Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 1 00:00:00,540 --> 00:00:04,710 Today we're going to be talking about how to calculate the first several terms in a sequence of partial 2 00:00:04,710 --> 00:00:05,460 sums. 3 00:00:05,700 --> 00:00:10,350 And in this particular problem we've been asked to calculate the first several terms of this sequence 4 00:00:10,470 --> 00:00:16,380 and divide by 1 plus the square root of N from N equals 1 to infinity. 5 00:00:16,590 --> 00:00:22,680 Now before we talk about how to calculate the first terms I want to as a reminder make sure that we 6 00:00:22,710 --> 00:00:28,260 understand the difference between a sequence and a sequence of partial sums. 7 00:00:28,260 --> 00:00:33,800 So in a regular series here we might have the series a Sabine. 8 00:00:33,960 --> 00:00:34,470 Right. 9 00:00:34,500 --> 00:00:39,110 That's different than how we denote the series of partial sums. 10 00:00:39,190 --> 00:00:47,790 Es seben So a sideband is our typical series as Serbin is our series or a sequence of partial sums. 11 00:00:47,790 --> 00:00:55,390 Now when it comes to a regular series we might have some regular series like for example one two three 12 00:00:55,420 --> 00:00:57,480 four five right. 13 00:00:57,690 --> 00:01:04,320 The series of partial sums will be each one of these terms added together with the terms before it. 14 00:01:04,500 --> 00:01:07,150 So the first terms will be the same right one in one. 15 00:01:07,170 --> 00:01:13,380 But then the second term I'm going to add two to the first term in my series of partial sum so one plus 16 00:01:13,380 --> 00:01:14,960 two gives me three. 17 00:01:14,970 --> 00:01:21,090 Then I add the next term to that three plus three gives me six to six I add four and I get 10 to 10 18 00:01:21,090 --> 00:01:24,510 I add 5 and I get 15 and I could keep going. 19 00:01:24,510 --> 00:01:26,720 So you can see how these series are different. 20 00:01:26,720 --> 00:01:30,570 We have the regular series a sidebar in the series of partial sums. 21 00:01:30,690 --> 00:01:32,790 We're taking partial sums of this series. 22 00:01:32,820 --> 00:01:39,630 In other words the sum of the series up to the given point so 10 is the sum of the series up until the 23 00:01:39,630 --> 00:01:41,450 fourth term of the series. 24 00:01:41,460 --> 00:01:44,610 So that's why we call it the series of partial sums. 25 00:01:44,610 --> 00:01:50,670 So in order to calculate the first terms in the sequence of partial sums here what we want to do is 26 00:01:50,670 --> 00:01:52,750 just start playing in values of n. 27 00:01:52,750 --> 00:01:57,570 Based on whatever we're given here so we were given in this particular problem and equals 1. 28 00:01:57,570 --> 00:01:59,220 So we start with an equals 1. 29 00:01:59,310 --> 00:02:08,790 So we say at any equals 1 we plug in 1 to our series here and we get 1 divided by 1 plus the square 30 00:02:08,790 --> 00:02:09,860 root of 1. 31 00:02:10,050 --> 00:02:15,990 And of course when we evaluate that we'll get the squared of one to be one one plus one is two. 32 00:02:16,050 --> 00:02:22,190 And so we get a value of 1 1/2 which is equal to zero point five. 33 00:02:22,260 --> 00:02:26,250 And we can just add a couple extra zeros because we know we're going to need them later. 34 00:02:26,250 --> 00:02:26,760 OK. 35 00:02:26,880 --> 00:02:33,600 So then we plug in a value and equals two and we can just keep doing this right we'll get 2 divided 36 00:02:33,600 --> 00:02:36,790 by 1 plus the square root of two. 37 00:02:37,050 --> 00:02:44,220 And if we evaluate that we'll get approximately point eight to a four that's an approximate value. 38 00:02:44,370 --> 00:02:49,770 But remember because we're dealing with the sequence of partial sums we have to add this value the value 39 00:02:49,770 --> 00:02:53,670 of the second term to the previous value that we got. 40 00:02:53,670 --> 00:03:00,450 So we have to add that two point five point eight to eight four plus point five is going to give us 41 00:03:00,450 --> 00:03:08,580 the value of the second term in a sequence of partial sums which is approximately 1 point 3 2 8 4 and 42 00:03:08,580 --> 00:03:12,380 we can round that to about 3 to 8. 43 00:03:12,690 --> 00:03:20,010 So let's just do one more term we'll get an equals three we get three over 1 plus the square root of 44 00:03:20,010 --> 00:03:20,790 3. 45 00:03:20,910 --> 00:03:28,000 When we evaluate that on our calculator we'll get approximately 1 point 0 9 8 or so. 46 00:03:28,470 --> 00:03:33,750 But remember that's only the third term of a seven in order to get the third term of 7. 47 00:03:33,780 --> 00:03:40,740 We need to add it to our previous value of 1.3 to 8 when we do that and we round we get approximately 48 00:03:40,770 --> 00:03:47,100 two point four to seven for the third term in our sequence of partial sum. 49 00:03:47,100 --> 00:03:53,610 So remember that value we get here when we just plug in our value for end directly is the value of a 50 00:03:53,610 --> 00:03:56,880 7 when we add it to the previous term. 51 00:03:56,880 --> 00:04:01,130 We get the value in the sequence of partial sums as Subban. 52 00:04:01,290 --> 00:04:05,440 So here's how we generate the values of S-band and we can just keep going. 53 00:04:05,460 --> 00:04:11,550 And he calls for unequals 5 and 6 and we just added to that previous sum that we found what we'd see 54 00:04:11,550 --> 00:04:18,780 is that we get approximately three point 6 0 for any calls four for N equals five. 55 00:04:18,780 --> 00:04:24,730 We get approximately five point three zero five for any six. 56 00:04:24,780 --> 00:04:29,910 We get approximately seven point zero for four and we keep going. 57 00:04:29,910 --> 00:04:36,360 One thing we can say just looking at this sequence of partial sums though is that the sequence appears 58 00:04:36,360 --> 00:04:41,090 to be divergent because this value just keeps getting larger and larger and larger. 59 00:04:41,100 --> 00:04:45,950 And in fact the change between each term keeps getting larger and larger. 60 00:04:45,960 --> 00:04:54,900 This sequence of partial sums as then appears to be divergent or just appears divergent and that's one 61 00:04:54,900 --> 00:05:00,710 conclusion that we can try to draw from the list of terms that we can it here. 6866