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These are the user uploaded subtitles that are being translated: 1 00:00:00,670 --> 00:00:04,820 In this you are going to be talking about how you use the root test to see whether or not a series converges 2 00:00:04,850 --> 00:00:06,410 or diverges. 3 00:00:06,410 --> 00:00:11,000 And in this particular problem we've been given this series here some from unequals went to infinity 4 00:00:11,170 --> 00:00:16,200 of three to the power divided by and plus one race to the power. 5 00:00:16,370 --> 00:00:19,010 And as a reminder this is the root test here. 6 00:00:19,040 --> 00:00:20,770 It all hinges on this value. 7 00:00:20,840 --> 00:00:25,970 This is the value that we need to find because once we find out we can say that the series converges 8 00:00:26,000 --> 00:00:32,360 absolutely of Eliz less than one diverges is greater than one or that the test is inconclusive as equal 9 00:00:32,360 --> 00:00:32,880 to one. 10 00:00:32,990 --> 00:00:35,990 So we have to find that value L and here's how we're going to do it. 11 00:00:36,090 --> 00:00:40,870 L is going to be the limit as and approaches infinity of the absolute value of a 7. 12 00:00:40,890 --> 00:00:44,550 Remember that this function here inside of our series that that's a 7. 13 00:00:44,600 --> 00:00:46,900 And we're going to raise that to the one over and power. 14 00:00:47,000 --> 00:00:52,760 Now you're not always going to be able to use root test to test for Convergence root test is useful 15 00:00:52,970 --> 00:01:00,350 when all of the terms inside of your series are raised to the power of N because raising everything 16 00:01:00,350 --> 00:01:04,050 to the one over and power is going to eliminate that exponent. 17 00:01:04,250 --> 00:01:06,900 So here's what you want to do with a series like this one. 18 00:01:06,980 --> 00:01:09,400 We want to identify that really have two terms here. 19 00:01:09,410 --> 00:01:15,230 We have three and we have and plus one since they're both raised to the power of n we know that all 20 00:01:15,230 --> 00:01:17,190 the terms in our series are raised to the power. 21 00:01:17,360 --> 00:01:20,260 And so root test is probably a good test to use. 22 00:01:20,270 --> 00:01:25,400 So we want to put this series Inside of our value for else we're going to say is going to be equal to 23 00:01:25,790 --> 00:01:31,100 the limit as and approaches infinity and we're going to take the absolute value of our series of the 24 00:01:31,100 --> 00:01:37,850 absolute value of three of the N divided by and plus one quantity at the end power and then we're raising 25 00:01:37,850 --> 00:01:39,230 that to the one over end. 26 00:01:39,440 --> 00:01:46,670 So then we can rewrite the value inside of our absolute value brackets we can rewrite our series as 27 00:01:46,670 --> 00:01:49,690 three over and plus one. 28 00:01:49,850 --> 00:01:53,130 We're going gonna put this whole thing in parentheses and raise it to the end power. 29 00:01:53,330 --> 00:01:59,420 And the reason is because since all the terms in this fraction are raised to the power we can pull that 30 00:01:59,480 --> 00:02:07,730 exponent outside of the fraction in the same way that if we had three squared over four squared we could 31 00:02:07,730 --> 00:02:11,200 call that three fourths quantity squared. 32 00:02:11,210 --> 00:02:16,750 The same thing we can pull that exponent out since we have that same exponent in the numerator and denominator. 33 00:02:16,850 --> 00:02:19,540 But the expansion has to be on every turn. 34 00:02:19,670 --> 00:02:20,750 So here it was on every term. 35 00:02:20,780 --> 00:02:21,650 We could pull it out. 36 00:02:21,680 --> 00:02:27,620 And now what we can say is that we really have the multiplication of exponents here and times 1 over 37 00:02:27,620 --> 00:02:28,290 n. 38 00:02:28,310 --> 00:02:33,120 Well of course that's just going to net to 1 because the end here in the numerator cancels with that. 39 00:02:33,120 --> 00:02:39,350 And there's nominator and the remaining exponent is just one so you can see how the root test makes 40 00:02:39,350 --> 00:02:40,910 that exponent disappear. 41 00:02:40,910 --> 00:02:44,060 So then we're really just left with three over and plus one. 42 00:02:44,060 --> 00:02:49,880 And now keep in mind that the terms of our series start at begin at end is equal to 1. 43 00:02:49,910 --> 00:02:53,360 So the smallest value we can ever have for N is 1. 44 00:02:53,360 --> 00:02:58,140 If we plug one in for and right here we're going to get 3 over 1 plus 1 or 3 over 2. 45 00:02:58,240 --> 00:03:02,750 And if we continued with the next value of ADD and the next value in the next value we'd have any was 46 00:03:02,750 --> 00:03:04,760 2 and it was three and equals four. 47 00:03:04,760 --> 00:03:10,700 Well no matter what positive integer we plug in here for N we're going to get a positive value for this 48 00:03:10,760 --> 00:03:12,780 fraction three over and plus 1. 49 00:03:12,860 --> 00:03:17,780 Which means that these absolute value bars are redundant and we can drop them because this is always 50 00:03:17,780 --> 00:03:19,310 going to be positive. 51 00:03:19,310 --> 00:03:23,450 If you don't have that situation where the terms of your series are not always positive you'll need 52 00:03:23,450 --> 00:03:24,670 to keep the absolute value bars. 53 00:03:24,680 --> 00:03:30,830 But for now we can drop ours because this fraction is always going to produce a positive result so then 54 00:03:30,830 --> 00:03:36,340 we just have the limit is and goes to infinity of three over and plus 1. 55 00:03:36,440 --> 00:03:41,330 And what we see here is that if we were to make a substitution for end of infinity if we plugged in 56 00:03:41,330 --> 00:03:47,060 a very very very very large number for end we'd have N plus 1 or you could think of it as infinity plus 57 00:03:47,060 --> 00:03:52,190 one that's still going to be infinity and we're going to have three divided by an extremely large number 58 00:03:52,490 --> 00:03:57,710 that value always converges to zero when we have some constant in the numerator and an infinitely large 59 00:03:57,710 --> 00:03:59,420 value in the denominator. 60 00:03:59,450 --> 00:04:01,920 Then this whole limit becomes zero. 61 00:04:02,000 --> 00:04:05,100 So we're going to go ahead and say then that L is equal to zero. 62 00:04:05,100 --> 00:04:10,980 They remember the value of L that we find is all based on its relationship to 1. 63 00:04:11,000 --> 00:04:18,440 So because 0 is less than 1 and we therefore have an l value that is less than 1 we can say that this 64 00:04:18,440 --> 00:04:23,960 series converges Absolutely and that's how you use the root test to find the convergence or divergence 65 00:04:24,230 --> 00:04:25,100 of A Series. 7315