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These are the user uploaded subtitles that are being translated: 1 00:00:00,500 --> 00:00:05,130 Too we're going to be talking about how to determine whether or not a series converges absolutely or 2 00:00:05,130 --> 00:00:06,910 converges conditionally. 3 00:00:06,990 --> 00:00:11,490 And in this particular video we're going to be doing two different examples trying to determine whether 4 00:00:11,490 --> 00:00:17,090 or not each of these infinite series converges absolutely or conditionally if at all. 5 00:00:17,100 --> 00:00:23,490 Now it's important to remember that a series is conditionally convergent if it converges but is not 6 00:00:23,550 --> 00:00:25,050 absolutely convergent. 7 00:00:25,110 --> 00:00:30,960 The series is absolutely convergent if the absolute value of the series is convergent And so basically 8 00:00:30,960 --> 00:00:37,770 what we can what we can remember is that if the absolute value of the series a seven and we would call 9 00:00:38,040 --> 00:00:42,600 this part of the series here this value this formula for the series. 10 00:00:42,600 --> 00:00:44,090 This is a subset. 11 00:00:44,090 --> 00:00:46,200 And this here is a subset. 12 00:00:46,200 --> 00:00:52,690 Another example of a series is the absolute value of a sub N is convergent. 13 00:00:52,860 --> 00:00:58,290 If this is convergent then we know that the series is absolutely convergent. 14 00:00:58,500 --> 00:01:01,220 Otherwise it may not be absolutely convergent. 15 00:01:01,230 --> 00:01:07,080 This particular part might not be true but this series in general without the absolute value bars may 16 00:01:07,080 --> 00:01:07,710 be convergent. 17 00:01:07,710 --> 00:01:11,460 And in that case it would be conditionally convergent. 18 00:01:11,460 --> 00:01:17,340 So when we're talking about absolute convergence the best way to test for it is using either the ratio 19 00:01:17,340 --> 00:01:19,320 test or the root test. 20 00:01:19,350 --> 00:01:26,640 If we can the ratio and root tests for Convergence have absolute convergence built right into the definition 21 00:01:26,970 --> 00:01:28,530 of those convergence tests. 22 00:01:28,530 --> 00:01:31,820 So if there's any way that we can use those we want to. 23 00:01:31,920 --> 00:01:37,950 The ratio test is often a little bit more flexible than the root test in terms of applying it to series 24 00:01:37,950 --> 00:01:38,880 in general. 25 00:01:38,910 --> 00:01:44,790 But either one will allow us to determine absolute convergence if the series is set up in a way such 26 00:01:44,790 --> 00:01:46,140 that we can use the tests. 27 00:01:46,140 --> 00:01:48,620 So let's go through one example of each. 28 00:01:48,720 --> 00:01:54,660 So we can start getting a feel for how we can use those tests to determine absolute convergence. 29 00:01:54,660 --> 00:02:00,720 So in this first example we have the infinite sum from equals 1 to infinity of this series which is 30 00:02:00,720 --> 00:02:09,060 10 to the power divided by the quantity and plus one at times 4 raised to the power to and plus 1. 31 00:02:09,060 --> 00:02:12,560 Now this is a perfect candidate for the ratio test. 32 00:02:12,570 --> 00:02:21,480 And the reason is because the ratio test tells us that the limit L is equal to the limit as an goes 33 00:02:21,480 --> 00:02:31,290 to infinity of the absolute value of basically a sub and plus 1 divided by a sub n and all that means 34 00:02:31,290 --> 00:02:36,780 is that we're going to be plugging in and plus 1 everywhere we have N in our original series here and 35 00:02:36,780 --> 00:02:40,710 we're going to put that in the numerator and these absolute value bars and then we're going to take 36 00:02:40,710 --> 00:02:44,190 the original function a Subban and put that in the denominator. 37 00:02:44,190 --> 00:02:48,730 So in other words let's go ahead and substitute and plus 1 everywhere are we have. 38 00:02:48,780 --> 00:02:53,840 And in our original series so instead of tens the end will get 10 to the end plus 1. 39 00:02:54,090 --> 00:02:59,540 And then in our denominator here instead of and plus one will substitute and plus 1 here for this end 40 00:02:59,540 --> 00:03:08,120 and get and plus one plus one or in other words and plus two and then here we'll get four to the two 41 00:03:08,180 --> 00:03:08,670 times. 42 00:03:08,700 --> 00:03:13,170 And plus one two times and plus one is two plus two. 43 00:03:13,440 --> 00:03:19,290 So we'll get to N plus two plus one or in other words two and plus three. 44 00:03:19,300 --> 00:03:22,660 So we got four race to the two end plus three power. 45 00:03:22,920 --> 00:03:28,210 Then we divide that by the original series a seven. 46 00:03:28,230 --> 00:03:38,220 So we divide it by 10 to the N divided by and plus 1 times 4 to the two and plus 1 power. 47 00:03:38,220 --> 00:03:42,930 Now this is sort of an intermediate step that people skip a lot of times because you're always going 48 00:03:42,930 --> 00:03:44,520 to get with this test. 49 00:03:44,520 --> 00:03:50,190 A fraction divided by a fraction and of course when we have that situation we can just take the numerator 50 00:03:50,190 --> 00:03:50,490 here. 51 00:03:50,490 --> 00:03:59,640 So ten to the end plus 1 divided by and plus two times 4 to the two and plus three. 52 00:03:59,820 --> 00:04:05,670 And instead of dividing by this fraction in the denominator we can multiply by it's reciprocal that's 53 00:04:05,670 --> 00:04:06,830 the same thing. 54 00:04:06,840 --> 00:04:09,000 So we just flip it upside down and we get. 55 00:04:09,030 --> 00:04:16,570 And plus 1 times 4 to the two and plus 1 all divided by 10 to the end. 56 00:04:16,770 --> 00:04:19,560 And we're taking the absolute value of this whole thing. 57 00:04:19,560 --> 00:04:24,450 Now as you can see knowing that we could have just gone straight to this step and skipped writing out 58 00:04:24,450 --> 00:04:25,730 this whole thing here. 59 00:04:25,740 --> 00:04:29,360 So either way however you want to do it is is fine. 60 00:04:29,370 --> 00:04:33,780 So in this case now that we have the fractions written this way we're just looking to match up values 61 00:04:33,780 --> 00:04:38,790 from the numerator and denominator and what that means is basically we're looking for like bases or 62 00:04:38,850 --> 00:04:40,350 terms that are similar to one another. 63 00:04:40,350 --> 00:04:45,560 So for example we have 10 to the end plus one here and we have 10 to the end. 64 00:04:45,660 --> 00:04:49,070 We've got the same base of 10 just a different exponent. 65 00:04:49,230 --> 00:04:54,360 So remember that when we have fractions like this we're going to kind of pair these up together when 66 00:04:54,360 --> 00:04:57,110 we have for example let's just take an easy one right. 67 00:04:57,120 --> 00:05:03,790 Tend to the third over 10 squared we're going to simplify this by subtracting the exponent in the denominator 68 00:05:03,790 --> 00:05:05,400 from the exponent in the numerator. 69 00:05:05,560 --> 00:05:10,900 So three minus two is one this becomes tend to the first power is the simplified version of tend to 70 00:05:10,900 --> 00:05:12,520 the third over 10 square. 71 00:05:12,520 --> 00:05:16,420 Same thing here we have tend to the end plus one over tend to the N. 72 00:05:16,420 --> 00:05:21,340 So the way that we simplify that tend to the end plus one over tend to the end. 73 00:05:21,490 --> 00:05:29,830 We subtract and from and plus once we get and plus one minus N and in that case and plus one minus N 74 00:05:30,220 --> 00:05:31,730 we get the ends to cancel. 75 00:05:31,780 --> 00:05:33,340 And we're just left with one. 76 00:05:33,340 --> 00:05:36,030 So this is 10 to the first power. 77 00:05:36,250 --> 00:05:37,410 So you can see how that. 78 00:05:37,420 --> 00:05:46,480 So what we're going to be left with for that particular term is just ten to the first power in the numerator 79 00:05:46,490 --> 00:05:48,910 so 10 of the first power there. 80 00:05:48,910 --> 00:05:55,930 Now let's go ahead and match up 4 to the two and plus one in the numerator and four to the two and plus 81 00:05:55,960 --> 00:05:57,780 three in the denominator. 82 00:05:57,970 --> 00:06:05,170 If we take two and plus one from the numerator and we subtract what's in the denominator 2 and plus 83 00:06:05,380 --> 00:06:13,300 3 what we'll get is to N plus one minus two and minus three are two ends cancel and we're left with 84 00:06:13,330 --> 00:06:16,630 1 minus three which is a negative 2. 85 00:06:16,720 --> 00:06:23,500 So what that tells us because this value is negative is that we're just left with four race to the negative 86 00:06:23,500 --> 00:06:24,030 two. 87 00:06:24,070 --> 00:06:28,930 Or in other words one over four to the positive to move that to the denominator. 88 00:06:28,930 --> 00:06:30,240 We get a positive value. 89 00:06:30,250 --> 00:06:33,030 We're just going we left with four squared in the denominator. 90 00:06:33,130 --> 00:06:36,770 So we put this in the denominator for squared like that. 91 00:06:36,780 --> 00:06:39,290 That's all that's left of this term. 92 00:06:39,340 --> 00:06:46,420 And then of course we have that multiplied by Plus 1 over and plus 2 which we can't simplify there's 93 00:06:46,420 --> 00:06:48,640 no exponents there to simplify. 94 00:06:48,700 --> 00:06:54,130 Those are just terms that are remaining and to evaluate this limit we can go ahead and pull out the 95 00:06:54,130 --> 00:06:57,990 10 over four squared remember that four squared is 16. 96 00:06:58,030 --> 00:07:03,050 So we essentially have 10 over 16 here or in other words 5 over 8. 97 00:07:03,070 --> 00:07:04,120 When we reduce it. 98 00:07:04,360 --> 00:07:13,000 So what we're left with is the limit is equal to five eighths times the limit as and goes to infinity 99 00:07:13,510 --> 00:07:17,740 of the absolute value of plus 1 over and plus 2. 100 00:07:17,920 --> 00:07:22,420 And when we have just a rational function like this polynomials in the numerator and denominator the 101 00:07:22,420 --> 00:07:28,660 easiest way to evaluate this infinite limit is to divide through both the numerator and denominator 102 00:07:28,930 --> 00:07:33,150 by the highest degree and Terman in this case that's And to the first power. 103 00:07:33,160 --> 00:07:40,700 So we want to multiply by 1 over into the first divided by 1 over to the first like this. 104 00:07:40,750 --> 00:07:49,420 And what we're left with then is our equals five eighths times the limit as and goes to infinity of. 105 00:07:49,540 --> 00:07:52,320 And at times 1 over end just gives us one. 106 00:07:52,360 --> 00:07:58,420 So we have one plus 1 times 1 over and gives us 1 over and. 107 00:07:58,560 --> 00:08:06,490 And then in the denominator end times 1 over and gives us 1 2 times 1 over and gives us 2 over and. 108 00:08:06,520 --> 00:08:12,580 And the reason that we do this is because now we have these values of n in the denominator. 109 00:08:12,730 --> 00:08:18,850 When we have a constant like one or two here divided by an end is going to infinity. 110 00:08:19,030 --> 00:08:24,850 This denominator becomes extremely large and these two terms here are eventually going to tend toward 111 00:08:25,050 --> 00:08:27,860 zero and becomes larger and larger and larger. 112 00:08:27,880 --> 00:08:33,250 So these two are both going to go away and become 0 and as you can see all that we're left with is just 113 00:08:33,310 --> 00:08:40,100 1 over 1 or 1 the limit as and goes to infinity of 1 is just 1 itself. 114 00:08:40,120 --> 00:08:46,150 So our limit is equal to five eighths times one or just five eighths. 115 00:08:46,150 --> 00:08:51,850 Now the ratio test tells us here's where the conclusion of the ratio test comes in the ratio test tells 116 00:08:51,850 --> 00:08:58,920 us that whatever result we have here if it is less than 1 then a series is absolutely convergent. 117 00:08:58,990 --> 00:09:03,670 If the value is equal to 1 the test is inconclusive the value is greater than 1. 118 00:09:03,790 --> 00:09:05,940 Then we know that the series diverges. 119 00:09:05,980 --> 00:09:10,110 But in this case the value is less than 1 5 8 is less than 1. 120 00:09:10,120 --> 00:09:15,020 So by the ratio test this series here is absolutely convergent. 121 00:09:15,030 --> 00:09:17,340 It converges Absolutely. 122 00:09:17,350 --> 00:09:22,830 So we looked at an example of how to use the ratio test to determine whether or not a series converges. 123 00:09:22,870 --> 00:09:23,800 Absolutely. 124 00:09:23,800 --> 00:09:28,300 Now let's take a look an example of how to use the root test to determine whether or not the series 125 00:09:28,300 --> 00:09:28,850 converges. 126 00:09:28,860 --> 00:09:29,810 Absolutely. 127 00:09:29,830 --> 00:09:34,810 We're going to look at this infinite sum here from unequals went to infinity of the quantity and squared 128 00:09:34,810 --> 00:09:40,000 plus one divided by two n squared plus one all raised to the end power. 129 00:09:40,000 --> 00:09:46,060 Now this is a perfect candidate for the root test because the root test tells us that the limit is equal 130 00:09:46,060 --> 00:09:56,420 to the limit as an goes to infinity of the end through of the absolute value of the series a 7. 131 00:09:56,590 --> 00:10:01,290 So in other words remember that our series A seben is represented by the function here. 132 00:10:01,390 --> 00:10:05,650 So we're going to put that inside absolute value bars and we're going to take the nth root of it. 133 00:10:05,650 --> 00:10:11,230 Keep in mind that when we take the and through this and through here it's just going to cancel out this 134 00:10:11,350 --> 00:10:16,450 and through right here and that's what makes this particular series a perfect candidate for the root 135 00:10:16,450 --> 00:10:17,090 test. 136 00:10:17,110 --> 00:10:22,720 When ever you can raise everything in the entire series whenever you can put the entire series in parentheses 137 00:10:22,720 --> 00:10:27,360 like this and raise it to the end power or the two and power or something like that. 138 00:10:27,520 --> 00:10:34,930 Then the root test is a great test to use because you can get this xponent here to cancel thereby significantly 139 00:10:34,930 --> 00:10:38,210 reducing the complexity of the series. 140 00:10:38,230 --> 00:10:45,490 So given that we're going to say that the limit is equal to the limit as and goes to infinity of the 141 00:10:45,490 --> 00:10:54,080 absolute value of our series so we have squared plus one divided by two and squared plus one. 142 00:10:54,310 --> 00:11:02,140 And keep in mind that the series here we had it raised to the power like this raised to the power well 143 00:11:02,140 --> 00:11:07,750 this and throughout the series is basically raising this whole thing to the one over and that's why 144 00:11:07,750 --> 00:11:08,500 they cancel. 145 00:11:08,520 --> 00:11:11,410 Because and times one over end is just one. 146 00:11:11,440 --> 00:11:17,860 So this goes away and we're just left with the absolute value of an squared plus one over two and squared 147 00:11:17,890 --> 00:11:19,360 plus one. 148 00:11:19,390 --> 00:11:25,060 Given that we're going to do the same kind of thing that we did with our ratio test where we multiply 149 00:11:25,060 --> 00:11:31,660 both numerator and denominator by the highest degree and variable in this case that end squared right. 150 00:11:31,690 --> 00:11:38,470 And to the power of two is the highest exponent on any and variable in this sequence here this series. 151 00:11:38,650 --> 00:11:46,180 So we're going to multiply by one over and squared divided by one over and squared. 152 00:11:46,180 --> 00:11:52,000 And what that's going to give us limit equals l equals limit as and goes to infinity. 153 00:11:52,000 --> 00:11:56,980 What that's going to give us and squared at times 1 over and square we get the squares to cancel and 154 00:11:56,980 --> 00:12:04,150 we're left with one plus one times one over and squared is one over and squared and then in the denominator 155 00:12:04,150 --> 00:12:11,700 same thing we get the squares here to cancel and we're left with two plus one times one over and squared 156 00:12:11,710 --> 00:12:13,450 is just won over and squared. 157 00:12:13,570 --> 00:12:20,270 And now we have that same situation where as and becomes very very very large goes towards infinity. 158 00:12:20,320 --> 00:12:27,610 These two small fractions here will become 0 1 if you on your calculator take one and you divide it 159 00:12:27,610 --> 00:12:30,410 by a very large number like one million or 10 million. 160 00:12:30,460 --> 00:12:36,760 Your calculator will actually give you an answer of zero because this number is going to become so incredibly 161 00:12:36,760 --> 00:12:40,930 small that it becomes insignificant and we can just call it zero. 162 00:12:41,110 --> 00:12:49,110 They both cancel as you can see obviously we're just left with L equals the limit as and goes to infinity 163 00:12:49,190 --> 00:12:51,610 of the absolute value of one half. 164 00:12:51,640 --> 00:12:55,350 Well the is of 1 1/2 is just one half. 165 00:12:55,450 --> 00:12:58,840 The limit is and goes to infinity of 1 half is still just one half. 166 00:12:58,840 --> 00:13:01,450 There is no value we have to plug in for more. 167 00:13:01,660 --> 00:13:06,680 So what that tells us is that our limit L is equal to one half. 168 00:13:06,940 --> 00:13:14,710 And similarly with the root test as with the ratio test when with the root test L is less than 1. 169 00:13:14,740 --> 00:13:17,980 We know that the series converges Absolutely. 170 00:13:18,100 --> 00:13:21,630 So we can call this series absolutely convergent. 171 00:13:21,630 --> 00:13:23,310 We'll label this up here also. 172 00:13:23,350 --> 00:13:24,810 Absolutely convergent. 173 00:13:25,090 --> 00:13:28,070 And keep in mind that it's going to be the same thing if we were to get out. 174 00:13:28,150 --> 00:13:28,940 Equals 1. 175 00:13:28,960 --> 00:13:35,110 The test would be inconclusive if we get a greater than 1 then we know by the root test that the series 176 00:13:35,200 --> 00:13:35,870 diverges. 177 00:13:35,890 --> 00:13:40,080 But because the value is the value of the limit L is less than 1. 178 00:13:40,120 --> 00:13:44,310 We know that by the root test the series is absolutely convergent. 20004