Would you like to inspect the original subtitles? These are the user uploaded subtitles that are being translated: 1 00:00:00,450 --> 00:00:04,590 In this series we're talking about how to expand some through its first few terms. 2 00:00:04,680 --> 00:00:12,210 We've been given the series one over and squared so this value right here is our series And oftentimes 3 00:00:12,210 --> 00:00:20,300 we write our series as a 7 so we could say the series is a seven and a 7 is equal to 1 divided by end 4 00:00:20,310 --> 00:00:26,040 squared so that's our series The function that defines our series and this sigma notation here the thing 5 00:00:26,040 --> 00:00:33,150 that looks like any tells us take the sum of this series the value below the sigma notation is an equals 6 00:00:33,240 --> 00:00:33,970 1. 7 00:00:33,990 --> 00:00:38,040 This tells us start at the term that corresponds with N equals 1. 8 00:00:38,040 --> 00:00:43,510 This tells us where to stop so it tells us to stop at a value of and equals six. 9 00:00:43,650 --> 00:00:50,460 So if we expand this series through its first six terms we would take the values and equals 1 2 3 4 10 00:00:50,460 --> 00:00:56,160 5 and 6 the first six terms and add them together because we have this summation notation. 11 00:00:56,160 --> 00:00:57,520 Here we're taking the sum. 12 00:00:57,630 --> 00:01:00,240 So we add all those terms together that we find. 13 00:01:00,240 --> 00:01:07,470 So we're looking at the term that corresponds with end equals one and equals two three four five and 14 00:01:07,470 --> 00:01:08,360 six. 15 00:01:08,400 --> 00:01:09,690 Those are the first six terms. 16 00:01:09,720 --> 00:01:12,490 That's where the index tells us to start and stop. 17 00:01:12,570 --> 00:01:15,290 And all we have to do is start with N equals 1. 18 00:01:15,300 --> 00:01:20,410 Plug that into our series 1 divided by and squared and simplify that value. 19 00:01:20,520 --> 00:01:22,970 So first we'll say plug in equals 1. 20 00:01:23,010 --> 00:01:25,530 Here we're going to get 1 divided by 1 squared. 21 00:01:25,530 --> 00:01:27,330 So let's just go ahead and write that out for now. 22 00:01:27,330 --> 00:01:29,340 So 1 divided by 1 squared. 23 00:01:29,340 --> 00:01:34,230 Then we're going to plug in an equal to and because we're taking the sum because we have this summation 24 00:01:34,230 --> 00:01:36,920 notation we want to add all these terms together. 25 00:01:37,140 --> 00:01:41,110 So plugging in N equals 2 we're going to get 1 over 2 squared. 26 00:01:41,280 --> 00:01:49,770 Plugging in N equals three will get 1 over 3 squared and we'll get one over 4 squared one over five 27 00:01:49,770 --> 00:01:50,320 squared. 28 00:01:50,340 --> 00:01:57,110 And finally one over six squared and we stop there because this value right here of six tells us stop 29 00:01:57,120 --> 00:01:58,770 when you get two and equal sex. 30 00:01:58,770 --> 00:02:04,080 In other words your last term will be the term where you plug in and equal sex which is the term that 31 00:02:04,080 --> 00:02:05,380 we just found here. 32 00:02:05,400 --> 00:02:06,860 So that's all we're going to stop. 33 00:02:06,930 --> 00:02:09,010 So now we just need to simplify. 34 00:02:09,030 --> 00:02:13,680 So here in the denominator we get one square which is just 1 1 divided by one is 1. 35 00:02:13,680 --> 00:02:20,100 So here we get one then we're going to get plus two squared is force we get one fourth three squared 36 00:02:20,100 --> 00:02:20,780 is nine. 37 00:02:20,790 --> 00:02:29,130 So we get one ninth for score to 16 so one over 16 and then we're going to get one over 25 and one over 38 00:02:29,160 --> 00:02:31,790 thirty six and that's all there is to it. 39 00:02:31,830 --> 00:02:39,000 This value that we just found this series this sum is the exact same thing as what we were originally 40 00:02:39,000 --> 00:02:39,840 given. 41 00:02:39,840 --> 00:02:43,260 This notation right here it's just a different way of writing it. 42 00:02:43,260 --> 00:02:48,450 So this whole thing here is exactly equal to this whole thing here. 43 00:02:48,480 --> 00:02:50,400 This is just a simpler way of writing it. 44 00:02:50,400 --> 00:02:55,770 So if you were given this Sammet to start with and you wanted to write it in summation notation you 45 00:02:55,770 --> 00:02:56,810 could write it this way. 46 00:02:56,910 --> 00:02:59,160 And we'll talk about how to do that a little bit later. 47 00:02:59,190 --> 00:03:04,350 But here we just started with summation notation we were given this series and we were given the index 48 00:03:04,680 --> 00:03:11,130 and we expanded this series we expanded the summation notation into an expanded sum and that's how you 49 00:03:11,130 --> 00:03:14,040 expand a series through its first few terms. 5146