackets here,"},{"Start":"03:11.210 ","End":"03:16.090","Text":"this will also be a convergent."},{"Start":"03:16.090 ","End":"03:21.080","Text":"It\u0027s convergent and it\u0027s equal to this."},{"Start":"03:21.080 ","End":"03:24.995","Text":"But if you look at this, this is precisely,"},{"Start":"03:24.995 ","End":"03:30.480","Text":"this is equal to a_n plus b_n minus a_n is b_n,"},{"Start":"03:30.480 ","End":"03:35.009","Text":"so this means that the sum of b_n is convergent."},{"Start":"03:35.009 ","End":"03:39.280","Text":"These 2 are equal."},{"Start":"03:39.670 ","End":"03:45.630","Text":"This is a contradiction, because b_n was assumed to be divergent,"},{"Start":"03:45.630 ","End":"03:47.580","Text":"and here, we see it\u0027s convergent."},{"Start":"03:47.580 ","End":"03:49.725","Text":"What does the contradiction mean?"},{"Start":"03:49.725 ","End":"03:53.100","Text":"That the suppose not is wrong."},{"Start":"03:53.100 ","End":"03:54.440","Text":"Then in other words,"},{"Start":"03:54.440 ","End":"03:56.360","Text":"that it has to be that it\u0027s true, because"},{"Start":"03:56.360 ","End":"03:59.975","Text":"opposing that it ain\u0027t true, gives a contradiction."},{"Start":"03:59.975 ","End":"04:02.370","Text":"That\u0027s all. We\u0027re done."}],"ID":10567},{"Watched":false,"Name":"Exercise 11 Part b","Duration":"","ChapterTopicVideoID":10241,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.485","Text":"This exercise is also 1 of those prove or disprove questions."},{"Start":"00:05.485 ","End":"00:09.020","Text":"We have basically 3 series."},{"Start":"00:09.020 ","End":"00:10.535","Text":"We have the series a_n."},{"Start":"00:10.535 ","End":"00:15.825","Text":"N goes from 1 to infinity,"},{"Start":"00:15.825 ","End":"00:17.385","Text":"it doesn\u0027t really matter."},{"Start":"00:17.385 ","End":"00:19.785","Text":"We have the series b_n,"},{"Start":"00:19.785 ","End":"00:22.545","Text":"and we have the series a_n plus b_n."},{"Start":"00:22.545 ","End":"00:28.800","Text":"Now, the background to this question is that we know that if this converges,"},{"Start":"00:28.800 ","End":"00:33.615","Text":"and this converges, then the series of the sum also converges."},{"Start":"00:33.615 ","End":"00:39.810","Text":"The question is, is it true if I replace the word converges with the word diverges?"},{"Start":"00:39.810 ","End":"00:44.495","Text":"Is a divergent series plus a divergent series, also divergent?"},{"Start":"00:44.495 ","End":"00:48.710","Text":"We\u0027d have to prove or give a counterexample."},{"Start":"00:48.710 ","End":"00:52.700","Text":"Now, I happen to know that this is not true,"},{"Start":"00:52.700 ","End":"00:55.710","Text":"and I\u0027m going to give a counterexample."},{"Start":"00:56.230 ","End":"01:02.885","Text":"In fact, it\u0027s easy to give a counterexample to this and show that it isn\u0027t true."},{"Start":"01:02.885 ","End":"01:05.885","Text":"All you need is 1 example to disprove something."},{"Start":"01:05.885 ","End":"01:09.260","Text":"Let\u0027s take for a_n any divergent series,"},{"Start":"01:09.260 ","End":"01:11.554","Text":"for example, the harmonic series."},{"Start":"01:11.554 ","End":"01:18.930","Text":"I know that the sum of 1, this 1 diverges."},{"Start":"01:18.930 ","End":"01:21.660","Text":"That\u0027s what I\u0027ll take for my a_n,"},{"Start":"01:21.660 ","End":"01:24.705","Text":"so this is like the a_n."},{"Start":"01:24.705 ","End":"01:32.310","Text":"For b_n, I\u0027ll take the sum of minus 1/ a_n."},{"Start":"01:32.310 ","End":"01:33.740","Text":"Whatever I took here,"},{"Start":"01:33.740 ","End":"01:35.495","Text":"I\u0027ll take the minus of it here."},{"Start":"01:35.495 ","End":"01:37.025","Text":"I\u0027ll even put it in brackets."},{"Start":"01:37.025 ","End":"01:40.795","Text":"Also, let\u0027s say from 1 to infinity."},{"Start":"01:40.795 ","End":"01:42.810","Text":"N is less important,"},{"Start":"01:42.810 ","End":"01:44.040","Text":"and if I forget to write it,"},{"Start":"01:44.040 ","End":"01:46.065","Text":"then not to worry."},{"Start":"01:46.065 ","End":"01:49.235","Text":"This 1 also diverges."},{"Start":"01:49.235 ","End":"01:57.570","Text":"Why? Because if you multiply a convergence series by a constant,"},{"Start":"01:57.570 ","End":"02:00.405","Text":"it converges and similarly for divergent."},{"Start":"02:00.405 ","End":"02:05.390","Text":"If you multiply a divergent by any non-0 constant,"},{"Start":"02:05.390 ","End":"02:13.880","Text":"this is my b_n but it is minus 1 times a_n."},{"Start":"02:13.880 ","End":"02:20.250","Text":"As I said, multiplying by a non-0 constant won\u0027t change convergence or divergence."},{"Start":"02:20.390 ","End":"02:24.045","Text":"That\u0027s my a_n, and that\u0027s my b_n."},{"Start":"02:24.045 ","End":"02:26.535","Text":"Let\u0027s see what is a_n plus b_n."},{"Start":"02:26.535 ","End":"02:36.375","Text":"The sum of a_n plus b_n is just the sum of a_n is 1,"},{"Start":"02:36.375 ","End":"02:43.290","Text":"b_n is minus 1."},{"Start":"02:43.290 ","End":"02:46.365","Text":"This is the sum of 0."},{"Start":"02:46.365 ","End":"02:50.360","Text":"The sum of the constant term 0 is always convergent,"},{"Start":"02:50.360 ","End":"02:52.969","Text":"and in fact, it converges to 0."},{"Start":"02:52.969 ","End":"02:55.400","Text":"The point is not the answer,"},{"Start":"02:55.400 ","End":"02:58.830","Text":"but the fact is that it converges."},{"Start":"02:58.910 ","End":"03:04.335","Text":"We gave an example where this diverges,"},{"Start":"03:04.335 ","End":"03:09.060","Text":"this diverges but the sum converges,"},{"Start":"03:09.060 ","End":"03:12.310","Text":"so this is false, and we\u0027re done."}],"ID":10568},{"Watched":false,"Name":"Exercise 11 Part c","Duration":"","ChapterTopicVideoID":10242,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"This exercise is 2 and 1."},{"Start":"00:03.600 ","End":"00:05.160","Text":"They\u0027re a bit similar,"},{"Start":"00:05.160 ","End":"00:06.450","Text":"Part 1 and Part 2."},{"Start":"00:06.450 ","End":"00:13.245","Text":"In Part 1, we\u0027re told that the sum of a_n squared converges."},{"Start":"00:13.245 ","End":"00:18.060","Text":"Does it mean that the series a_n converges?"},{"Start":"00:18.060 ","End":"00:20.160","Text":"If so you have to prove it,"},{"Start":"00:20.160 ","End":"00:23.085","Text":"and to disprove it, you need to give a counter-example."},{"Start":"00:23.085 ","End":"00:27.810","Text":"Well, it turns out that 1 is false,"},{"Start":"00:27.810 ","End":"00:30.940","Text":"and I\u0027ll give a counter-example."},{"Start":"00:32.450 ","End":"00:35.280","Text":"The counter-example is this,"},{"Start":"00:35.280 ","End":"00:42.210","Text":"let\u0027s take a_n to be 1 over n. Now,"},{"Start":"00:42.210 ","End":"00:46.540","Text":"the sum of a_n squared,"},{"Start":"00:47.240 ","End":"00:55.625","Text":"which is the sum of 1 over n squared, definitely converges."},{"Start":"00:55.625 ","End":"00:57.845","Text":"We\u0027ve seen it many times."},{"Start":"00:57.845 ","End":"01:03.320","Text":"It\u0027s also a p-series with p bigger than 1, so it converges,"},{"Start":"01:03.320 ","End":"01:08.625","Text":"but the sum of just a_n,"},{"Start":"01:08.625 ","End":"01:15.125","Text":"2nd 1, is the sum of 1 over n. This is the harmonic series."},{"Start":"01:15.125 ","End":"01:18.905","Text":"It\u0027s a p series where p is not bigger than 1, it\u0027s equal to 1."},{"Start":"01:18.905 ","End":"01:24.850","Text":"This 1 diverges, so this is the counter-example."},{"Start":"01:24.850 ","End":"01:31.075","Text":"Now, Part 2 is trying to get clever."},{"Start":"01:31.075 ","End":"01:35.530","Text":"Suppose that this converges and suppose that this also"},{"Start":"01:35.530 ","End":"01:40.890","Text":"converges because that can happen also. Does it converge absolutely?"},{"Start":"01:40.890 ","End":"01:44.110","Text":"Absolutely, remember means putting the absolute value."},{"Start":"01:44.110 ","End":"01:48.535","Text":"You think maybe because it\u0027s squared, that\u0027s always positive."},{"Start":"01:48.535 ","End":"01:54.200","Text":"Turns out that number 2 is also false."},{"Start":"01:54.200 ","End":"01:58.750","Text":"Even if this converges and this converges,"},{"Start":"01:58.750 ","End":"02:02.500","Text":"this could still have conditional convergence."},{"Start":"02:04.430 ","End":"02:07.395","Text":"The answer is false,"},{"Start":"02:07.395 ","End":"02:11.470","Text":"and I\u0027m going to give a counter-example."},{"Start":"02:13.730 ","End":"02:19.220","Text":"In this counter-example, very similar to the above."},{"Start":"02:19.220 ","End":"02:27.895","Text":"I\u0027ll take a_n to be minus 1 to the power of n plus 1,"},{"Start":"02:27.895 ","End":"02:33.970","Text":"1 over n. This is the harmonic series,"},{"Start":"02:34.640 ","End":"02:38.190","Text":"1 plus 1/2 plus 1/3 and so on,"},{"Start":"02:38.190 ","End":"02:39.630","Text":"and this is the alternating,"},{"Start":"02:39.630 ","End":"02:43.260","Text":"1 minus 1/2 plus 1/3 minus 1/4 and so on."},{"Start":"02:43.260 ","End":"02:45.290","Text":"We know that this 1,"},{"Start":"02:45.290 ","End":"02:49.920","Text":"because it\u0027s an alternating series, converges."},{"Start":"02:49.920 ","End":"02:51.740","Text":"Well, I\u0027m getting ahead of myself."},{"Start":"02:51.740 ","End":"02:53.825","Text":"Let\u0027s do this methodically."},{"Start":"02:53.825 ","End":"03:01.190","Text":"Let\u0027s take that. Now, when I square this,"},{"Start":"03:01.190 ","End":"03:03.890","Text":"the minus 1 to whatever power it is,"},{"Start":"03:03.890 ","End":"03:06.425","Text":"when I square it, it\u0027s going to be 1."},{"Start":"03:06.425 ","End":"03:08.615","Text":"I get 1 over n squared,"},{"Start":"03:08.615 ","End":"03:13.670","Text":"exactly the same as above, and this converges."},{"Start":"03:13.670 ","End":"03:21.180","Text":"However, the sum of a_n,"},{"Start":"03:23.000 ","End":"03:34.360","Text":"which is the sum of minus 1 to the n plus 1 times 1 over n also converges."},{"Start":"03:35.030 ","End":"03:37.260","Text":"This is by the"},{"Start":"03:37.260 ","End":"03:47.010","Text":"alternating series test"},{"Start":"03:47.010 ","End":"03:49.300","Text":"due to Leibniz, but"},{"Start":"04:00.470 ","End":"04:04.215","Text":"I can\u0027t say that it converges absolutely,"},{"Start":"04:04.215 ","End":"04:09.385","Text":"but the sum of the absolute value of a_n,"},{"Start":"04:09.385 ","End":"04:13.400","Text":"which is the sum of 1 over n because the absolute value"},{"Start":"04:13.400 ","End":"04:18.240","Text":"just throws out the sign, diverges."},{"Start":"04:19.790 ","End":"04:24.690","Text":"This means if this diverges,"},{"Start":"04:24.690 ","End":"04:27.210","Text":"this doesn\u0027t converge absolutely."},{"Start":"04:27.210 ","End":"04:33.690","Text":"It does converge, so it implies that this converges conditionally."},{"Start":"04:36.350 ","End":"04:40.950","Text":"This just means that this is"},{"Start":"04:40.950 ","End":"04:46.860","Text":"not absolute convergence because the absolute value doesn\u0027t converge."},{"Start":"04:47.260 ","End":"04:53.730","Text":"In both cases, both of these statements turned out to be false."},{"Start":"04:53.730 ","End":"04:56.930","Text":"Done with this 1."}],"ID":10569},{"Watched":false,"Name":"Exercise 11 Part d","Duration":"","ChapterTopicVideoID":10243,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.055","Text":"In this exercise, we\u0027re given a convergent positive series,"},{"Start":"00:05.055 ","End":"00:09.900","Text":"and we have to prove or disprove"},{"Start":"00:09.900 ","End":"00:15.990","Text":"that the sum of 1/a_n is divergent."},{"Start":"00:15.990 ","End":"00:20.250","Text":"First of all, it\u0027s defined, because we\u0027re strictly positive,"},{"Start":"00:20.250 ","End":"00:23.470","Text":"so 1 over, it\u0027s not 0."},{"Start":"00:23.870 ","End":"00:28.230","Text":"What we do is the regular limit test."},{"Start":"00:28.230 ","End":"00:30.930","Text":"We know that if a series is convergent,"},{"Start":"00:30.930 ","End":"00:40.090","Text":"we know that the limit, as n goes to infinity of a_n, is 0."},{"Start":"00:43.180 ","End":"00:47.810","Text":"If I show that the limit of this is not 0,"},{"Start":"00:47.810 ","End":"00:51.270","Text":"then it\u0027s not going to be convergent."},{"Start":"00:52.930 ","End":"00:57.935","Text":"In a way it\u0027s a proof by contradiction,"},{"Start":"00:57.935 ","End":"01:00.920","Text":"could be looked at, but we\u0027ll see."},{"Start":"01:00.920 ","End":"01:04.805","Text":"So if a_n, limit goes to 0,"},{"Start":"01:04.805 ","End":"01:09.770","Text":"it actually goes to 0 from above,"},{"Start":"01:09.770 ","End":"01:14.105","Text":"you could say it\u0027s 0 plus,"},{"Start":"01:14.105 ","End":"01:16.400","Text":"because it\u0027s always positive,"},{"Start":"01:16.400 ","End":"01:18.395","Text":"it goes to 0 from above."},{"Start":"01:18.395 ","End":"01:28.725","Text":"That means that the limit of 1/a_n, this is always positive."},{"Start":"01:28.725 ","End":"01:35.105","Text":"If we have 1 over something that goes to 0 plus this is 1/0 plus,"},{"Start":"01:35.105 ","End":"01:37.640","Text":"which is plus infinity."},{"Start":"01:37.640 ","End":"01:41.660","Text":"Normally 1/0, you wouldn\u0027t know if it has"},{"Start":"01:41.660 ","End":"01:46.610","Text":"a limit or not, even including infinity and minus infinity,"},{"Start":"01:46.610 ","End":"01:48.905","Text":"it could zigzag back and forth,"},{"Start":"01:48.905 ","End":"01:51.665","Text":"but here, because it goes to 0, and it\u0027s only positive,"},{"Start":"01:51.665 ","End":"01:55.055","Text":"it has a limit of infinity."},{"Start":"01:55.055 ","End":"01:57.740","Text":"Well, amongst other things,"},{"Start":"01:57.740 ","End":"02:06.425","Text":"it means that the limit as n goes to infinity of 1/a_n is not equal to 0."},{"Start":"02:06.425 ","End":"02:09.830","Text":"You can say it doesn\u0027t exist, or it\u0027s not equal to 0,"},{"Start":"02:09.830 ","End":"02:11.030","Text":"or you can say it\u0027s infinity,"},{"Start":"02:11.030 ","End":"02:12.140","Text":"but whatever it is,"},{"Start":"02:12.140 ","End":"02:17.010","Text":"we cannot say the statement that this equals 0 is false."},{"Start":"02:17.010 ","End":"02:22.550","Text":"Because 1/a_n does not go to 0,"},{"Start":"02:24.260 ","End":"02:26.880","Text":"then it must be divergent,"},{"Start":"02:26.880 ","End":"02:28.430","Text":"because if it were convergent,"},{"Start":"02:28.430 ","End":"02:30.725","Text":"the general term would go to 0,"},{"Start":"02:30.725 ","End":"02:37.985","Text":"so the series 1/a_n is divergent."},{"Start":"02:37.985 ","End":"02:43.010","Text":"So this statement is true. We\u0027ve proved it."}],"ID":10570},{"Watched":false,"Name":"Exercise 11 Part e","Duration":"","ChapterTopicVideoID":10239,"CourseChapterTopicPlaylistID":52098,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/default.jpeg","UploadDate":null,"DurationForVideoObject":"PT0S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"This is another 1 of those proof questions."},{"Start":"00:04.080 ","End":"00:07.020","Text":"We\u0027re given a convergent series,"},{"Start":"00:07.020 ","End":"00:08.430","Text":"and then we get a new series"},{"Start":"00:08.430 ","End":"00:10.380","Text":"by squaring the terms."},{"Start":"00:10.380 ","End":"00:11.250","Text":"The question is,"},{"Start":"00:11.250 ","End":"00:14.190","Text":"is that series also convergent?"},{"Start":"00:14.190 ","End":"00:17.940","Text":"It turns out the answer is no."},{"Start":"00:17.940 ","End":"00:20.670","Text":"I\u0027ll just write that down, no,"},{"Start":"00:20.670 ","End":"00:26.415","Text":"and I need to provide a counterexample."},{"Start":"00:26.415 ","End":"00:28.920","Text":"For my counterexample,"},{"Start":"00:28.920 ","End":"00:33.240","Text":"I\u0027ll take the sum of a_n to be the series,"},{"Start":"00:33.240 ","End":"00:37.980","Text":"the sum of, I\u0027ll make it an alternating series,"},{"Start":"00:37.980 ","End":"00:43.046","Text":"minus 1^n times 1"},{"Start":"00:43.046 ","End":"00:47.030","Text":"over the square root of n."},{"Start":"00:47.030 ","End":"00:53.750","Text":"Now, this actually converges,"},{"Start":"00:53.750 ","End":"00:57.890","Text":"and the easiest way to see this is to use"},{"Start":"00:57.890 ","End":"01:02.180","Text":"the Leibniz\u0027s alternating series test."},{"Start":"01:02.180 ","End":"01:05.060","Text":"This part is decreasing."},{"Start":"01:05.060 ","End":"01:07.045","Text":"Obviously, when n increases,"},{"Start":"01:07.045 ","End":"01:11.040","Text":"1 over n decreases with the square root also,"},{"Start":"01:11.040 ","End":"01:14.295","Text":"and it converges to 0."},{"Start":"01:14.295 ","End":"01:17.570","Text":"By the Leibniz\u0027s test, this is convergent."},{"Start":"01:17.570 ","End":"01:23.265","Text":"However, the sum of a_n squared is the sum,"},{"Start":"01:23.265 ","End":"01:25.320","Text":"the minus 1^n when you square,"},{"Start":"01:25.320 ","End":"01:27.230","Text":"it disappears because plus"},{"Start":"01:27.230 ","End":"01:30.170","Text":"or minus 1 squared is always 1,"},{"Start":"01:30.170 ","End":"01:32.240","Text":"and we\u0027re just left with 1 over"},{"Start":"01:32.240 ","End":"01:33.650","Text":"the square root of n squared,"},{"Start":"01:33.650 ","End":"01:35.120","Text":"which is just 1 over n."},{"Start":"01:35.120 ","End":"01:37.610","Text":"This is the harmonic series,"},{"Start":"01:37.610 ","End":"01:39.970","Text":"and it diverges."},{"Start":"01:39.970 ","End":"01:42.970","Text":"That\u0027s a counterexample."},{"Start":"01:43.520 ","End":"01:47.450","Text":"We\u0027re basically done,"},{"Start":"01:47.450 ","End":"01:50.030","Text":"but I\u0027d like to give you a bonus question."},{"Start":"01:50.030 ","End":"01:52.880","Text":"Suppose, I added an extra condition"},{"Start":"01:52.880 ","End":"01:57.260","Text":"and I said, \"Suppose that a_n"},{"Start":"01:57.260 ","End":"02:01.110","Text":"was a positive series or non-negative."},{"Start":"02:01.110 ","End":"02:02.660","Text":"Suppose a_n is bigger"},{"Start":"02:02.660 ","End":"02:05.520","Text":"or equal to 0 for all n,"},{"Start":"02:05.520 ","End":"02:07.860","Text":"what then?\""},{"Start":"02:08.360 ","End":"02:11.265","Text":"This is just bonus, and you can stop now."},{"Start":"02:11.265 ","End":"02:14.490","Text":"But then it would be true."},{"Start":"02:14.490 ","End":"02:16.290","Text":"If I added this condition,"},{"Start":"02:16.290 ","End":"02:20.325","Text":"then the answer would be yes."},{"Start":"02:20.325 ","End":"02:22.820","Text":"The reason that this would be true"},{"Start":"02:22.820 ","End":"02:27.290","Text":"is because I could use either"},{"Start":"02:27.290 ","End":"02:29.150","Text":"the comparison test"},{"Start":"02:29.150 ","End":"02:31.235","Text":"or the limit comparison test."},{"Start":"02:31.235 ","End":"02:34.340","Text":"Basically, the reason is"},{"Start":"02:34.340 ","End":"02:37.320","Text":"that because a_n goes to 0,"},{"Start":"02:37.320 ","End":"02:39.150","Text":"in any convergent series,"},{"Start":"02:39.150 ","End":"02:42.140","Text":"at some point, it\u0027s going to be less than 1"},{"Start":"02:42.140 ","End":"02:43.715","Text":"from some point onwards."},{"Start":"02:43.715 ","End":"02:47.075","Text":"So a_n squared is less than a_n,"},{"Start":"02:47.075 ","End":"02:48.830","Text":"at least from some point on"},{"Start":"02:48.830 ","End":"02:51.015","Text":"or except for a finite number."},{"Start":"02:51.015 ","End":"02:57.020","Text":"If this is smaller than the convergent series and it\u0027s positive,"},{"Start":"02:57.020 ","End":"02:59.300","Text":"then it\u0027s also convergent."},{"Start":"02:59.300 ","End":"03:02.855","Text":"Or you could use the ratio test"},{"Start":"03:02.855 ","End":"03:04.820","Text":"that the ratio of this over this"},{"Start":"03:04.820 ","End":"03:07.085","Text":"is a_n and it tends to 0."},{"Start":"03:07.085 ","End":"03:09.800","Text":"If I added an extra condition that a_n"},{"Start":"03:09.800 ","End":"03:12.635","Text":"is a non-negative convergent series,"},{"Start":"03:12.635 ","End":"03:15.154","Text":"then it would be true otherwise not,"},{"Start":"03:15.154 ","End":"03:16.925","Text":"because here we have an example,"},{"Start":"03:16.925 ","End":"03:19.625","Text":"and in fact, we have to use negatives."},{"Start":"03:19.625 ","End":"03:23.880","Text":"That\u0027s it, we\u0027re done with this exercise."}],"ID":10571}],"Thumbnail":null,"ID":52098}]

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