1 00:00:00,104 --> 00:00:04,580 We need general techniques for determining when a series converges. 2 00:00:04,580 --> 00:00:10,119 [MUSIC]. 3 00:00:10,119 --> 00:00:14,820 We've already seen an example of what's usually called the Comparison Test. 4 00:00:14,820 --> 00:00:17,170 The original series that we looked at was this series. 5 00:00:17,170 --> 00:00:19,710 It has the sum, n goes from 0 to infinity. 6 00:00:19,710 --> 00:00:24,090 Sine squared n, divided by 2 to the n. And that series converges. 7 00:00:24,090 --> 00:00:25,010 But, remember why. 8 00:00:25,010 --> 00:00:28,990 Let's recall how we prove that this series converges. 9 00:00:28,990 --> 00:00:32,460 And what we did was to compare it to a geometric series. 10 00:00:32,460 --> 00:00:35,120 So, 0 is less than or equal to sine 11 00:00:35,120 --> 00:00:40,670 squared n over 2 to the n, is less than or equal to 1 over 2 to the n. 12 00:00:40,670 --> 00:00:43,164 And these inner qualities just follow from the fact 13 00:00:43,164 --> 00:00:45,540 that sine squared is trapped between 0 and 1. 14 00:00:46,820 --> 00:00:50,591 And what do I know about the sum of 1 over 2 to the n? 15 00:00:50,591 --> 00:00:55,130 The sum 1 over 2 to the n, n goes from 0 to infinity, converges. 16 00:00:56,220 --> 00:00:58,450 Right, this is a geometric series, I can even evaluate it. 17 00:00:59,720 --> 00:01:00,149 Now, what does 18 00:01:00,149 --> 00:01:01,850 that tell me about the original series that I'm studying? 19 00:01:01,850 --> 00:01:03,670 What about the sine squared n over 2 to the n? 20 00:01:04,690 --> 00:01:07,630 Well the sequence or partial sums for this series are 21 00:01:07,630 --> 00:01:11,530 all bounded above by the vast value of this series. 22 00:01:11,530 --> 00:01:14,088 And the sequence of partial sums is non 23 00:01:14,088 --> 00:01:17,620 decreasing because these terms are all none negative. 24 00:01:17,620 --> 00:01:20,960 So, I've got a monotone sequence which is bounded above. 25 00:01:20,960 --> 00:01:25,150 That means the sequence of partial sums converges, and that's just what it means 26 00:01:25,150 --> 00:01:28,580 to say that this series converges. That was a bit quick. 27 00:01:28,580 --> 00:01:30,380 So let's generalize this argument, try to 28 00:01:30,380 --> 00:01:34,070 formulate it as a broadly useful technique. 29 00:01:34,070 --> 00:01:35,730 So here's the usual setup. 30 00:01:35,730 --> 00:01:39,950 I'm, imagine that I've got two series. I'll use K as the index, K goes from 0 to 31 00:01:39,950 --> 00:01:46,840 infinity of a sub k, and another series, K goes from 0 to infinity of b sub k. 32 00:01:46,840 --> 00:01:50,340 And I'm supposing that 0 is less than or equal 33 00:01:50,340 --> 00:01:54,670 to a sub k, is less than or equal to b sub k, for all K. 34 00:01:54,670 --> 00:01:59,170 I'm imagining that the sum of the b sub k's converges. 35 00:01:59,170 --> 00:01:59,670 Right. 36 00:01:59,670 --> 00:02:02,510 So I'm going to be assuming that this series, 37 00:02:02,510 --> 00:02:05,270 converges, and then I'm wondering about this series. 38 00:02:05,270 --> 00:02:10,190 Does that series converge, or diverge? Well, to study that question, supposed to 39 00:02:10,190 --> 00:02:16,040 look at the sequence of partial sums. So that's S sub n is the sum 40 00:02:16,040 --> 00:02:21,840 of the terms, K goes from 0 to n, of a sub k. 41 00:02:21,840 --> 00:02:25,885 And now, the whole question is whether this sequence converges? 42 00:02:25,885 --> 00:02:28,130 Because the convergence of this sequence is exactly 43 00:02:28,130 --> 00:02:31,420 what it means to say that this series converges. 44 00:02:31,420 --> 00:02:34,410 What kind of sequence is that? Well, here's what I know. 45 00:02:36,230 --> 00:02:41,020 I know that a sub k is less than or equal to b sub k. 46 00:02:42,090 --> 00:02:45,920 And that means if I add up a whole bunch of a sub k's. 47 00:02:45,920 --> 00:02:48,760 Say, the a sub k is between k equals 0 and n. 48 00:02:48,760 --> 00:02:53,600 That's less than or equal to adding up the b sub k's, K goes from 0 to n. 49 00:02:54,600 --> 00:02:58,100 Now this thing here is just the nth partial sum. 50 00:02:59,670 --> 00:03:02,040 Now what do I know about this sum? 51 00:03:02,040 --> 00:03:07,242 Well the rest of the b sub k's are all non negative, so if I add up even more b sub 52 00:03:07,242 --> 00:03:11,505 k's, I'm just making this even bigger. So what I can conclude, 53 00:03:11,505 --> 00:03:15,510 is this. That s sub n is less than or 54 00:03:15,510 --> 00:03:19,765 equal to the value of the series, K goes from 0 to 55 00:03:19,765 --> 00:03:24,740 infinity, of b sub k. In other words, what I can conclude is 56 00:03:24,740 --> 00:03:32,660 that S sub n, is bounded above. And it's monotone. 57 00:03:32,660 --> 00:03:34,190 So let's check that. 58 00:03:34,190 --> 00:03:37,070 I'm going to show that if m is bigger than n, then 59 00:03:37,070 --> 00:03:41,720 S sub m is at least as big as S sub n. 60 00:03:41,720 --> 00:03:43,340 Well, what's S sub m? 61 00:03:43,340 --> 00:03:48,770 S sub m is the sum, K goes from zero to m, of a sub k. 62 00:03:48,770 --> 00:03:55,605 And I'm sure that's bigger than or equal to the sum K goes from 0 to n, a sub k. 63 00:03:55,605 --> 00:03:57,780 But, m is bigger than n, 64 00:03:57,780 --> 00:04:00,670 so I can re-write this sum. I can split this sum up. 65 00:04:00,670 --> 00:04:04,570 This is the sum K goes from 0 to n, of 66 00:04:04,570 --> 00:04:08,820 a sub k, plus the terms between n plus 1 and m. 67 00:04:08,820 --> 00:04:16,695 So, K goes from n plus 1 to m of a sub k, and that's at least as big, I hope, 68 00:04:16,695 --> 00:04:22,830 as the sum K goes from 0 to n of a sub k. Now, why is this true? 69 00:04:22,830 --> 00:04:22,980 Well, 70 00:04:22,980 --> 00:04:25,870 I've got the sum K goes from 0 to n of a 71 00:04:25,870 --> 00:04:29,690 sub k on both sides, but I've got this extra term over here. 72 00:04:29,690 --> 00:04:32,750 What I need to know is that this extra term is at least 0. 73 00:04:32,750 --> 00:04:37,810 But that's true, because all the a sub k's are non negative, and if you 74 00:04:37,810 --> 00:04:39,400 add up a whole bunch of non negative 75 00:04:39,400 --> 00:04:42,580 numbers, well, then, the result is non negative. 76 00:04:42,580 --> 00:04:47,560 So this is telling me that the sequence of partial sums, is non decreasing. 77 00:04:47,560 --> 00:04:48,160 So the 78 00:04:48,160 --> 00:04:55,200 sequences of partial sums is non decreasing and bounded above. 79 00:04:55,200 --> 00:04:58,760 So with a monotone convergence theorem, it converges. 80 00:04:58,760 --> 00:05:00,090 Let's summarize that. 81 00:05:00,090 --> 00:05:05,760 So here's what we've proved. If we've got two sequences, which are 82 00:05:05,760 --> 00:05:13,180 related in this way, that a sub k is between 0 and b sub k, and the sum, K goes 83 00:05:13,180 --> 00:05:19,888 from 0 to infinity, of b sub k, converges. Then, 84 00:05:19,888 --> 00:05:27,390 the sum K goes from 0 to infinity of a sub k, also converges. 85 00:05:27,390 --> 00:05:32,138 We can also say something when the sum of the a sub k's, diverges. 86 00:05:32,138 --> 00:05:38,190 Suppose that 0 is less than or equal to a sub k, is less than or equal to b sub k. 87 00:05:38,190 --> 00:05:41,360 And let's suppose that the sum of the a 88 00:05:41,360 --> 00:05:46,000 sub k's, K goes from 0 to infinity, diverges. 89 00:05:46,000 --> 00:05:49,045 And I don't know, what can I say about the sum of the b sub k's? 90 00:05:50,740 --> 00:05:56,620 Well, if this series diverges, that means that the sequence of partial sums, 91 00:05:56,620 --> 00:06:03,660 the sum, K goes from 0 to n of a sub k. That can't converge, 92 00:06:03,660 --> 00:06:05,820 right, because to say this thing converges is to say the 93 00:06:05,820 --> 00:06:09,620 series converges, so if this series diverges, this sequence must diverge. 94 00:06:10,670 --> 00:06:14,910 But that sequence, is monotone for the same reason as before, right. 95 00:06:14,910 --> 00:06:17,310 I'm adding up non negative terms, so as I 96 00:06:17,310 --> 00:06:21,120 add up more terms, at least is non decreasing. 97 00:06:21,120 --> 00:06:25,670 So the sequence is a monotone sequence, and yet, that sequence doesn't converge. 98 00:06:25,670 --> 00:06:29,100 So that means that the sequence of partial sums must be 99 00:06:29,100 --> 00:06:32,910 unbounded, because if it were bounded, and it was 100 00:06:32,910 --> 00:06:35,530 monotone, then it would converge, but it doesn't converge. 101 00:06:36,710 --> 00:06:38,629 So I've got an unbounded sequence. 102 00:06:40,200 --> 00:06:43,080 What about the sequence of partial sums for 103 00:06:43,080 --> 00:06:46,370 the series involving the b sub k's, right? 104 00:06:46,370 --> 00:06:50,800 So the sequence of partial sums there as the sum K goes from 0 to n of b sub k. 105 00:06:51,840 --> 00:06:54,599 This is even bigger than the sequence 106 00:06:54,599 --> 00:07:01,794 of partial sums for the a sub k's. This sequence, is likewise unbounded. 107 00:07:01,794 --> 00:07:06,583 Well, if this sequence is unbounded then that sequence doesn't converge. 108 00:07:06,583 --> 00:07:09,565 And if that sequence doesn't converge that means 109 00:07:09,565 --> 00:07:12,083 that the sum of the b sub k's, diverges. 110 00:07:12,083 --> 00:07:12,988 And that was fast. 111 00:07:12,988 --> 00:07:17,437 So we hope to see the convergence and divergence statements together. 112 00:07:17,437 --> 00:07:19,795 Well, here's our standing assumption. That 0 is less than or 113 00:07:19,795 --> 00:07:22,200 equal to a sub k, is less than or equal to b sub k. 114 00:07:22,200 --> 00:07:25,190 And the first thing that we saw, is that if the series of 115 00:07:25,190 --> 00:07:30,070 the b sub k's converges, then the series of the a sub k's converges. 116 00:07:31,810 --> 00:07:34,330 On the other hand, if the series of the a sub 117 00:07:34,330 --> 00:07:38,498 k's diverges, then the series of the b sub k's diverges. 118 00:07:38,498 --> 00:07:43,040 It's super important to keep track of the directions of these implications. 119 00:07:43,040 --> 00:07:45,050 Alright, here I've got the convergence 120 00:07:45,050 --> 00:07:48,040 of b sub k implying the convergence of a sub k. 121 00:07:48,040 --> 00:07:54,130 And here I've got the divergence of a sub k implying the divergence of b sub k. 122 00:07:54,130 --> 00:07:55,560 And hopefully that makes sense, right? 123 00:07:55,560 --> 00:07:57,737 This is saying that as long as you've got non 124 00:07:57,737 --> 00:08:02,600 negative terms, if you're below a convergent series, you converge. 125 00:08:02,600 --> 00:08:04,098 And this is saying, as long as you've got non 126 00:08:04,098 --> 00:08:09,020 negative terms, if you're above a divergent series, you also diverge. 127 00:08:09,020 --> 00:08:10,369 And let me end by saying 128 00:08:10,369 --> 00:08:13,187 the Comparison Test is just a ton of fun to use. 129 00:08:13,187 --> 00:08:15,491 It's definitely my favorite test. 130 00:08:15,491 --> 00:08:25,491 [SOUND].