1 00:00:00,310 --> 00:00:01,057 The Root Test. 2 00:00:01,057 --> 00:00:07,608 [MUSIC] 3 00:00:07,608 --> 00:00:11,260 Here's another Conversions Test. So here's the Root Test. 4 00:00:11,260 --> 00:00:14,080 You want to analyze a series with positive terms. 5 00:00:14,080 --> 00:00:16,670 Your suppose to calculate this limit, call it big L. 6 00:00:16,670 --> 00:00:20,655 Big L is the limit as n goes to infinity of the nth root of the nth term. 7 00:00:20,655 --> 00:00:27,390 Now, if big L is less than 1 then the series, the sum of these events converges. 8 00:00:27,390 --> 00:00:32,700 If big L is bigger than 1, then the series diverges, and if big L is 9 00:00:32,700 --> 00:00:34,350 equal to 1 then the Root Test 10 00:00:34,350 --> 00:00:36,970 is inconclusive, we'll have to try something else. 11 00:00:36,970 --> 00:00:40,734 You might think that there are some reasons to really love the Root Test. 12 00:00:40,734 --> 00:00:42,340 Here's an example. 13 00:00:42,340 --> 00:00:47,802 Let's look at the sum angles from 1 to infinity of 1 over n to the nth power. 14 00:00:47,802 --> 00:00:51,555 I don't know whether that converges or diverges 15 00:00:51,555 --> 00:00:53,020 [LAUGH]. 16 00:00:53,020 --> 00:00:58,640 I am going to use against my better judgement, the Root Test. 17 00:00:58,640 --> 00:00:59,780 So what am I supposed to do? 18 00:00:59,780 --> 00:01:02,134 Well, this is a series all of whose terms are positive. 19 00:01:02,134 --> 00:01:08,683 So I'll calculate big L, the limit as n approaches infinity of the nth 20 00:01:08,683 --> 00:01:15,318 root of the nth term, well this is the limit as n approaches infinity. 21 00:01:15,318 --> 00:01:18,695 What's the nth root of nth power or just n. 22 00:01:18,695 --> 00:01:24,246 So this is the limit 1 over n as n approaches infinity, that is 0, and 0 23 00:01:24,246 --> 00:01:29,805 is less than 1. So by the Root Test 24 00:01:29,805 --> 00:01:30,920 [SOUND] 25 00:01:30,920 --> 00:01:36,800 the series converges. But in that case 26 00:01:36,800 --> 00:01:41,190 I could have just used a Comparison Test, so here is the thing to notice, okay. 27 00:01:42,870 --> 00:01:46,972 1 over n to the nth power is smaller than 1 over n squared. 28 00:01:46,972 --> 00:01:51,847 may be the only confusing case when n is 1, in which case these are equal, but 29 00:01:51,847 --> 00:01:56,122 then when n is 2, it's 1 over 2 square is less than equal to 1 over 30 00:01:56,122 --> 00:01:57,042 2 squared. 31 00:01:57,042 --> 00:02:03,370 When n equals 3, this is one over 3 cubed which is way less than 1 over 3 squared. 32 00:02:03,370 --> 00:02:07,900 Well, 1 over n to the n, as I've already pointed out, is positive, and I know that 33 00:02:07,900 --> 00:02:14,060 the sum n goes from 1 to infinity of 1 over n squared converges, that's 34 00:02:14,060 --> 00:02:20,200 a P series with P equals 2. So, by, just the Comparison Test. 35 00:02:21,270 --> 00:02:25,350 Right? I've got a series, now, which is, term 36 00:02:25,350 --> 00:02:30,890 wise, less than a convergent series. So by the Comparison Test, the 37 00:02:30,890 --> 00:02:36,410 sum, 1 over n to the n, n goes from 1 to infinity, converges as well. 38 00:02:36,410 --> 00:02:39,500 And that's why i'm not so impressed with the Root Test. 39 00:02:39,500 --> 00:02:40,500 Well here's what happens right? 40 00:02:40,500 --> 00:02:43,250 People go out into the world and they're given a series 41 00:02:43,250 --> 00:02:47,590 problems and sometimes those series involved something to the nth power. 42 00:02:47,590 --> 00:02:50,070 And people see you with something to the nth power, I'm going to 43 00:02:50,070 --> 00:02:54,170 apply the Root Test because then I can get rid of the nth power. 44 00:02:54,170 --> 00:02:56,090 And yeah, that's true. 45 00:02:56,090 --> 00:02:59,470 But you could also have applied the Ratio Test. 46 00:02:59,470 --> 00:03:01,360 And the Ratio Test is good not just when you've 47 00:03:01,360 --> 00:03:04,580 got powers but also when you've got factorials floating around. 48 00:03:04,580 --> 00:03:09,160 And that's not just to say that the Root Test is entirely useless. 49 00:03:09,160 --> 00:03:13,210 It's just, it's not as useful I think as people make it out to 50 00:03:13,210 --> 00:03:13,360 be, right? 51 00:03:13,360 --> 00:03:16,140 The Ratio Test is often easier to apply, 52 00:03:16,140 --> 00:03:19,940 and it's more likely to cause some useful cancellation. 53 00:03:19,940 --> 00:03:22,075 We're gonig to see in the future some more 54 00:03:22,075 --> 00:03:24,332 instances of where the Root Test does come in 55 00:03:24,332 --> 00:03:26,589 handy, but for the time being, I think your 56 00:03:26,589 --> 00:03:29,549 first inclination should be to reach for the Ratio Test. 57 00:03:29,549 --> 00:03:39,549 [NOISE].