1 00:00:00,340 --> 00:00:02,730 Let's think about a more complicated series. 2 00:00:02,730 --> 00:00:02,731 [SOUND]. 3 00:00:02,731 --> 00:00:11,725 How can we approach this series? It's 4 00:00:11,725 --> 00:00:17,870 the sum, k goes from 0 to infinity, of sine of k, squared, divided by 2 to the k. 5 00:00:17,870 --> 00:00:21,980 Does this series diverge or converge? 6 00:00:21,980 --> 00:00:26,940 Look suspiciously similar to a series that we do understand. 7 00:00:26,940 --> 00:00:27,889 Yeah, it looks 8 00:00:27,889 --> 00:00:29,890 a bit like this series, right? 9 00:00:29,890 --> 00:00:32,258 If I just got rid of this sin of k squared 10 00:00:32,258 --> 00:00:36,780 term and replaced it with a 1, I'd have this geometric series. 11 00:00:36,780 --> 00:00:41,630 This series converges and its value is 2. 12 00:00:41,630 --> 00:00:46,322 Let's be careful, I want a precise relationship between the mysterious series 13 00:00:46,322 --> 00:00:51,150 that I don't understand, and the geometric series that I do understand. 14 00:00:51,150 --> 00:00:53,810 Precisely how are these related? 15 00:00:53,810 --> 00:01:02,720 Well, sine of k is between minus 1 and 1, so sine of k squared is between 0 and 1. 16 00:01:02,720 --> 00:01:07,680 And now if I were to divide all of this by 2 to the k. 17 00:01:09,210 --> 00:01:14,208 Look, I now I have, sine of k squared divided by 2 18 00:01:14,208 --> 00:01:19,140 to the k is between 0 and 1 over 2 to the k. 19 00:01:19,140 --> 00:01:23,170 What's the original goal, what am I actually trying to understand here. 20 00:01:23,170 --> 00:01:27,890 I'm trying to understand whether this series converges or diverges. 21 00:01:27,890 --> 00:01:29,580 What does that question mean? 22 00:01:29,580 --> 00:01:34,620 That question really means, that I'm supposed to be looking at a sequence 23 00:01:34,620 --> 00:01:39,210 of partial sums, should add up the terms between k equals 0 and n. 24 00:01:39,210 --> 00:01:43,070 And then ask about the limit of the partial sum. 25 00:01:43,070 --> 00:01:44,588 And if this limit exists, 26 00:01:44,588 --> 00:01:49,750 well that's exactly what it means to say that this original series converges. 27 00:01:49,750 --> 00:01:52,205 What sort of sequence is S sub n? 28 00:01:52,205 --> 00:01:53,815 One thing I know about sequence of partial 29 00:01:53,815 --> 00:01:56,230 sums is the sequence of partials sums is non-decreasing. 30 00:01:56,230 --> 00:01:58,150 How do I know that? 31 00:01:58,150 --> 00:02:00,870 Well, the terms that I'm adding up look 32 00:02:00,870 --> 00:02:04,630 like this and those terms are non negative. 33 00:02:04,630 --> 00:02:09,598 So, if you add a non-negative number, the result's not getting any lower, all 34 00:02:09,598 --> 00:02:10,200 right? 35 00:02:10,200 --> 00:02:13,790 And that means that the sequence of partial sums is not getting any smaller. 36 00:02:13,790 --> 00:02:13,930 It's non-decreasing. 37 00:02:13,930 --> 00:02:20,540 Not only that, the sequence of partial sums, s of n, is also bounded. 38 00:02:20,540 --> 00:02:21,600 So, why bounded? 39 00:02:21,600 --> 00:02:24,540 Well, it again boils down to this inequality. 40 00:02:24,540 --> 00:02:27,096 Because this is less than this, if I add up a 41 00:02:27,096 --> 00:02:30,870 bunch of these, that's less than adding up a bunch of these. 42 00:02:30,870 --> 00:02:34,655 It's exactly what I'm saying here. If I sum all of these terms, 43 00:02:34,655 --> 00:02:38,100 it's less than or equal to the sum of these terms, because each 44 00:02:38,100 --> 00:02:41,800 of these terms is less than or equal to each of these terms. 45 00:02:41,800 --> 00:02:46,460 And that's just because sine square of k is less than or equal to 1. 46 00:02:46,460 --> 00:02:48,460 But I know something else here, right? 47 00:02:48,460 --> 00:02:52,440 This is a convergent geometric series. 48 00:02:52,440 --> 00:02:56,080 As I let n drift off to infinity, this is approaching 2. 49 00:02:56,080 --> 00:03:00,079 So putting this all together, I've got that these 50 00:03:00,079 --> 00:03:06,010 partial sums are bounded by 2. So I've got a bounded monotone sequence. 51 00:03:06,010 --> 00:03:09,706 But, a bounded monotone sequence converges, so 52 00:03:09,706 --> 00:03:13,720 we know something about the original series. 53 00:03:13,720 --> 00:03:17,256 And because the sequence of partial sums converges, that's 54 00:03:17,256 --> 00:03:20,940 exactly what it means to say that the series converges. 55 00:03:20,940 --> 00:03:25,101 This idea, this idea of taking a series we do understand, and using 56 00:03:25,101 --> 00:03:29,090 that series to explore a series we don't understand. 57 00:03:29,090 --> 00:03:30,180 This idea has a name. 58 00:03:30,180 --> 00:03:35,512 It's called the comparison test. This idea is important enough that, in a 59 00:03:35,512 --> 00:03:40,441 future video, we're going to write down a specific statement of the comparison test. 60 00:03:40,441 --> 00:03:50,441 [SOUND].