1 00:00:00,320 --> 00:00:03,490 The ratio test is looking like a great test. 2 00:00:03,490 --> 00:00:04,583 Does it always work? 3 00:00:04,583 --> 00:00:10,932 [SOUND]. 4 00:00:10,932 --> 00:00:13,760 Let's recall what the ratio test says. 5 00:00:13,760 --> 00:00:16,548 I'm going to be considering a series the sum of 6 00:00:16,548 --> 00:00:19,760 a sub n and goes from zero to infinity. 7 00:00:19,760 --> 00:00:21,827 And I'm going to assume that all of the terms that 8 00:00:21,827 --> 00:00:24,430 I'm adding up, all of the a sub ns, are non-negative. 9 00:00:24,430 --> 00:00:26,340 So a sub n is greater than or equal to 0. 10 00:00:26,340 --> 00:00:30,552 And the ratio test tells me to consider the limit of the ratio of 11 00:00:30,552 --> 00:00:35,979 subsequent terms, so I'm taking the limit, as n approaches infinity of a sub n 12 00:00:35,979 --> 00:00:39,490 plus 1 over a sub n, and I'm calling that limit L. 13 00:00:40,630 --> 00:00:42,760 And then what does the ratio test tell me? 14 00:00:42,760 --> 00:00:45,169 Well the ratio test tells me that if that 15 00:00:45,169 --> 00:00:48,920 limit is less than 1, then the series converges. 16 00:00:48,920 --> 00:00:53,380 If that limit is bigger than 1, then the series diverges. 17 00:00:53,380 --> 00:00:59,180 But if the limit is equal to 1 then the ratio test is inconclusive. 18 00:00:59,180 --> 00:01:01,630 So what happens when l equals one? 19 00:01:01,630 --> 00:01:05,400 Well supposedly the ratio test doesn't say anything in that case. 20 00:01:05,400 --> 00:01:09,144 But if we listen really carefully, is there something that we 21 00:01:09,144 --> 00:01:12,840 can get out of the ratio test even when l equals one? 22 00:01:12,840 --> 00:01:14,320 No. 23 00:01:14,320 --> 00:01:20,450 Whoa, okay, but why not? Well there are cases where l equals 1 and 24 00:01:20,450 --> 00:01:27,056 the series converges. But then there are also cases where 25 00:01:27,056 --> 00:01:31,110 L equals one and the series diverges. 26 00:01:31,110 --> 00:01:33,530 Can we actually see some of these examples? 27 00:01:33,530 --> 00:01:38,518 Here's an example but maybe a silly example where L 28 00:01:38,518 --> 00:01:43,529 is equal to 1 but the series will end up diverging. 29 00:01:44,560 --> 00:01:45,660 So let's try this. 30 00:01:45,660 --> 00:01:52,020 Here's the example, it's the sum, n goes from 0 to infinity of the number 1. 31 00:01:52,020 --> 00:01:52,330 [LAUGH] 32 00:01:52,330 --> 00:01:55,510 well, I mean this, this definitely divergences, right? 33 00:01:55,510 --> 00:02:01,840 If you keep adding up 1 plus 1 plus 1, right, that's not a finite number. 34 00:02:01,840 --> 00:02:04,080 So this is a divergent series. 35 00:02:04,080 --> 00:02:06,470 But of course, in this case, what's a sub n? 36 00:02:06,470 --> 00:02:10,070 Well a sub n doesn't depend on n, it's just always 1. 37 00:02:10,070 --> 00:02:17,330 And so the limit of the n + 1th term over the nth term as n goes to infinity 38 00:02:17,330 --> 00:02:23,550 is just 1, because a sub n plus 1 and a sub n are both 1. 39 00:02:23,550 --> 00:02:28,180 What about an example where the series converges even though L equals 1? 40 00:02:28,180 --> 00:02:33,342 So on an example where the L in the ratio test is 1, but the series ends 41 00:02:33,342 --> 00:02:39,150 up converging even though the ratio test doesn't tell us that. 42 00:02:39,150 --> 00:02:42,712 So let's see an example. An example of this is the sum and 43 00:02:42,712 --> 00:02:46,890 goes from 1 to infinity of 1 over n squared. 44 00:02:46,890 --> 00:02:49,710 THis series we've already seen to converge. 45 00:02:49,710 --> 00:02:57,170 It converges by, let's say, the p series test. 46 00:02:57,170 --> 00:03:02,880 It's a p series where p equals 2. But, what is l in this case. 47 00:03:02,880 --> 00:03:08,560 Well, the nth term in the series a sub n, is 1 over n squared. 48 00:03:08,560 --> 00:03:14,320 So l is the limit of a sub n plus1 over a sub n, as n 49 00:03:14,320 --> 00:03:20,350 goes to infinity. And in that case, that's 50 00:03:20,350 --> 00:03:27,350 the limit as n goes to infinity of 1 over n plus 1 squared. 51 00:03:27,350 --> 00:03:32,870 Divided by 1 over n squared and that limit is indeed 1. 52 00:03:32,870 --> 00:03:33,774 But how do 53 00:03:33,774 --> 00:03:39,488 I know that the limit is 1? Well, to compute this limit, I 54 00:03:39,488 --> 00:03:45,416 could first rewrite this as the limit as n approaches infinity 55 00:03:45,416 --> 00:03:50,888 1 over what, I must gotta move the n plus 1 squared in the 56 00:03:50,888 --> 00:03:56,588 denominator so it will be n plus 1 squared over n squared now 57 00:03:56,588 --> 00:03:57,614 [INAUDIBLE] 58 00:03:57,614 --> 00:04:03,314 expand it out to the n plus 1 squared this be the limit and as n 59 00:04:03,314 --> 00:04:09,248 approaches infinity 1 over n squared plus 2n plus 1 that's what I 60 00:04:09,248 --> 00:04:15,488 got expanded n plus 1 squared, divided by n squared, And this is the sum here 61 00:04:15,488 --> 00:04:22,010 in the numerator and I got split this up into three separate fractions. 62 00:04:22,010 --> 00:04:22,856 This is 63 00:04:22,856 --> 00:04:28,530 the limit as n goes to infinity 1 over n 64 00:04:28,530 --> 00:04:34,270 squared over n squared, plus two n over n squared 65 00:04:34,270 --> 00:04:40,260 plus one over n squared. Now I could simplify this a bit more too. 66 00:04:40,260 --> 00:04:46,590 This is the limit n goes to infinity of one over n squared over n squared is one. 67 00:04:46,590 --> 00:04:48,018 2 n over n squared is 68 00:04:48,018 --> 00:04:53,770 2 over and and 1 over n squared I'll just write as one over n squared. 69 00:04:53,770 --> 00:04:55,020 Well what's the limit? 70 00:04:55,020 --> 00:05:00,360 When and is very large, this is very close to 0 and this is very close to 0. 71 00:05:00,360 --> 00:05:03,318 So when n is very large, this is 1 over 1 72 00:05:03,318 --> 00:05:07,000 plus a number close to 0 plus a number close to 0. 73 00:05:07,000 --> 00:05:11,510 This limit Is 1, which is exactly what I'm claiming. 74 00:05:11,510 --> 00:05:13,097 So when we say that the ratio 75 00:05:13,097 --> 00:05:17,730 test is silent when l equals 1, it's not for a lack of understanding. 76 00:05:17,730 --> 00:05:21,100 There really are series that converge and series that diverge. 77 00:05:21,100 --> 00:05:23,756 In both cases, l equals 1. 78 00:05:23,756 --> 00:05:33,756 [NOISE]