WEBVTT 00:00:06.890 --> 00:00:08.520 PROFESSOR: Welcome back to recitation. 00:00:08.520 --> 00:00:11.180 In this video, I want us to compute the following limit. 00:00:11.180 --> 00:00:15.390 It's the limit is n goes to infinity of the sum for i 00:00:15.390 --> 00:00:19.160 equals 0 to n minus 1 of the following, 2 over n times 00:00:19.160 --> 00:00:23.850 the quantity 2i over n quantity squared minus 1. 00:00:23.850 --> 00:00:25.630 Now this might look a little intimidating 00:00:25.630 --> 00:00:28.600 to try and take a limit of this, but what 00:00:28.600 --> 00:00:30.770 I'd like you to do, as a hint to you, 00:00:30.770 --> 00:00:34.800 is that you should think about this as being potentially 00:00:34.800 --> 00:00:38.680 a Riemann sum of a certain function. 00:00:38.680 --> 00:00:40.470 So if you can figure out the function, 00:00:40.470 --> 00:00:43.370 and you can figure out the appropriate interval 00:00:43.370 --> 00:00:46.090 that you're taking a Riemann sum over, as n goes to infinity, 00:00:46.090 --> 00:00:49.220 you should be able to write this as an integral. 00:00:49.220 --> 00:00:52.260 We know how to use the fundamental theorem of calculus 00:00:52.260 --> 00:00:55.100 to determine that a definite integral in many cases. 00:00:55.100 --> 00:01:00.599 Hopefully this is a function for which we know a way to do that. 00:01:00.599 --> 00:01:01.640 So that's my hint to you. 00:01:01.640 --> 00:01:04.947 Think about it, it's a Riemann sum approximating an integral, 00:01:04.947 --> 00:01:07.360 and I'll give you a while to work on it, 00:01:07.360 --> 00:01:08.570 and then I'll be back. 00:01:15.730 --> 00:01:17.221 OK, welcome back. 00:01:17.221 --> 00:01:19.720 Well hopefully it's been fun for you to look at this problem 00:01:19.720 --> 00:01:20.591 so far. 00:01:20.591 --> 00:01:22.340 Let me just remind you what we were doing. 00:01:22.340 --> 00:01:23.714 We were trying to compute a limit 00:01:23.714 --> 00:01:27.540 as n goes to infinity of the sum from i equals 0 to n minus 1 00:01:27.540 --> 00:01:33.320 of 2 over n times 2i over n squared minus 1. 00:01:33.320 --> 00:01:36.860 So I gave you the big hint that this is probably going 00:01:36.860 --> 00:01:38.220 to be written as an integral. 00:01:38.220 --> 00:01:42.170 So let me show you some pieces of this sum that 00:01:42.170 --> 00:01:46.132 should help us see what the integral is, and then I'll 00:01:46.132 --> 00:01:47.840 make a guess about what this is, and then 00:01:47.840 --> 00:01:52.430 I'll try to give an educated way to check my guess. 00:01:52.430 --> 00:01:55.450 So the first thing we noticed is that there is one thing, 00:01:55.450 --> 00:01:58.680 this is a product of two functions, and one of them-- 00:01:58.680 --> 00:01:59.920 well, of n, I guess. 00:01:59.920 --> 00:02:01.910 But this is a product of two things. 00:02:01.910 --> 00:02:05.310 One thing appears in every single term 00:02:05.310 --> 00:02:06.480 that you have for i. 00:02:06.480 --> 00:02:09.280 So the sum has n terms, and they're all 00:02:09.280 --> 00:02:13.080 going to be 2 over n times this, and the i is going to change. 00:02:13.080 --> 00:02:16.260 But this does not change, this 2 over n does not change, right? 00:02:16.260 --> 00:02:18.780 In fact, I could even pull that out if I wanted. 00:02:18.780 --> 00:02:21.030 But, I don't want to pull it out of the sum right now. 00:02:21.030 --> 00:02:22.860 I want us to look at what's actually 00:02:22.860 --> 00:02:24.667 going on in this product. 00:02:24.667 --> 00:02:26.750 So if this thing is appearing over and over again, 00:02:26.750 --> 00:02:28.940 and we know this is probably a Riemann sum, 00:02:28.940 --> 00:02:32.790 then chances are this is our delta x. 00:02:32.790 --> 00:02:34.820 So delta x being equal to 2 over n, 00:02:34.820 --> 00:02:41.120 we know delta x equals b minus a over n, where b and a are 00:02:41.120 --> 00:02:43.520 our left endpoint-- oh, sorry, our right endpoint-- 00:02:43.520 --> 00:02:44.925 and our left endpoint. 00:02:44.925 --> 00:02:45.425 Right? 00:02:45.425 --> 00:02:47.040 We integrate from a to b. 00:02:47.040 --> 00:02:50.160 So b minus a is the length of the interval. 00:02:50.160 --> 00:02:52.310 So this is really dividing up whatever 00:02:52.310 --> 00:02:56.550 interval we're integrating over, into n equal subintervals. 00:02:56.550 --> 00:02:59.540 So that's my first thought, is that b minus a over n 00:02:59.540 --> 00:03:01.797 is equal to 2 over n. 00:03:01.797 --> 00:03:03.630 And now we want to try and figure out, well, 00:03:03.630 --> 00:03:05.735 what the heck is this. 00:03:05.735 --> 00:03:07.110 Well, when we take a Riemann sum, 00:03:07.110 --> 00:03:09.270 remember when we take a Riemann sum what we get. 00:03:09.270 --> 00:03:14.840 We get the sum of delta x times f of x sub i, 00:03:14.840 --> 00:03:18.110 and i is what's varying from 0 to n minus 1. 00:03:18.110 --> 00:03:19.910 Let me put a little curve in here 00:03:19.910 --> 00:03:21.850 so we see those are two different things. 00:03:21.850 --> 00:03:23.630 So this is i equals 0 to n minus 1. 00:03:23.630 --> 00:03:25.420 I have this delta x here. 00:03:25.420 --> 00:03:28.310 I'm anticipating this is some f of x sub i. 00:03:28.310 --> 00:03:30.330 And so the question is, what f is it? 00:03:30.330 --> 00:03:31.330 Right? 00:03:31.330 --> 00:03:34.900 If I know what f it is, than I know that this sum will 00:03:34.900 --> 00:03:37.690 be equal to something, the integral from a 00:03:37.690 --> 00:03:42.350 to b of f of x dx, and a and b will differ by 2. 00:03:42.350 --> 00:03:43.520 So that's where I'm heading. 00:03:43.520 --> 00:03:46.750 So now the question is, what is this a function of? 00:03:46.750 --> 00:03:50.870 What function is this, I should say. 00:03:50.870 --> 00:03:53.036 Now first guess would be something 00:03:53.036 --> 00:03:57.270 like, well, I'm taking some quantity, I'm squaring it, 00:03:57.270 --> 00:03:58.630 and subtracting 1. 00:03:58.630 --> 00:04:02.330 So my first guess for this function is x squared minus 1. 00:04:02.330 --> 00:04:04.640 I mean, that seems easy to me. 00:04:04.640 --> 00:04:07.690 Let's see if this would actually even make sense just 00:04:07.690 --> 00:04:12.180 by looking at the subscripts, or sorry, the index, the indices 00:04:12.180 --> 00:04:14.170 I have here. 00:04:14.170 --> 00:04:14.940 So what do I have? 00:04:14.940 --> 00:04:17.990 Well, when I put in i equals 0-- let's 00:04:17.990 --> 00:04:20.720 put down some of these values-- when I put in i equals 0, 00:04:20.720 --> 00:04:25.450 I get 2 times 0 over n squared minus 1. 00:04:25.450 --> 00:04:29.030 When I put in i equals 1, I get 2 times 1 00:04:29.030 --> 00:04:32.470 over n squared minus 1. 00:04:32.470 --> 00:04:37.100 And I go all the way up, to 2 times n 00:04:37.100 --> 00:04:41.770 minus 1 over n squared minus 1. 00:04:41.770 --> 00:04:43.770 So it's kind of a long sum there, but these are, 00:04:43.770 --> 00:04:46.220 this is what our sum of these things 00:04:46.220 --> 00:04:49.230 looks like if I pull out the 2 over n. 00:04:49.230 --> 00:04:53.380 So here I get 0 squared minus 1. 00:04:53.380 --> 00:04:54.760 That looks pretty good. 00:04:54.760 --> 00:04:58.170 Here I get 2 times 1 over n squared minus 1. 00:04:58.170 --> 00:05:00.900 So it does look like I'm doing something, taking something, 00:05:00.900 --> 00:05:02.576 squaring it, subtracting 1. 00:05:02.576 --> 00:05:03.950 Does it make sense that these are 00:05:03.950 --> 00:05:05.780 the kind of x values I would expect 00:05:05.780 --> 00:05:10.000 to get if this were the Riemann sum of x squared minus 1? 00:05:10.000 --> 00:05:13.090 It does, and let's think about why. 00:05:13.090 --> 00:05:17.620 I'm starting at x equals 0 here, it sure looks like. 00:05:17.620 --> 00:05:21.230 Let's look at what happens when I go all the way over here. 00:05:21.230 --> 00:05:24.760 What happens when n gets really, really big, is it 00:05:24.760 --> 00:05:28.140 this ratio approaches 2. 00:05:28.140 --> 00:05:32.120 So it's 2 times n minus 1 over n. n minus 1 over n, 00:05:32.120 --> 00:05:34.512 as n gets arbitrarily large, as n gets really big, 00:05:34.512 --> 00:05:35.470 then this approaches 2. 00:05:35.470 --> 00:05:38.090 So this is approaching 2 squared minus 1. 00:05:38.090 --> 00:05:40.910 So it's giving me more evidence that this is probably 00:05:40.910 --> 00:05:43.420 the function x squared minus 1. 00:05:43.420 --> 00:05:46.530 And now I'm starting to believe the interval is 0 to 2. 00:05:46.530 --> 00:05:49.480 I know it's a length 2 interval, and it's looking 00:05:49.480 --> 00:05:51.250 like the interval is 0 to 2. 00:05:51.250 --> 00:05:54.340 Let's come back and talk about one more thing. 00:05:54.340 --> 00:05:56.480 The one other thing that you should notice 00:05:56.480 --> 00:05:58.260 is that how does this value differ 00:05:58.260 --> 00:06:01.310 from this value, and the next, and the next, and the next? 00:06:01.310 --> 00:06:04.050 They differ by 2 over n. 00:06:04.050 --> 00:06:06.410 So each time whatever input I had previously, 00:06:06.410 --> 00:06:09.554 I'm now adding 2 over n to the next input. 00:06:09.554 --> 00:06:10.970 And that should make sense of what 00:06:10.970 --> 00:06:12.940 we know about Riemann sums, because what I do, 00:06:12.940 --> 00:06:16.620 is I divide my interval into these subintervals 00:06:16.620 --> 00:06:18.440 of length 2 over n. 00:06:18.440 --> 00:06:20.300 I'm evaluating it at one x-value, 00:06:20.300 --> 00:06:22.550 that-- I'm starting, in this case, at 0. 00:06:22.550 --> 00:06:24.840 Then the next interval is 2 over n more. 00:06:24.840 --> 00:06:27.225 Then I evaluate at that x-value. 00:06:27.225 --> 00:06:31.260 The next one is 2 over n more, and I evaluate at that x-value. 00:06:31.260 --> 00:06:33.640 So this is looking like-- I'm going to write it here, 00:06:33.640 --> 00:06:41.950 this is my guess-- integral from 0 to 2 of x squared minus 1 dx. 00:06:41.950 --> 00:06:45.050 And now to make myself feel good about this-- 00:06:45.050 --> 00:06:46.510 I'm pretty sure it's that. 00:06:46.510 --> 00:06:48.410 To make you feel good about this, 00:06:48.410 --> 00:06:51.807 I'm going to divide this into four subintervals, 00:06:51.807 --> 00:06:54.140 and I'm going to show you what the Riemann sum with four 00:06:54.140 --> 00:06:55.880 intervals looks like, and then we 00:06:55.880 --> 00:07:00.380 can talk about how it relates to this one over here. 00:07:00.380 --> 00:07:02.510 OK, so let me draw a graph. 00:07:02.510 --> 00:07:04.260 Actually, I'll use just white chalk again. 00:07:04.260 --> 00:07:10.490 Let me draw a graph of x squared minus 1 from 0 to 2. 00:07:10.490 --> 00:07:18.080 So 0, 1, 2, minus 1. 00:07:18.080 --> 00:07:22.490 OK, so at 0, x squared minus 1 is negative 1. 00:07:22.490 --> 00:07:26.820 At x equals 1, x squared minus 1 is 0. 00:07:26.820 --> 00:07:31.460 And at 2, x squared minus 1 is 3. 00:07:31.460 --> 00:07:36.770 So hopefully, this is all going to go into the video, 00:07:36.770 --> 00:07:39.370 and-- in the video screen, I mean. 00:07:39.370 --> 00:07:41.120 And there we go, something like that. 00:07:41.120 --> 00:07:43.410 So this is, you know, it continues over here, 00:07:43.410 --> 00:07:45.870 but I'm really only interested in this part. 00:07:45.870 --> 00:07:47.840 So now let's look at what the subintervals are. 00:07:47.840 --> 00:07:50.252 And now I'm going to get some colored chalk. 00:07:50.252 --> 00:07:51.460 So what are the subintervals? 00:07:51.460 --> 00:07:52.798 I'm taking 1 over 4, OK? 00:07:56.020 --> 00:08:01.204 And so delta x, in this case, is 2 over 4, 00:08:01.204 --> 00:08:02.120 which is equal to 1/2. 00:08:02.120 --> 00:08:04.730 Right? 00:08:04.730 --> 00:08:07.122 And so what are my, what are-- so what 00:08:07.122 --> 00:08:08.330 is my sum going to look like? 00:08:08.330 --> 00:08:10.329 Well, I am going to tell you that I'm also going 00:08:10.329 --> 00:08:11.680 to use left-handed endpoints. 00:08:11.680 --> 00:08:14.130 And I mentioned earlier why that is, I believe. 00:08:14.130 --> 00:08:15.580 Maybe I didn't. 00:08:15.580 --> 00:08:18.300 But, I started off at i equals 0, 00:08:18.300 --> 00:08:20.350 and my first input value was 0. 00:08:20.350 --> 00:08:24.510 My last input value had an n minus 1 in it instead of an n. 00:08:24.510 --> 00:08:28.380 So I was doing, somehow, the second-to-last place 00:08:28.380 --> 00:08:29.630 that I was interested in here. 00:08:29.630 --> 00:08:32.700 So it's definitely more of a left-hand endpoint thing. 00:08:32.700 --> 00:08:35.020 So I'm going to do this with left-hand endpoints. 00:08:35.020 --> 00:08:37.300 And I'm not going to simplify as I go. 00:08:37.300 --> 00:08:40.740 I'm going to write it out in sort of an expanded form. 00:08:40.740 --> 00:08:42.559 OK, so let's write it out in expanded form. 00:08:42.559 --> 00:08:46.700 So the Riemann sum-- this is y equals x squared minus 1. 00:08:46.700 --> 00:08:51.470 The Riemann sum is, the first term is 1/2 times what? 00:08:51.470 --> 00:08:54.520 It's the value, this x-value, which 00:08:54.520 --> 00:09:01.240 is 0, evaluated on this curve, so 0 squared minus 1. 00:09:01.240 --> 00:09:04.010 The next term-- I'll just write them right below each other-- 00:09:04.010 --> 00:09:04.905 is 1/2. 00:09:04.905 --> 00:09:08.170 'Cause again, let's draw a picture of what 00:09:08.170 --> 00:09:09.170 the first one is, sorry. 00:09:09.170 --> 00:09:12.570 It's this rectangle. 00:09:12.570 --> 00:09:13.460 Right? 00:09:13.460 --> 00:09:17.810 It's evaluated-- It's length 1/2, 00:09:17.810 --> 00:09:20.410 and it's the function evaluated at 0. 00:09:20.410 --> 00:09:21.920 The next one is length 1/2, and it's 00:09:21.920 --> 00:09:23.570 going to be the function evaluated 00:09:23.570 --> 00:09:26.280 at whatever this left-hand endpoint is here. 00:09:26.280 --> 00:09:29.780 So it's going to be this area. 00:09:29.780 --> 00:09:33.080 So it's going to be length 1/2, and then 00:09:33.080 --> 00:09:36.520 the height is going to be at x equals 1/2, 00:09:36.520 --> 00:09:41.790 so 1/2 quantity squared minus 1. 00:09:41.790 --> 00:09:45.420 The next one is going to be this interval. 00:09:45.420 --> 00:09:49.190 Well, there's no rectangle to draw because it's just, 00:09:49.190 --> 00:09:51.960 the output is zero at the left endpoint here. 00:09:51.960 --> 00:09:54.770 So it's just going to be-- it's going 00:09:54.770 --> 00:09:58.750 to have output equal to 0, at length 1/2 and height 0. 00:09:58.750 --> 00:10:00.020 But we'll write it out anyway. 00:10:00.020 --> 00:10:03.102 It's going to be 1/2 times the quantity-- now, 00:10:03.102 --> 00:10:09.370 I went up 1/2 more, so it's going to be two 1/2's, 00:10:09.370 --> 00:10:11.590 two times 1/2 squared minus 1. 00:10:11.590 --> 00:10:14.710 Let me just show you why I did this. 00:10:14.710 --> 00:10:16.610 OK, if we look at the picture, here I'd 00:10:16.610 --> 00:10:18.900 gone up 1/2 from my initial value. 00:10:18.900 --> 00:10:21.710 Here I'd gone up another 1/2 from my initial value. 00:10:21.710 --> 00:10:24.496 So that's two 1/2's from my initial value of 0. 00:10:24.496 --> 00:10:26.496 The next one is going to be three 1/2's, so this 00:10:26.496 --> 00:10:29.501 is three 1/2's away, or commonly known as 3/2. 00:10:29.501 --> 00:10:30.000 OK? 00:10:32.710 --> 00:10:37.970 So that one is going to be-- 1/2 is the base length again, 00:10:37.970 --> 00:10:44.000 times the quantity 3 times 1/2 squared minus 1. 00:10:44.000 --> 00:10:47.265 And that is in the picture, this rectangle. 00:10:51.951 --> 00:10:52.450 Great. 00:10:52.450 --> 00:10:53.780 So what are we see here? 00:10:53.780 --> 00:10:57.590 If we look at this, these four pieces, what do we have? 00:10:57.590 --> 00:10:59.890 We have a 1/2 in front each time. 00:10:59.890 --> 00:11:01.100 Which, what was the 1/2? 00:11:01.100 --> 00:11:03.460 It was b minus a over n. 00:11:03.460 --> 00:11:07.230 So b minus a was 2, n was 4. 00:11:07.230 --> 00:11:10.127 So maybe I should have kept that as 2 over 4. 00:11:10.127 --> 00:11:11.960 But, it's a little easier to write it as 1/2 00:11:11.960 --> 00:11:13.350 because of what I'm doing next. 00:11:13.350 --> 00:11:15.470 I square something, I subtract 1. 00:11:15.470 --> 00:11:19.590 I go up by the value that this is from the initial one here. 00:11:19.590 --> 00:11:23.000 And so now I'm taking the output of what was in here. 00:11:23.000 --> 00:11:27.410 I now take the output at 1/2 more than what was here. 00:11:27.410 --> 00:11:31.210 Now I take it at two 1/2's more than what was here, or 1/2 more 00:11:31.210 --> 00:11:33.950 than what was there, and then three 1/2's 00:11:33.950 --> 00:11:36.430 more than what was here, or one more than what was there. 00:11:36.430 --> 00:11:38.846 That's kind of confusing, but let's go back to the picture 00:11:38.846 --> 00:11:41.090 and see what it is. 00:11:41.090 --> 00:11:43.480 My delta x was 1/2, right? 00:11:43.480 --> 00:11:44.930 So I evaluate at the first place, 00:11:44.930 --> 00:11:49.600 and I evaluate one more up, and then I evaluate one more up, 00:11:49.600 --> 00:11:52.630 and I evaluate one more up, which gives me outputs here, 00:11:52.630 --> 00:11:53.750 there, there, and there. 00:11:53.750 --> 00:11:55.370 Right? 00:11:55.370 --> 00:11:57.710 So really if you go back and you look 00:11:57.710 --> 00:12:04.090 at the formulation of the sum, this was 2 over n times 00:12:04.090 --> 00:12:09.250 quantity 2i over n squared minus 1, 00:12:09.250 --> 00:12:13.210 you can see the 2 over n is my 1/2, 00:12:13.210 --> 00:12:15.850 and then this is maybe the hardest part to see, 00:12:15.850 --> 00:12:18.200 but that's the 2 over n is my 1/2 again, 00:12:18.200 --> 00:12:24.400 and the i is this thing that's coming in as 1, 2, 3. 00:12:24.400 --> 00:12:27.810 And so that i was going from 0 to n minus 1-- so I 00:12:27.810 --> 00:12:31.480 should have said 0, 1, 2, 3. 00:12:31.480 --> 00:12:31.980 Right? 00:12:31.980 --> 00:12:35.705 So that i is the 0 to n minus 1, and then I'm evaluating that, 00:12:35.705 --> 00:12:37.740 and then I add them all up. 00:12:37.740 --> 00:12:44.620 So when I take the sum, I get, for n equals 4, I get this. 00:12:44.620 --> 00:12:46.512 So in fact, this is just a guess, 00:12:46.512 --> 00:12:47.886 still maybe you should, maybe you 00:12:47.886 --> 00:12:49.480 should convince yourself more. 00:12:49.480 --> 00:12:52.150 I'm actually convinced at this point, 00:12:52.150 --> 00:12:54.280 based on not just this evidence, but the evidence I 00:12:54.280 --> 00:12:57.030 understood before about how the function works. 00:12:57.030 --> 00:13:00.140 Maybe you want to compare it when n equals 00:13:00.140 --> 00:13:01.330 6, or something like that. 00:13:01.330 --> 00:13:02.788 You may need a little more evidence 00:13:02.788 --> 00:13:05.797 to make you understand this particular piece. 00:13:05.797 --> 00:13:07.380 But, hopefully that makes sense to you 00:13:07.380 --> 00:13:10.150 that this is really just i times delta x, and then 00:13:10.150 --> 00:13:11.780 evaluated somewhere. 00:13:11.780 --> 00:13:15.900 That's the main idea of this component. 00:13:15.900 --> 00:13:17.989 OK, well hopefully this is informative to you. 00:13:17.989 --> 00:13:19.780 If you want to know the exact answer of how 00:13:19.780 --> 00:13:22.720 to compute the sum, obviously you just take the integral. 00:13:22.720 --> 00:13:24.570 So I know you can do that. 00:13:24.570 --> 00:13:26.106 So that's where I'll stop.