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These are the user uploaded subtitles that are being translated: 1 00:00:00,630 --> 00:00:04,800 Today we're going to be talking about how to determine whether or not the telescoping series is convergent 2 00:00:04,830 --> 00:00:05,830 or divergent. 3 00:00:06,150 --> 00:00:08,990 And if it is convergent how to find it some. 4 00:00:09,120 --> 00:00:13,920 And in this particular problem we've been given this telescoping series which is the infinite sum from 5 00:00:13,920 --> 00:00:20,880 unequals 1 to infinity of our series here which is either the one or end power minus either the one 6 00:00:20,910 --> 00:00:23,040 over and plus one power. 7 00:00:23,040 --> 00:00:27,780 Now the best way to tell whether or not you actually have a telescoping series on your hands in the 8 00:00:27,780 --> 00:00:33,210 first place is to plug in the first several terms of the series starting with whatever we've been given 9 00:00:33,210 --> 00:00:35,720 here which in our case is an equals one. 10 00:00:35,760 --> 00:00:40,680 So because unequals one we're going to be starting with one and we're just going to plug in the first 11 00:00:40,740 --> 00:00:46,310 several terms of our series to see what kind of series we end up with. 12 00:00:46,320 --> 00:00:53,250 So when we plug in one we get either the one over 1 or just eat the one so that's easy. 13 00:00:53,880 --> 00:01:00,280 When we plug in one to this second term here we get one plus one in our denominator which is two. 14 00:01:00,390 --> 00:01:08,280 In other words either the one half so we get minus the race to the one half power that's our first term 15 00:01:08,280 --> 00:01:09,200 in the series. 16 00:01:09,330 --> 00:01:14,200 Now what do we do we plug in an equal to what we get here for our first term. 17 00:01:14,290 --> 00:01:21,960 Either one half so easy to the one half minus when we plug into two plus one is three. 18 00:01:21,960 --> 00:01:27,270 So we get to the third So minus the two the one third. 19 00:01:27,300 --> 00:01:31,020 Now what happens if we plug in and equals three. 20 00:01:31,050 --> 00:01:40,890 So Yet here we have a one third minus the two the one fourth and you can start to see a pattern a little 21 00:01:40,890 --> 00:01:41,310 bit. 22 00:01:41,340 --> 00:01:47,760 Now it should become obvious fairly quickly that this is a telescoping series because what we see if 23 00:01:47,760 --> 00:01:55,110 we look at this series is that the second value in the first term is always going to be the negative 24 00:01:55,170 --> 00:01:58,020 version of the first value in the second term. 25 00:01:58,020 --> 00:02:00,540 So here we have negative IID of the one half. 26 00:02:00,540 --> 00:02:05,700 Here we have positive either the one half if we added these all up together they would cancel with one 27 00:02:05,700 --> 00:02:08,130 another right they would net to zero. 28 00:02:08,130 --> 00:02:12,040 Here we have negative E to the one third and positive to the one third. 29 00:02:12,060 --> 00:02:16,530 So those would cancel with each other they'd also net to zero and we could keep going. 30 00:02:16,530 --> 00:02:23,010 We'd cancel every term in between if we continue on with this series until the end of the series if 31 00:02:23,010 --> 00:02:28,320 it hadn't and what we would get to for our end term is our original series here right. 32 00:02:28,320 --> 00:02:36,660 Our last term in the series would be the one over n minus the the one over and plus one that would be 33 00:02:36,660 --> 00:02:38,320 our last term. 34 00:02:38,460 --> 00:02:39,160 OK. 35 00:02:39,360 --> 00:02:44,640 So if we have that what we see is that we're going to end up canceling when you have a telescoping series 36 00:02:44,940 --> 00:02:49,920 and this is happening and these terms are canceling like this you're always going to end up canceling 37 00:02:50,610 --> 00:02:55,410 every term in the middle of the series because this one is going to cancel with the first value in the 38 00:02:55,410 --> 00:02:56,540 next term. 39 00:02:56,600 --> 00:03:02,280 This either the one minus and value is going to cancel with the second value in the previous term that's 40 00:03:02,280 --> 00:03:03,880 going to go away as well. 41 00:03:04,170 --> 00:03:09,610 And all you're ever going to be left with is the very first term and the very last term. 42 00:03:09,660 --> 00:03:18,690 And so what we have here for the value of the partial sum or the series SLBM is equal to our first term 43 00:03:18,690 --> 00:03:19,080 here. 44 00:03:19,090 --> 00:03:27,850 E minus E to the one over and plus one minus E to the one over and plus one. 45 00:03:27,870 --> 00:03:32,550 Now a few of them were from before we talked about a series of partial sums remember that our original 46 00:03:32,550 --> 00:03:40,260 series here we call a sub that's the original series to the one over and minus the one over end plus 47 00:03:40,260 --> 00:03:45,750 one that's our original series a seben when we're talking about the series of partial sums we denoted 48 00:03:45,810 --> 00:03:46,670 as Subhan. 49 00:03:46,820 --> 00:03:51,990 And what we're talking about is adding all of these terms together and getting a model for that. 50 00:03:51,990 --> 00:03:55,760 Notice that this is a different value than our original series. 51 00:03:55,770 --> 00:04:01,010 A7 this is a representation of the series of partial sums as Subban. 52 00:04:01,080 --> 00:04:06,570 The cool thing about a telescoping series is that if you can find an explicit value for the end partial 53 00:04:06,570 --> 00:04:12,500 sum that alone proves that the telescoping series is convergent that's all you need. 54 00:04:12,600 --> 00:04:18,570 So as long as you can find this and you can easily find a value by cancelling all these terms in the 55 00:04:18,570 --> 00:04:25,430 center and being left with just the first term and the last term like this and you get a model for this 56 00:04:25,430 --> 00:04:32,100 series of partial sums you get this as value as long as you find this you know that the telescoping 57 00:04:32,100 --> 00:04:37,600 series is convergent and that's essentially your convergence test for telescoping series. 58 00:04:37,620 --> 00:04:40,970 So that's going to be our proof that the series is convergence. 59 00:04:40,980 --> 00:04:43,080 We can say it's convergent now. 60 00:04:43,200 --> 00:04:46,940 The only thing we need to know is what the some of the series is. 61 00:04:46,950 --> 00:04:52,950 Well it's really easy once we have a model for the series of partial sums because if you'll remember 62 00:04:53,370 --> 00:05:02,880 the infinite sum from any quolls 1 to infinity of our series Sabayon this is just a definition. 63 00:05:02,890 --> 00:05:08,290 When we have a series A7 and we're looking for the infinite sum here that's going to be equal to the 64 00:05:08,290 --> 00:05:15,120 limit as NGOs to infinity of the series of partial sums as Subban. 65 00:05:15,280 --> 00:05:21,370 So this is a convenient sometimes convenient easy way to find the sum of the series if we can find a 66 00:05:21,370 --> 00:05:23,920 model for sub in the series of partial sums. 67 00:05:24,090 --> 00:05:29,710 All we have to do is take its limit and approaches infinity and that'll give us the sambar original 68 00:05:29,710 --> 00:05:30,370 series. 69 00:05:30,520 --> 00:05:34,890 So we're going to change this year the limit is and goes to infinity of Esben. 70 00:05:35,080 --> 00:05:42,340 We're going to change that into the limit as and goes to infinity or our actual value for s 7 which 71 00:05:42,340 --> 00:05:48,870 is B minus the to the one over and plus 1 power. 72 00:05:48,910 --> 00:05:49,780 So we get that. 73 00:05:49,970 --> 00:05:57,450 OK so now what happens if we evaluate this limit if we plug in a very very very large value for n where 74 00:05:57,640 --> 00:06:04,180 when we do that here in this xponent we'll get essentially infinity or a very large number plus one. 75 00:06:04,180 --> 00:06:11,020 So in other words just still a very large number in this denominator here one or any constant over an 76 00:06:11,020 --> 00:06:14,800 extremely large number that's always just going to tend to zero. 77 00:06:14,800 --> 00:06:18,200 This is going to become zero here when that happens. 78 00:06:18,200 --> 00:06:23,190 We were left with E to the zero power zero power is just one. 79 00:06:23,200 --> 00:06:26,210 So this becomes our first value here. 80 00:06:26,220 --> 00:06:32,600 E t minus E to the zero power or in other words the minus one. 81 00:06:32,740 --> 00:06:39,580 And this is the sum of our series which we just found by taking the limit as and goes to infinity of 82 00:06:39,580 --> 00:06:41,740 our series of partial sums as 7. 83 00:06:41,860 --> 00:06:47,980 The value we get for that in our case minus one is the sum of our original series The sum of the series 84 00:06:48,370 --> 00:06:49,060 a Subhan. 9639