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These are the user uploaded subtitles that are being translated: 1 00:00:00,750 --> 00:00:05,950 Today we're going to be talking about how to use the term test to try to show whether a series diverges. 2 00:00:05,970 --> 00:00:10,170 Now keep in mind that the term test is also called the divergence test. 3 00:00:10,170 --> 00:00:12,370 And you also hear it called the zero test. 4 00:00:12,390 --> 00:00:14,280 It's all the same test. 5 00:00:14,310 --> 00:00:17,600 It's just called different things so you'll see it by different names. 6 00:00:17,700 --> 00:00:19,960 Just know that it's the same test here. 7 00:00:20,100 --> 00:00:24,600 And in this particular video we're going to be doing three separate examples I've written three different 8 00:00:24,660 --> 00:00:30,850 infinite sums here and we're going to be using the divergence test to test each one of them for divergence. 9 00:00:30,850 --> 00:00:36,450 Now here's what the divergence test says and I like to call it the divergence test as opposed to the 10 00:00:36,600 --> 00:00:44,340 term test or the zero test because it helps remind me that this test is really only good for talking 11 00:00:44,340 --> 00:00:45,570 about divergence. 12 00:00:45,570 --> 00:00:49,910 It tells us nothing about whether or not a series converges. 13 00:00:49,920 --> 00:00:51,860 So here's what the test says. 14 00:00:51,960 --> 00:00:56,260 It says that if we take any series and we'll call it a 7. 15 00:00:56,430 --> 00:01:02,970 And any of these theories that we have here can be a sudden we can call this a seben here this natural 16 00:01:02,970 --> 00:01:06,390 log of this rational function here could be a 7. 17 00:01:06,600 --> 00:01:09,910 Or this quantity here this series could be a subset. 18 00:01:09,920 --> 00:01:16,530 So any of these series if we have a series a seven and we take its limit and goes to infinity if that 19 00:01:16,620 --> 00:01:22,560 infinite limit is not equal to zero if the result of that limit is not equal to zero then this test 20 00:01:22,560 --> 00:01:24,990 tells us that the series diverges. 21 00:01:24,990 --> 00:01:29,940 That doesn't mean that if this is equal to zero that the series converges. 22 00:01:29,940 --> 00:01:35,760 It just means that if the limit is equal to zero that the test is inconclusive so the test really only 23 00:01:35,760 --> 00:01:39,830 can show us whether or not a series diverges. 24 00:01:39,930 --> 00:01:43,370 If the limit is and goes to infinity of the series is not equal to zero. 25 00:01:43,380 --> 00:01:46,040 Then we've proven that the series diverges. 26 00:01:46,110 --> 00:01:47,780 But if it is equal to zero. 27 00:01:47,940 --> 00:01:49,920 We haven't shown that the series converges. 28 00:01:49,920 --> 00:01:52,740 This particular test is just inconclusive. 29 00:01:52,780 --> 00:01:58,110 We can't get any information from the test and we would have to use a different test to see whether 30 00:01:58,110 --> 00:02:03,510 or not the series converges or diverges so that's why I like to call it the divergence test because 31 00:02:03,540 --> 00:02:10,260 it helps remind me that we're only making conclusions about divergence that we can't say anything about 32 00:02:10,260 --> 00:02:12,320 whether or not a series is convergent. 33 00:02:12,630 --> 00:02:17,250 So let's go ahead and take a look at our first example the infinite sum from N equals 1 to infinity 34 00:02:17,600 --> 00:02:21,350 of and minus 1 over 3 and minus 1. 35 00:02:21,360 --> 00:02:26,550 So what we're going to do to use the divergence test to test this series divergence we're going to say 36 00:02:27,000 --> 00:02:30,820 the limit as an ant goes to infinity. 37 00:02:30,900 --> 00:02:32,470 Of a sudden which is. 38 00:02:32,480 --> 00:02:36,640 In this case and minus one over three and minus one. 39 00:02:36,780 --> 00:02:39,560 We're just going to evaluate this and see what the result is. 40 00:02:39,600 --> 00:02:44,400 Now in this case it's really easy because we just have a rational function and remember that when we 41 00:02:44,400 --> 00:02:49,340 have a rational function we can evaluate this infinite limit in one of two ways. 42 00:02:49,560 --> 00:02:55,860 We can either look at the coefficients on the largest degree n terms in both the numerator and denominator 43 00:02:55,860 --> 00:03:00,520 so the largest degree term in the numerator is this and variable right here this and term. 44 00:03:00,750 --> 00:03:03,450 The largest degree turn the denominator is 3 n. 45 00:03:03,450 --> 00:03:05,870 So you basically have 1 and over 3 n. 46 00:03:06,030 --> 00:03:10,460 And if we look at these coefficients here then we know that the limit is one third. 47 00:03:10,710 --> 00:03:17,460 The other way we can do it is by multiplying both numerator and denominator by 1 divided by the largest 48 00:03:17,460 --> 00:03:20,310 degree the highest degree and variable. 49 00:03:20,310 --> 00:03:26,520 If we had an n squared anywhere in this rational function we would have to multiply by 1 over and squared 50 00:03:26,580 --> 00:03:33,420 divided by 1 over and squared but N itself and to the first power is the largest degree a value of and 51 00:03:33,440 --> 00:03:34,020 that we have. 52 00:03:34,020 --> 00:03:37,570 So we multiply by 1 over and divided by 1 over 10. 53 00:03:37,710 --> 00:03:46,740 And what that gives us is an at times 1 over N is just one negative 1 times 1 over n is minus 1 over 54 00:03:46,740 --> 00:03:51,930 N divided by 3 End Times 1 over and the ends cancel. 55 00:03:51,930 --> 00:03:58,590 We're just left with three minus one over end because negative 1 times one over end is negative 1 over 56 00:03:58,580 --> 00:03:59,330 N.. 57 00:03:59,730 --> 00:04:05,520 Now if we take the limit as and goes to infinity of this function here what we'll get is these little 58 00:04:05,520 --> 00:04:09,520 fractions one over and negative one over in these become zero. 59 00:04:09,530 --> 00:04:15,660 Because when we have a small constant like this or really any constant divided by an infinitely large 60 00:04:15,720 --> 00:04:22,950 value that value that whole fraction itself goes to zero and therefore as you can see what we're left 61 00:04:22,950 --> 00:04:26,290 with is just this one third value here. 62 00:04:26,370 --> 00:04:34,800 So we know that the limit as and goes to infinity of our series A seben is equal to one third because 63 00:04:34,800 --> 00:04:37,740 one third does not equal zero. 64 00:04:37,740 --> 00:04:45,890 The divergence test by the divergence test we can conclude that this series a sub in a sudden diverges 65 00:04:47,720 --> 00:04:48,770 series diverges. 66 00:04:48,890 --> 00:04:54,800 So the divergence test is conclusive because this ONE-THIRD value is not equal to zero. 67 00:04:55,070 --> 00:05:00,590 OK so that was a sub n Let's go ahead and call this series B sidebands so we can distinguish between 68 00:05:00,590 --> 00:05:03,100 them and we'll call this one see. 69 00:05:03,170 --> 00:05:09,590 So if we do another example if we look at B Subhan here we're going to take the limit as and goes to 70 00:05:09,680 --> 00:05:19,220 infinity of the natural log of end squared plus 1 divided by 2 and squared plus 1 and using limit laws 71 00:05:19,220 --> 00:05:28,190 we can say that this limit is actually the natural log of the limit as and goes to infinity and squared 72 00:05:28,190 --> 00:05:32,490 plus 1 divided by 2 and squared plus 1. 73 00:05:32,660 --> 00:05:39,950 And from here we have the same type of rational function that we had in our first example and what we're 74 00:05:39,950 --> 00:05:44,000 going to get is just the coefficients of the highest degree and values. 75 00:05:44,000 --> 00:05:46,430 Here we get one and squared over two n squared. 76 00:05:46,430 --> 00:05:51,650 Everything else will cancel and we're just left with this one half these two coefficients on our end 77 00:05:51,650 --> 00:05:52,870 squared terms. 78 00:05:52,970 --> 00:05:57,200 The limit as and goes to infinity of 1 a half is just one half. 79 00:05:57,260 --> 00:06:00,200 So we get the natural log of one half. 80 00:06:00,200 --> 00:06:08,420 This of course is not equal to zero and therefore by the divergence test our series B saw then diverges 81 00:06:11,170 --> 00:06:16,780 Now if we look at our third example here because we have the summer of two separate terms like this 82 00:06:17,050 --> 00:06:19,890 we can break apart each term put it in its own. 83 00:06:19,900 --> 00:06:25,510 Some you can always do that if you have two terms like this added together or subtracted from one another 84 00:06:25,510 --> 00:06:33,220 you can break them apart so we'll say this some from N equals one to infinity of one over to the N plus 85 00:06:33,220 --> 00:06:40,570 the sum from an equals 1 to infinity of 1 over n times and plus 1. 86 00:06:40,570 --> 00:06:44,210 Now we can test these separately for divergence. 87 00:06:44,350 --> 00:06:51,790 So we'll test this first term here one over either the N will stay the limit as and goes to infinity 88 00:06:52,090 --> 00:06:54,380 of one over E to the n. 89 00:06:54,580 --> 00:07:00,970 Well if we evaluate this at some very very large number for N we can get some extremely large number 90 00:07:01,000 --> 00:07:06,700 and an infinitely large number here in the denominator and when we just have a constant divided by an 91 00:07:06,700 --> 00:07:09,970 infinitely large number this is going to go to zero. 92 00:07:09,970 --> 00:07:16,040 This is going to be equal to zero or by the divergence test remember the divergence test is inconclusive. 93 00:07:16,060 --> 00:07:18,940 If the limit is and goes to infinity is equal to zero. 94 00:07:19,120 --> 00:07:23,630 So the divergence test for this particular term is inconclusive. 95 00:07:23,650 --> 00:07:26,470 We would actually have to test it a different way. 96 00:07:26,680 --> 00:07:31,690 And what we would find if we rolled out the first several terms of this series is actually that it's 97 00:07:31,750 --> 00:07:37,380 a geometric series and we could use the geometric series test to test this for convergence. 98 00:07:37,510 --> 00:07:41,130 Let's go ahead and look at the second series here. 99 00:07:41,140 --> 00:07:49,490 One over end times the quantity and plus one will say the limit as it goes to infinity of one over. 100 00:07:49,600 --> 00:07:52,380 And at times the quantity and plus one. 101 00:07:52,390 --> 00:07:59,520 Again if we just plug in a very very large value for n we'll get an infinitely large value in the denominator. 102 00:07:59,530 --> 00:08:02,670 We've got a small constant here in the numerator. 103 00:08:02,680 --> 00:08:05,860 This value is also going to tend toward zero. 104 00:08:05,860 --> 00:08:11,170 We're just going to get zero for that value and therefore the divergence test for this particular series 105 00:08:11,530 --> 00:08:13,720 is also inconclusive. 106 00:08:13,900 --> 00:08:20,250 The way that we would test it is we use partial fractions to break apart this series here. 107 00:08:20,260 --> 00:08:28,060 What we'd end up with is one over n minus one over and plus 1 which would replace the series we have 108 00:08:28,060 --> 00:08:28,550 here. 109 00:08:28,600 --> 00:08:33,310 And then if we write out the first several terms of this series we realize that it's a telescoping series 110 00:08:33,550 --> 00:08:39,340 and treating it as a telescoping series it becomes very easy to find the sum of this series and therefore 111 00:08:39,340 --> 00:08:41,380 to conclude that it converges. 112 00:08:41,530 --> 00:08:47,330 So that's a couple examples of how we use the divergence test to determine whether or not a series diverges 113 00:08:47,680 --> 00:08:53,550 or just to say that the divergence test is inconclusive and use a different test instead. 12842