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These are the user uploaded subtitles that are being translated: 1 00:00:00,300 --> 00:00:04,170 Today we're going to be talking about how to find the sum of the series of partial sums. 2 00:00:04,260 --> 00:00:10,380 And in this particular problem we've been given a series of partial sums as Subhan is equal to two minus 3 00:00:10,380 --> 00:00:14,370 three times 0.8 raised to the power. 4 00:00:14,370 --> 00:00:19,320 Now before we talk about how to find the sum of the series I want to talk briefly about the difference 5 00:00:19,320 --> 00:00:24,580 between a series and a series of partial sums just to illustrate this concept quickly. 6 00:00:24,600 --> 00:00:31,080 Let's say that we have a regular series we'll call it a Subhan and we're going to go ahead and list 7 00:00:31,170 --> 00:00:35,820 the terms in this series and then I want to show you the difference between a regular series and the 8 00:00:35,820 --> 00:00:41,660 partial sums of the series partial sums are usually denoted by Subhan to indicate. 9 00:00:41,660 --> 00:00:43,580 As for some. 10 00:00:43,650 --> 00:00:52,590 So if the terms in our series for example are just one two three four five six right like this those 11 00:00:52,590 --> 00:00:59,250 are the terms in our regular series the series of partial sums will be the sum of the associated term 12 00:00:59,520 --> 00:01:01,560 and every value in the series before it. 13 00:01:01,560 --> 00:01:05,080 So for the first term it's just exactly the same right. 14 00:01:05,090 --> 00:01:07,540 We have one here we have one here. 15 00:01:07,740 --> 00:01:13,530 But for our second term instead of just two we take two and we added to the value of the series before 16 00:01:13,530 --> 00:01:18,660 that which is once a two plus one is three for the third term. 17 00:01:18,660 --> 00:01:20,670 We take the third term and a second term. 18 00:01:20,700 --> 00:01:25,200 And the first term and add those together three plus two plus one is six. 19 00:01:25,200 --> 00:01:26,420 We get six there. 20 00:01:26,550 --> 00:01:29,120 When we add 4 to that we get 10. 21 00:01:29,160 --> 00:01:34,890 We had five to 10 we get 15 and we had six to 15 and we hit 21. 22 00:01:34,890 --> 00:01:42,780 This is the series of partial sums of the series a sub n so we would denote this series 1 2 3 4 5 6 23 00:01:42,840 --> 00:01:50,700 as the series A in the series of partial sums of this series would be denoted as Subban as 1 3 6 10 24 00:01:50,700 --> 00:01:52,470 15 21. 25 00:01:52,500 --> 00:01:57,770 So that's the difference and in this particular case obviously we have this series of partial sums. 26 00:01:57,780 --> 00:02:00,580 Esteban in this equation here. 27 00:02:00,640 --> 00:02:04,010 Now how do we find the sum of the series of partial sums. 28 00:02:04,020 --> 00:02:10,500 Well what we need to know is that what we're really talking about here is the infinite sum of the series 29 00:02:10,620 --> 00:02:14,690 a sub Subban right from end equals one to infinity. 30 00:02:14,880 --> 00:02:16,860 That's the sum of the series. 31 00:02:16,890 --> 00:02:19,020 But we're dealing here with partial sums. 32 00:02:19,020 --> 00:02:25,710 Well it turns out that the sum of the series as we take an out to infinity is just equal to the limit 33 00:02:26,370 --> 00:02:32,630 as and goes to infinity of as sub n the series of partial sums. 34 00:02:32,640 --> 00:02:39,570 So this is a convenient tool for us as a convenient equation for us to you often because if we have 35 00:02:39,630 --> 00:02:47,070 the series of partial sums here then we can just take the limit as that series goes to infinity and 36 00:02:47,070 --> 00:02:50,700 it gives us the sum of our original series. 37 00:02:50,700 --> 00:02:56,000 So what we're going to be evaluating here is the limit as and goes to infinity of s n. 38 00:02:56,100 --> 00:03:00,580 Now to do that in place of S N we can just plug in our equation here. 39 00:03:00,580 --> 00:03:10,860 So we'll get the limit as and goes to infinity of our equation to minus three times 0.8 raised to the 40 00:03:10,880 --> 00:03:12,060 end power. 41 00:03:12,060 --> 00:03:17,850 Now if we essentially plug in an infinite value here for n if we plug in an extremely large value for 42 00:03:17,850 --> 00:03:26,160 and what we find is that this quantity here 0.8 raised to the power continues to get smaller and smaller 43 00:03:26,160 --> 00:03:33,600 and smaller and smaller because if you take point eight and you multiply it by itself what you get of 44 00:03:33,600 --> 00:03:38,880 course is point 6 4 which is a smaller value than point eight. 45 00:03:39,000 --> 00:03:44,570 If you multiply it again by point eight point six four by point eight or in other words point eight 46 00:03:44,610 --> 00:03:51,450 cubed you get Point 5 1 to the value continues to decrease and get less and less and less and less. 47 00:03:51,450 --> 00:03:58,860 So what we can say is that goes to infinity this point eight to the n becomes zero and it's just going 48 00:03:58,860 --> 00:04:00,980 to go away it's going to become 0. 49 00:04:00,980 --> 00:04:06,140 So what we're left with here is two minus three times zero. 50 00:04:06,230 --> 00:04:10,380 Of course that takes away the three as well we get three times zero here that goes away. 51 00:04:10,620 --> 00:04:13,860 And we're just left with a value of two. 52 00:04:13,860 --> 00:04:19,770 What this tells us is that the sum of the series the sum of the series a Sabahan whatever it is and 53 00:04:19,770 --> 00:04:24,090 we didn't have the series a seven to model we only had seven. 54 00:04:24,360 --> 00:04:26,750 But the sum of the series is two. 55 00:04:26,970 --> 00:04:34,480 And we know that using the equation for the series of partial sums which we denoted as 7. 56 00:04:34,530 --> 00:04:40,410 So that's how we use the series of partial sums to find the sum of the series a 7. 57 00:04:40,410 --> 00:04:41,730 In this case two. 6427