WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.144 Commons license. 00:00:03.144 --> 00:00:05.460 Your support will help MIT OpenCourseWare 00:00:05.460 --> 00:00:09.646 continue to offer high quality educational resources for free. 00:00:09.646 --> 00:00:12.260 To make a donation or to view additional materials 00:00:12.260 --> 00:00:16.150 from hundreds of MIT courses visit MIT OpenCourseWare 00:00:16.150 --> 00:00:17.090 at ocw.mit.edu. 00:00:21.344 --> 00:00:22.760 PROFESSOR: Today we are continuing 00:00:22.760 --> 00:00:24.730 with improper integrals. 00:00:24.730 --> 00:00:27.371 I still have a little bit more to tell you about them. 00:00:30.080 --> 00:00:33.510 What we were discussing at the very end of last time 00:00:33.510 --> 00:00:36.195 was improper integrals. 00:00:41.560 --> 00:00:44.570 Now and these are going to be improper integrals 00:00:44.570 --> 00:00:48.220 of the second kind. 00:00:48.220 --> 00:00:50.990 By second kind I mean that they have a singularity 00:00:50.990 --> 00:00:52.095 at a finite place. 00:00:54.800 --> 00:00:57.260 That would be something like this. 00:00:57.260 --> 00:01:00.140 So here's the definition if you like. 00:01:00.140 --> 00:01:03.380 Same sort of thing as we did when the singularity 00:01:03.380 --> 00:01:04.120 was at infinity. 00:01:04.120 --> 00:01:09.440 So if you have the integral from 0 to 1 of f(x). 00:01:09.440 --> 00:01:12.600 This is going to be the same thing as the limit, 00:01:12.600 --> 00:01:17.780 as a goes to 0 from above, the integral from a to 1 00:01:17.780 --> 00:01:21.150 of f(x) dx. 00:01:21.150 --> 00:01:25.920 And the idea here is the same one that we had at infinity. 00:01:25.920 --> 00:01:27.320 Let me draw a picture of it. 00:01:27.320 --> 00:01:30.030 You have, imagine a function which is coming down like this 00:01:30.030 --> 00:01:32.020 and here's the point 1. 00:01:32.020 --> 00:01:35.180 And we don't know whether the area enclosed is 00:01:35.180 --> 00:01:39.040 going to be infinite or finite and so we cut it off 00:01:39.040 --> 00:01:40.680 at some place a. 00:01:40.680 --> 00:01:44.220 And we let a go to 0 from above. 00:01:44.220 --> 00:01:46.800 So really it's 0+. 00:01:46.800 --> 00:01:49.820 So we're coming in from the right here. 00:01:49.820 --> 00:01:52.810 And we're counting up the area in this chunk. 00:01:52.810 --> 00:01:56.520 And we're seeing as it expands whether it goes to infinity 00:01:56.520 --> 00:01:59.290 or whether it tends to some finite limit. 00:01:59.290 --> 00:02:02.990 Right, so this is the example and this is the definition. 00:02:02.990 --> 00:02:07.000 And just as we did for the other kind of improper integral, 00:02:07.000 --> 00:02:13.747 we say that this converges -- so that's the key word here -- 00:02:13.747 --> 00:02:24.220 if the limit is finite, exists maybe I should just say 00:02:24.220 --> 00:02:30.773 and diverges if not. 00:02:35.650 --> 00:02:38.945 Let's just take care of the basic examples. 00:02:42.110 --> 00:02:44.740 First of all I wrote this one down last time. 00:02:44.740 --> 00:02:47.390 We're going to evaluate this one. 00:02:47.390 --> 00:02:51.565 The integral from 0 to 1 of 1 over the square root of x. 00:02:55.402 --> 00:02:57.360 And this just, you almost don't notice the fact 00:02:57.360 --> 00:02:59.920 that it goes to infinity. 00:02:59.920 --> 00:03:02.570 This goes to infinity as x goes to 0. 00:03:02.570 --> 00:03:05.860 But if you evaluate it -- first of all we always write this 00:03:05.860 --> 00:03:06.860 as a power. 00:03:06.860 --> 00:03:08.010 Right? 00:03:08.010 --> 00:03:09.870 To get the evaluation. 00:03:09.870 --> 00:03:13.060 And then I'm not even going to replace the 0 by an a. 00:03:13.060 --> 00:03:14.490 I'm just going to leave it as 0. 00:03:14.490 --> 00:03:18.660 The antiderivative here is x^(1/2) times 2. 00:03:21.540 --> 00:03:23.750 And then I evaluate that at 0 and 1. 00:03:23.750 --> 00:03:25.080 And I get 2. 00:03:25.080 --> 00:03:29.090 2 minus 0, which is 2. 00:03:29.090 --> 00:03:31.330 All right so this one is convergent. 00:03:31.330 --> 00:03:34.180 And not only is it convergent but we can evaluate it. 00:03:38.310 --> 00:03:42.140 The second example, being not systematic 00:03:42.140 --> 00:03:44.300 but really giving you the principal examples 00:03:44.300 --> 00:03:51.650 that we'll be thinking about, is this one here, dx / x. 00:03:51.650 --> 00:03:54.200 And this one gives you the antiderivative 00:03:54.200 --> 00:03:56.070 as the logarithm. 00:03:56.070 --> 00:03:58.180 Evaluated at 0 and 1. 00:03:58.180 --> 00:04:00.190 And now again you have to have this thought 00:04:00.190 --> 00:04:02.770 process in your mind that you're really taking the limit. 00:04:02.770 --> 00:04:06.440 But this is going to be the log of 1 minus the log of 0. 00:04:06.440 --> 00:04:07.830 Really the log of 0 from above. 00:04:07.830 --> 00:04:10.970 There is no such thing as the log of 0 from below. 00:04:10.970 --> 00:04:12.820 And this is minus infinity. 00:04:12.820 --> 00:04:19.020 So it's 0 minus minus infinity, which is plus infinity. 00:04:19.020 --> 00:04:20.213 And so this one diverges. 00:04:29.710 --> 00:04:32.750 All right so what's the general-- 00:04:32.750 --> 00:04:39.070 So more or less in general, let's just, for powers anyway, 00:04:39.070 --> 00:04:45.220 if you work out this thing for dx / x^p from 0 to 1. 00:04:45.220 --> 00:04:51.820 What you're going to find is that it's 1/(1-p) when p is 00:04:51.820 --> 00:04:53.020 less than 1. 00:04:53.020 --> 00:05:00.240 And it diverges for p >= 1. 00:05:02.750 --> 00:05:07.880 Now that's the final result. If you carry out this integration 00:05:07.880 --> 00:05:10.920 it's not difficult. 00:05:10.920 --> 00:05:14.850 All right so now I just want to try 00:05:14.850 --> 00:05:17.270 to help you to remember this. 00:05:17.270 --> 00:05:20.500 And to think about how you should think about it. 00:05:20.500 --> 00:05:23.890 So I'm going to say it in a few more ways. 00:05:23.890 --> 00:05:26.850 All right just repeat what I've said already 00:05:26.850 --> 00:05:32.020 but try to get it to percolate and absorb itself. 00:05:32.020 --> 00:05:34.270 And in order to do that I have to make 00:05:34.270 --> 00:05:37.750 the contrast between the kind of improper integral 00:05:37.750 --> 00:05:39.410 that I was dealing with before. 00:05:39.410 --> 00:05:43.720 Which was not as x goes to 0 here but as x goes to infinity, 00:05:43.720 --> 00:05:45.920 the other side. 00:05:45.920 --> 00:05:47.023 Let's make this contrast. 00:05:52.470 --> 00:05:55.610 First of all, if I look at the angle 00:05:55.610 --> 00:05:57.610 that we have been paying attention to right now. 00:05:57.610 --> 00:06:00.580 We've just considered things like this. 00:06:00.580 --> 00:06:02.520 1 over x to the 1/2. 00:06:02.520 --> 00:06:06.600 Which is a lot smaller than 1/x. 00:06:06.600 --> 00:06:10.740 Which is a lot smaller than say 1/x^2. 00:06:10.740 --> 00:06:12.050 Which would be another example. 00:06:12.050 --> 00:06:14.560 This is as x goes to 0. 00:06:19.990 --> 00:06:21.935 So this one's the smallest one. 00:06:21.935 --> 00:06:23.310 This one's the next smallest one. 00:06:23.310 --> 00:06:26.820 And this one is very large. 00:06:26.820 --> 00:06:30.985 On the other hand it goes the other way at infinity. 00:06:36.670 --> 00:06:39.936 As x tends to infinity. 00:06:39.936 --> 00:06:43.140 All right so try to keep that in mind. 00:06:43.140 --> 00:06:48.760 And now I'm going to put a little box around the bad guys 00:06:48.760 --> 00:06:50.100 here. 00:06:50.100 --> 00:06:54.710 This one is divergent. 00:06:54.710 --> 00:06:57.870 And this one is divergent. 00:06:57.870 --> 00:06:59.950 And this one is divergent. 00:06:59.950 --> 00:07:01.610 And this one is divergent. 00:07:01.610 --> 00:07:03.510 The crossover point is 1/x. 00:07:03.510 --> 00:07:05.680 When we get smaller than that, we 00:07:05.680 --> 00:07:07.390 get to things which are convergent. 00:07:07.390 --> 00:07:10.590 When we get smaller than it on this other scale, 00:07:10.590 --> 00:07:12.412 it's convergent. 00:07:12.412 --> 00:07:13.995 All right so these guys are divergent. 00:07:20.300 --> 00:07:23.430 So they're associated with divergent integrals. 00:07:23.430 --> 00:07:25.390 The functions themselves are just 00:07:25.390 --> 00:07:27.585 tending towards-- well these tend to infinity, 00:07:27.585 --> 00:07:29.310 and these tend toward 0. 00:07:29.310 --> 00:07:33.830 So I'm not talking about the functions themselves 00:07:33.830 --> 00:07:35.190 but the integrals. 00:07:35.190 --> 00:07:40.040 Now I want to draw this again here, not small enough. 00:07:43.757 --> 00:07:44.840 I want to draw this again. 00:07:48.484 --> 00:07:50.150 And, so I'm just going to draw a picture 00:07:50.150 --> 00:07:51.760 of what it is that I have here. 00:07:51.760 --> 00:07:54.850 But I'm going to combine these two pictures. 00:07:54.850 --> 00:08:01.330 So here's the picture for example of y = 1/x. 00:08:04.714 --> 00:08:06.700 All right. 00:08:06.700 --> 00:08:08.770 That's y y = 1/x. 00:08:08.770 --> 00:08:10.470 And that picture is very balanced. 00:08:10.470 --> 00:08:12.600 It's symmetric on the two ends. 00:08:12.600 --> 00:08:17.660 If I cut it in half then what I get here is two halves. 00:08:17.660 --> 00:08:23.700 And this one has infinite area. 00:08:23.700 --> 00:08:27.730 That corresponds to the integral from 1 to infinity, 00:08:27.730 --> 00:08:30.530 dx / x being infinite. 00:08:30.530 --> 00:08:34.810 And the other piece, which -- this one we calculated last 00:08:34.810 --> 00:08:37.230 time, this is the one that we just calculated over here 00:08:37.230 --> 00:08:42.719 at Example 2 -- has the same property. 00:08:42.719 --> 00:08:45.720 It's infinite. 00:08:45.720 --> 00:08:48.102 And that's the fact that the integral from 0 to 1 of dx 00:08:48.102 --> 00:08:52.100 / x is infinite. 00:08:52.100 --> 00:08:56.570 Right, so both, we lose on both ends. 00:08:56.570 --> 00:09:02.390 On the other hand if I take something like -- 00:09:02.390 --> 00:09:05.480 I'm drawing it the same way but it's really not the same -- 00:09:05.480 --> 00:09:08.490 y = 1 over the square root of x. 00:09:08.490 --> 00:09:11.010 y = 1 / x^(1/2). 00:09:11.010 --> 00:09:17.980 And if I cut that in half here then the x^(1/2) is actually 00:09:17.980 --> 00:09:19.890 bigger than this guy. 00:09:19.890 --> 00:09:21.740 So this piece is infinite. 00:09:26.830 --> 00:09:29.555 And this part over here actually is going 00:09:29.555 --> 00:09:31.310 to give us an honest number. 00:09:31.310 --> 00:09:34.810 In fact this one is finite. 00:09:34.810 --> 00:09:36.550 And we just checked what the number is. 00:09:36.550 --> 00:09:38.340 It actually happens to have area 2. 00:09:46.250 --> 00:09:49.210 And what's happening here is if you would superimpose 00:09:49.210 --> 00:09:51.710 this graph on the other graph what you 00:09:51.710 --> 00:09:54.580 would see is that they cross. 00:09:54.580 --> 00:09:58.970 And this one sits on top. 00:09:58.970 --> 00:10:04.660 So if I drew this one in let's have another color here, 00:10:04.660 --> 00:10:05.840 orange let's say. 00:10:05.840 --> 00:10:08.630 If this were orange if I set it on top here 00:10:08.630 --> 00:10:11.130 it would go this way. 00:10:11.130 --> 00:10:14.790 OK and underneath the orange is still infinite. 00:10:14.790 --> 00:10:16.070 So both of these are infinite. 00:10:16.070 --> 00:10:18.028 On here on the other hand underneath the orange 00:10:18.028 --> 00:10:21.700 is infinite but underneath where the green is is finite. 00:10:21.700 --> 00:10:23.770 That's a smaller quantity. 00:10:23.770 --> 00:10:25.290 Infinity is a lot bigger than 2. 00:10:25.290 --> 00:10:27.826 2 is a lot less than infinity. 00:10:27.826 --> 00:10:30.835 All right so that's reflected in these comparisons here. 00:10:30.835 --> 00:10:33.610 Now if you like if I want to do these in green. 00:10:33.610 --> 00:10:39.900 This guy is good and this guy is good. 00:10:39.900 --> 00:10:42.960 Well let me just repeat that idea over here in this sort 00:10:42.960 --> 00:10:47.930 of reversed picture with y = 1/x^2. 00:10:47.930 --> 00:10:53.640 If I chop that in half then the good part is this end here. 00:10:53.640 --> 00:10:54.440 This is finite. 00:10:56.990 --> 00:10:59.570 And the bad part is this part of here 00:10:59.570 --> 00:11:01.310 which is way more singular. 00:11:01.310 --> 00:11:02.060 And it's infinite. 00:11:07.070 --> 00:11:10.080 All right so again what I've just tried to do 00:11:10.080 --> 00:11:16.710 is to give you some geometric sense and also 00:11:16.710 --> 00:11:18.580 some visceral sense. 00:11:18.580 --> 00:11:23.040 This guy, its tail as it goes out to infinity is much lower. 00:11:23.040 --> 00:11:25.470 It's much smaller than 1/x. 00:11:25.470 --> 00:11:28.080 And these guys trapped an infinite amount of area. 00:11:28.080 --> 00:11:30.110 This one traps only a finite amount of area. 00:11:36.676 --> 00:11:38.860 All right so now I'm just going to give one 00:11:38.860 --> 00:11:43.090 last example which combines these two types of pictures. 00:11:43.090 --> 00:11:45.420 It's really practically the same as what 00:11:45.420 --> 00:11:53.320 I've said before but I-- oh have to erase this one too. 00:12:01.340 --> 00:12:05.770 So here's another example: if you're in-- So let's 00:12:05.770 --> 00:12:08.132 take the following example. 00:12:08.132 --> 00:12:09.840 This is somewhat related to the first one 00:12:09.840 --> 00:12:11.600 that I gave last time. 00:12:11.600 --> 00:12:16.090 If you take a function y = 1/(x-3)^2. 00:12:18.770 --> 00:12:21.160 And you think about its integral. 00:12:21.160 --> 00:12:24.750 So let's think about the integral from 0 to infinity, 00:12:24.750 --> 00:12:27.090 dx / (x-3)^2. 00:12:27.090 --> 00:12:30.420 And suppose you were faced with this integral. 00:12:30.420 --> 00:12:32.630 In order to understand what it's doing 00:12:32.630 --> 00:12:36.310 you have to pay attention to two places where it can go wrong. 00:12:36.310 --> 00:12:39.530 We're going to split into two pieces. 00:12:39.530 --> 00:12:44.349 I'm going say break it up into this one here up to 5, 00:12:44.349 --> 00:12:45.390 for the sake of argument. 00:12:48.090 --> 00:12:49.715 And say from 5 to infinity. 00:12:53.672 --> 00:12:54.960 All right. 00:12:54.960 --> 00:12:56.260 So these are the two chunks. 00:12:56.260 --> 00:12:58.810 Now why did I break it up into those two pieces? 00:12:58.810 --> 00:13:00.670 Because what's happening with this function 00:13:00.670 --> 00:13:05.250 is that it's going up like this at 3. 00:13:05.250 --> 00:13:08.210 And so if I look at the two halves here. 00:13:08.210 --> 00:13:09.710 I'm going to draw them again and I'm 00:13:09.710 --> 00:13:12.560 going to illustrate them with the colors we've chosen, 00:13:12.560 --> 00:13:15.350 which are I guess red and green. 00:13:15.350 --> 00:13:20.660 What you'll discover is that this one 00:13:20.660 --> 00:13:30.540 here, which corresponds to this piece here, is infinite. 00:13:30.540 --> 00:13:32.320 And it's infinite because there's 00:13:32.320 --> 00:13:34.270 a square in the denominator. 00:13:34.270 --> 00:13:39.190 And as x goes to 3 this is very much like if we 00:13:39.190 --> 00:13:40.670 shifted the 3 to 0. 00:13:40.670 --> 00:13:42.560 Very much like this 1/x^2 here. 00:13:42.560 --> 00:13:44.247 But not in this context. 00:13:44.247 --> 00:13:46.330 In the other context where it's going to infinity. 00:13:49.460 --> 00:13:52.360 This is the same as at the picture 00:13:52.360 --> 00:13:57.398 directly above with the infinite part in red. 00:13:57.398 --> 00:13:59.650 All right. 00:13:59.650 --> 00:14:04.915 And this part here, this part is finite. 00:14:04.915 --> 00:14:06.240 All right. 00:14:06.240 --> 00:14:08.960 So since we have an infinite part plus a finite part 00:14:08.960 --> 00:14:15.070 the conclusion is that this thing, well this guy converges. 00:14:15.070 --> 00:14:18.850 And this one diverges. 00:14:21.470 --> 00:14:23.930 But the total unfortunately diverges. 00:14:23.930 --> 00:14:25.760 Right, because it's got one infinity in it. 00:14:25.760 --> 00:14:28.370 So this thing diverges. 00:14:31.964 --> 00:14:33.380 And that's what happened last time 00:14:33.380 --> 00:14:34.870 when we got a crazy number. 00:14:34.870 --> 00:14:37.480 If you integrated this you would get some negative number. 00:14:37.480 --> 00:14:39.850 If you wrote down the formulas carelessly. 00:14:39.850 --> 00:14:42.250 And the reason is that the calculation actually 00:14:42.250 --> 00:14:44.710 is nonsense. 00:14:44.710 --> 00:14:47.890 So you've gotta be aware, if you encounter 00:14:47.890 --> 00:14:51.600 a singularity in the middle, not to ignore it. 00:14:51.600 --> 00:14:52.100 Yeah. 00:14:52.100 --> 00:14:52.620 Question. 00:14:52.620 --> 00:14:53.786 AUDIENCE: [INAUDIBLE PHRASE] 00:14:56.430 --> 00:14:59.710 PROFESSOR: Why do we say that the whole thing diverges? 00:14:59.710 --> 00:15:02.490 The reason why we say that is the area under the whole curve 00:15:02.490 --> 00:15:03.690 is infinite. 00:15:03.690 --> 00:15:06.130 It's the sum of this piece plus this piece. 00:15:06.130 --> 00:15:08.218 And so the total is infinite. 00:15:08.218 --> 00:15:09.384 AUDIENCE: [INAUDIBLE PHRASE] 00:15:17.032 --> 00:15:17.990 PROFESSOR: We're stuck. 00:15:17.990 --> 00:15:19.323 This is an ill-defined integral. 00:15:19.323 --> 00:15:22.092 It's one where your red flashing warning sign should be on. 00:15:22.092 --> 00:15:23.800 Because you're not going to get the right 00:15:23.800 --> 00:15:24.758 answer by computing it. 00:15:24.758 --> 00:15:26.860 You'll never get an answer. 00:15:26.860 --> 00:15:29.230 Similarly you'll never get an answer with this. 00:15:29.230 --> 00:15:32.700 And you will get an answer with that. 00:15:32.700 --> 00:15:33.200 OK? 00:15:37.220 --> 00:15:38.944 Yeah another question. 00:15:38.944 --> 00:15:40.110 AUDIENCE: [INAUDIBLE PHRASE] 00:15:45.760 --> 00:15:47.690 PROFESSOR: So the question is, if you 00:15:47.690 --> 00:15:50.010 have a little glance at an integral, 00:15:50.010 --> 00:15:54.750 how are you going to decide where you should be heading? 00:15:54.750 --> 00:15:58.280 So I'm going to answer that orally. 00:15:58.280 --> 00:16:04.010 Although you know, but I'll say one little hint here. 00:16:04.010 --> 00:16:08.070 So you always have to check x going to infinity and x 00:16:08.070 --> 00:16:10.740 going to minus infinity, if they're in there. 00:16:10.740 --> 00:16:15.460 And you also have to check any singularity, like x going to 3 00:16:15.460 --> 00:16:16.419 for sure in this case. 00:16:16.419 --> 00:16:18.210 You have to pay attention to all the places 00:16:18.210 --> 00:16:19.376 where the thing is infinite. 00:16:19.376 --> 00:16:22.330 And then you want to focus in on each one separately. 00:16:22.330 --> 00:16:26.920 And decide what's going on it at that particular place. 00:16:26.920 --> 00:16:34.530 When it's a negative power-- So remember dx / x as x goes to 0 00:16:34.530 --> 00:16:36.980 is bad. 00:16:36.980 --> 00:16:39.510 And dx / x^2 is bad. 00:16:39.510 --> 00:16:40.490 dx / x^3 is bad. 00:16:40.490 --> 00:16:42.600 All of them are even worse. 00:16:42.600 --> 00:16:49.650 So anything of this form is bad: n = 1, 2, 3. 00:16:49.650 --> 00:16:51.980 These are the red box kinds. 00:16:51.980 --> 00:16:55.030 All right. 00:16:55.030 --> 00:16:56.940 That means that any of the integrals 00:16:56.940 --> 00:16:59.990 that we did in partial fractions which 00:16:59.990 --> 00:17:02.080 had a root, which had a factor of something 00:17:02.080 --> 00:17:03.160 in the denominator. 00:17:03.160 --> 00:17:04.790 Those are all divergent integrals 00:17:04.790 --> 00:17:06.670 if you cross the singularly. 00:17:06.670 --> 00:17:09.357 Not a single one of them makes sense across the singularity. 00:17:09.357 --> 00:17:09.856 Right? 00:17:12.692 --> 00:17:14.150 If you have square roots and things 00:17:14.150 --> 00:17:16.108 like that then you can repair things like that. 00:17:16.108 --> 00:17:18.030 And there's some interesting examples of that. 00:17:18.030 --> 00:17:21.080 Such as with the arcsine function and so forth. 00:17:21.080 --> 00:17:25.940 Where you have an improper integral which is really OK. 00:17:25.940 --> 00:17:26.730 All right. 00:17:26.730 --> 00:17:29.880 So that's the best I can do. 00:17:29.880 --> 00:17:32.260 It's obviously something you get experience with. 00:17:32.260 --> 00:17:34.110 All right. 00:17:34.110 --> 00:17:39.350 Now I'm going to move on and this is more or less 00:17:39.350 --> 00:17:42.150 our last topic. 00:17:42.150 --> 00:17:43.900 Yay, but not quite. 00:17:43.900 --> 00:17:46.730 Well, so I should say it's our penultimate topic. 00:17:46.730 --> 00:17:49.430 Right because we have one more lecture. 00:17:49.430 --> 00:17:52.010 All right. 00:17:52.010 --> 00:17:54.750 So that our next topic is series. 00:17:54.750 --> 00:17:58.810 Now we'll do it in a sort of a concrete way today. 00:17:58.810 --> 00:18:02.775 And then we'll do what are known as power series tomorrow. 00:18:05.400 --> 00:18:06.760 So let me tell you about series. 00:18:20.490 --> 00:18:22.604 Remember we're talking about infinity 00:18:22.604 --> 00:18:23.687 and dealing with infinity. 00:18:26.772 --> 00:18:28.730 So we're not just talking about any old series. 00:18:28.730 --> 00:18:30.230 We're talking about infinite series. 00:18:32.650 --> 00:18:37.400 There is one infinite series which is probably, 00:18:37.400 --> 00:18:39.000 which is without question the most 00:18:39.000 --> 00:18:41.980 important and useful series. 00:18:41.980 --> 00:18:44.690 And that's the geometric series but I'm 00:18:44.690 --> 00:18:48.405 going to introduce it concretely first in a particular case. 00:18:51.990 --> 00:18:54.260 If I draw a picture of this sum. 00:18:54.260 --> 00:18:56.480 Which in principle goes on forever. 00:18:56.480 --> 00:18:59.670 You can see that it goes someplace fairly easily 00:18:59.670 --> 00:19:02.730 by marking out what's happening on the number line. 00:19:02.730 --> 00:19:06.630 The first step takes us to 1 from 0. 00:19:06.630 --> 00:19:11.790 And then if I add this half, I get to 3/2. 00:19:11.790 --> 00:19:16.240 Right, so the first step was 1 and the second step was 1/2. 00:19:16.240 --> 00:19:20.940 Now if I add this quarter in, which is the next piece then 00:19:20.940 --> 00:19:22.400 I get some place here. 00:19:22.400 --> 00:19:28.440 But what I want to observe is that I got, 00:19:28.440 --> 00:19:31.130 I can look at it from the other point of view. 00:19:31.130 --> 00:19:36.240 I got, when I move this quarter I got half way to 2 here. 00:19:36.240 --> 00:19:39.420 I'm putting 2 in green because I want you to think 00:19:39.420 --> 00:19:42.950 of it as being the good kind. 00:19:42.950 --> 00:19:43.970 Right. 00:19:43.970 --> 00:19:45.420 The kind that has a number. 00:19:45.420 --> 00:19:47.280 And not one of the red kinds. 00:19:47.280 --> 00:19:49.920 We're getting there and we're almost there. 00:19:49.920 --> 00:19:52.620 So the next stage we get half way again. 00:19:52.620 --> 00:19:54.605 That's the eighth and so forth. 00:19:54.605 --> 00:19:56.490 And eventually we get to 2. 00:19:56.490 --> 00:20:00.476 So this sum we write equals two. 00:20:00.476 --> 00:20:02.590 All right that's kind of a paradox 00:20:02.590 --> 00:20:04.130 because we never get to 2. 00:20:04.130 --> 00:20:08.080 This is the paradox that Zeno fussed with. 00:20:08.080 --> 00:20:12.807 And his conclusion, you know, with the rabbit and the hare. 00:20:12.807 --> 00:20:14.140 No, the rabbit and the tortoise. 00:20:14.140 --> 00:20:18.890 Sorry hare chasing-- anyway, the rabbit chasing the tortoise. 00:20:18.890 --> 00:20:21.050 His conclusion-- you know, I don't 00:20:21.050 --> 00:20:23.770 know if you're aware of this, but he understood this paradox. 00:20:23.770 --> 00:20:25.811 And he said you know it doesn't look like it ever 00:20:25.811 --> 00:20:29.180 gets there because they're infinitely many times 00:20:29.180 --> 00:20:32.330 between the time-- you know that the tortoise is always behind, 00:20:32.330 --> 00:20:34.580 always behind, always behind, always behind. 00:20:34.580 --> 00:20:36.970 So therefore it's impossible that the tortoise 00:20:36.970 --> 00:20:38.710 catches up right. 00:20:38.710 --> 00:20:41.600 So do you know what his conclusion was? 00:20:41.600 --> 00:20:45.280 Time does not exist. 00:20:45.280 --> 00:20:48.007 That was actually literally his conclusion. 00:20:48.007 --> 00:20:49.840 Because he didn't understand the possibility 00:20:49.840 --> 00:20:50.799 of a continuum of time. 00:20:50.799 --> 00:20:53.090 Because there were infinitely many things that happened 00:20:53.090 --> 00:20:56.610 before the tortoise caught up. 00:20:56.610 --> 00:20:57.790 So that was the reasoning. 00:20:57.790 --> 00:21:00.430 I mean it's a long time ago but you know people didn't-- he 00:21:00.430 --> 00:21:02.430 didn't believe in continuum. 00:21:02.430 --> 00:21:03.480 All right. 00:21:03.480 --> 00:21:06.940 So anyway that's a small point. 00:21:06.940 --> 00:21:19.410 Now the general case here of a geometric series 00:21:19.410 --> 00:21:22.820 is where I put in a number a instead of 1/2 here. 00:21:22.820 --> 00:21:23.870 So what we had before. 00:21:23.870 --> 00:21:26.890 So that's 1 + a + a^2... 00:21:26.890 --> 00:21:31.900 Isn't quite the most general but anyway I'll write this down. 00:21:31.900 --> 00:21:34.630 And you're certainly going to want to remember that 00:21:34.630 --> 00:21:39.510 the formula for this in the limit is 1/(1-a). 00:21:39.510 --> 00:21:44.000 And I remind you that this only works when the absolute value 00:21:44.000 --> 00:21:45.810 is strictly less than 1. 00:21:45.810 --> 00:21:47.970 In other words when -1 is strictly less than a 00:21:47.970 --> 00:21:51.130 is less than 1. 00:21:51.130 --> 00:21:53.090 And that's really the issue that we're going 00:21:53.090 --> 00:21:54.310 to want to worry about now. 00:21:54.310 --> 00:21:58.270 What we're worrying about is this notion of convergence. 00:21:58.270 --> 00:22:05.300 And what goes wrong when there isn't convergence, 00:22:05.300 --> 00:22:07.360 when there's a divergence. 00:22:07.360 --> 00:22:13.220 So let me illustrate the divergences before going on. 00:22:13.220 --> 00:22:15.900 And this is what we have to avoid if we're 00:22:15.900 --> 00:22:18.990 going to understand series. 00:22:18.990 --> 00:22:21.890 So here's an example when a = 1. 00:22:21.890 --> 00:22:26.620 You get 1 + 1 + 1 plus et cetera. 00:22:26.620 --> 00:22:29.990 And that's equal to 1/(1-1). 00:22:29.990 --> 00:22:32.370 Which is 1 over 0. 00:22:32.370 --> 00:22:33.760 So this is not bad. 00:22:33.760 --> 00:22:34.861 It's almost right. 00:22:34.861 --> 00:22:35.360 Right? 00:22:35.360 --> 00:22:37.750 It's sort of infinity equals infinity. 00:22:37.750 --> 00:22:39.850 At the edge here we managed to get something 00:22:39.850 --> 00:22:42.340 which is sort of almost right. 00:22:42.340 --> 00:22:46.100 But you know, it's, we don't consider this to be logically 00:22:46.100 --> 00:22:47.540 to make complete sense. 00:22:47.540 --> 00:22:51.200 So it's a little dangerous. 00:22:51.200 --> 00:22:52.960 And so we just say that it diverges. 00:22:52.960 --> 00:22:54.100 And we get rid of this. 00:22:54.100 --> 00:22:55.990 So we're still putting it in red. 00:22:55.990 --> 00:22:58.540 All right. 00:22:58.540 --> 00:22:59.975 The bad guy here. 00:22:59.975 --> 00:23:00.850 So this one diverges. 00:23:04.670 --> 00:23:12.650 Similarly if I take a equals -1, I get 1 - 1 + 1 - 1 + 1... 00:23:12.650 --> 00:23:15.450 Because the odd and the even powers in that formula 00:23:15.450 --> 00:23:17.270 alternate sign. 00:23:17.270 --> 00:23:19.750 And this bounces back and forth. 00:23:19.750 --> 00:23:21.640 It never settles down. 00:23:21.640 --> 00:23:23.346 It starts at 1. 00:23:23.346 --> 00:23:25.720 And then it gets down to 0 and then it goes back up to 1, 00:23:25.720 --> 00:23:28.400 down to 0, back up to 1. 00:23:28.400 --> 00:23:29.517 It doesn't settle down. 00:23:29.517 --> 00:23:30.600 It bounces back and forth. 00:23:30.600 --> 00:23:31.580 It oscillates. 00:23:31.580 --> 00:23:34.740 On the other hand if you compare the right hand side. 00:23:34.740 --> 00:23:35.980 What's the right hand side? 00:23:35.980 --> 00:23:36.970 It's 1 / (1-(-1)). 00:23:39.730 --> 00:23:41.515 Which is 1/2. 00:23:41.515 --> 00:23:42.460 All right. 00:23:42.460 --> 00:23:45.169 So if you just paid attention to the formula, which 00:23:45.169 --> 00:23:47.710 is what we were doing when we integrated without thinking too 00:23:47.710 --> 00:23:50.070 hard about this, you get a number 00:23:50.070 --> 00:23:51.320 here but in fact that's wrong. 00:23:51.320 --> 00:23:53.153 Actually it's kind of an interesting number. 00:23:53.153 --> 00:23:56.380 It's halfway between the two, between 0 and 1. 00:23:56.380 --> 00:23:58.505 So again there's some sort of vague sense 00:23:58.505 --> 00:24:01.966 in which this is trying to be this answer. 00:24:01.966 --> 00:24:04.580 All right. 00:24:04.580 --> 00:24:08.744 It's not so bad but we're still going to put this in a red box. 00:24:08.744 --> 00:24:10.030 All right. 00:24:10.030 --> 00:24:12.710 because this is what we called divergence. 00:24:12.710 --> 00:24:16.130 So both of these cases are divergent. 00:24:16.130 --> 00:24:20.490 It only really works when alpha-- when a is less than 1. 00:24:20.490 --> 00:24:23.480 I'm going to add one more case just 00:24:23.480 --> 00:24:30.040 to see that mathematicians are slightly curious about what 00:24:30.040 --> 00:24:32.020 goes on in other cases. 00:24:32.020 --> 00:24:37.570 So this is 1 + 2 + 2^2 + 2^3 plus etc.. 00:24:37.570 --> 00:24:41.980 And that should be equal to -- according to this formula -- 00:24:41.980 --> 00:24:44.750 1/(1-2). 00:24:44.750 --> 00:24:48.242 Which is -1. 00:24:48.242 --> 00:24:49.860 All right. 00:24:49.860 --> 00:24:53.460 Now this one is clearly wrong, right? 00:24:53.460 --> 00:24:55.530 This one is totally wrong. 00:24:58.170 --> 00:24:59.465 It certainly diverges. 00:24:59.465 --> 00:25:02.370 The left hand side is obviously infinite. 00:25:02.370 --> 00:25:04.070 The right hand side is way off. 00:25:04.070 --> 00:25:05.960 It's -1. 00:25:05.960 --> 00:25:10.350 On the other hand it turns out actually 00:25:10.350 --> 00:25:13.360 that mathematicians have ways of making sense out of these. 00:25:13.360 --> 00:25:15.590 In number theory there's a strange system 00:25:15.590 --> 00:25:18.160 where this is actually true. 00:25:18.160 --> 00:25:21.530 And what happens in that system is 00:25:21.530 --> 00:25:24.030 that what you have to throw out is the idea 00:25:24.030 --> 00:25:27.050 that 0 is less than 1. 00:25:27.050 --> 00:25:29.980 There is no such thing as negative numbers. 00:25:29.980 --> 00:25:32.090 So this number exists. 00:25:32.090 --> 00:25:35.700 And it's the additive inverse of 1. 00:25:35.700 --> 00:25:39.780 It has this arithmetic property but the statement 00:25:39.780 --> 00:25:43.330 that this is, that 1 is bigger than 0 does not make sense. 00:25:43.330 --> 00:25:45.440 So you have your choice, either this diverges 00:25:45.440 --> 00:25:48.630 or you have to throw out something like this. 00:25:48.630 --> 00:25:51.510 So that's a very curious thing in higher mathematics. 00:25:51.510 --> 00:25:56.410 Which if you get to number theory there's fun stuff there. 00:25:56.410 --> 00:25:58.920 All right. 00:25:58.920 --> 00:26:02.740 OK but for our purposes these things are all out. 00:26:02.740 --> 00:26:03.300 All right. 00:26:03.300 --> 00:26:04.010 They're gone. 00:26:04.010 --> 00:26:05.160 We're not considering them. 00:26:05.160 --> 00:26:09.550 Only a between -1 and 1. 00:26:09.550 --> 00:26:10.070 All right. 00:26:13.910 --> 00:26:18.190 Now I want to do something systematic. 00:26:18.190 --> 00:26:21.530 And it's more or less on the lines of the powers 00:26:21.530 --> 00:26:23.090 that I'm erasing right now. 00:26:26.630 --> 00:26:28.590 I want to tell you about series which are 00:26:28.590 --> 00:26:30.420 kind of borderline convergent. 00:26:30.420 --> 00:26:33.720 And then next time when we talk about powers series we'll 00:26:33.720 --> 00:26:35.810 come back to this very important series which 00:26:35.810 --> 00:26:37.080 is the most important one. 00:26:40.680 --> 00:26:47.400 So now let's talk about some series-- er, general notations. 00:26:47.400 --> 00:26:49.970 And this will help you with the last bit. 00:26:53.810 --> 00:26:56.760 This is going to be pretty much the same as what 00:26:56.760 --> 00:27:00.890 we did for improper integrals. 00:27:00.890 --> 00:27:04.320 Namely, first of all I'm going to have S_N which 00:27:04.320 --> 00:27:09.760 is the sum of a_n, n equals 0 to capital N. 00:27:09.760 --> 00:27:12.220 And this is what we're calling a partial sum. 00:27:18.050 --> 00:27:23.980 And then the full limit, which is capital S, if you like. 00:27:23.980 --> 00:27:29.980 a_n, n equals 0 to infinity, is just the limit as N goes 00:27:29.980 --> 00:27:32.010 to infinity of the S_N's. 00:27:36.240 --> 00:27:39.770 And then we have the same kind of notation that we had before. 00:27:39.770 --> 00:27:42.860 Which is there are these two choices which 00:27:42.860 --> 00:27:45.890 is that if the limit exists. 00:27:50.450 --> 00:27:51.830 That's the green choice. 00:27:51.830 --> 00:27:54.460 And we say it converges. 00:27:54.460 --> 00:28:00.830 So we say the series converges. 00:28:00.830 --> 00:28:06.616 And then the other case which is the limit does not exist. 00:28:10.782 --> 00:28:12.240 And we can say the series diverges. 00:28:20.560 --> 00:28:21.343 Question. 00:28:21.343 --> 00:28:22.509 AUDIENCE: [INAUDIBLE PHRASE] 00:28:26.480 --> 00:28:29.290 PROFESSOR: The question was how did I get to this? 00:28:29.290 --> 00:28:31.840 And I will do that next time but in fact of course 00:28:31.840 --> 00:28:33.240 you've seen it in high school. 00:28:33.240 --> 00:28:35.930 Right this is-- Yeah. 00:28:35.930 --> 00:28:36.860 Yeah. 00:28:36.860 --> 00:28:40.000 We'll do that next time. 00:28:40.000 --> 00:28:42.222 The question was how did we arrive-- sorry I 00:28:42.222 --> 00:28:43.430 didn't tell you the question. 00:28:43.430 --> 00:28:44.360 The question was how do we arrive 00:28:44.360 --> 00:28:46.590 at this formula on the right hand side here. 00:28:46.590 --> 00:28:48.240 But we'll talk about that next time. 00:28:53.060 --> 00:28:54.020 All right. 00:28:54.020 --> 00:28:59.930 So here's the basic definition and what 00:28:59.930 --> 00:29:02.430 we're going to recognize about series. 00:29:02.430 --> 00:29:06.835 And I'm going to give you a few examples 00:29:06.835 --> 00:29:08.460 and then we'll do something systematic. 00:29:12.070 --> 00:29:14.495 So the first example-- well the first example 00:29:14.495 --> 00:29:16.290 is the geometric series. 00:29:16.290 --> 00:29:19.430 But the first example that I'm going to discuss now 00:29:19.430 --> 00:29:23.352 and in a little bit of detail is this sum 1/n^2, 00:29:23.352 --> 00:29:24.310 n equals 1 to infinity. 00:29:28.580 --> 00:29:34.050 It turns out that this series is very analogous -- 00:29:34.050 --> 00:29:36.690 and we'll develop this analogy carefully -- 00:29:36.690 --> 00:29:41.065 the integral from 1 to x, dx / x^2. 00:29:41.065 --> 00:29:46.220 And we're going to develop this analogy in detail later 00:29:46.220 --> 00:29:47.920 in this lecture. 00:29:47.920 --> 00:29:49.770 And this one is one of the ones-- 00:29:49.770 --> 00:29:51.890 so now you have to go back and actually remember, 00:29:51.890 --> 00:29:54.335 this is one of the ones you really want to memorize. 00:29:54.335 --> 00:29:56.460 And you should especially pay attention to the ones 00:29:56.460 --> 00:29:58.507 with an infinity in them. 00:29:58.507 --> 00:29:59.465 This one is convergent. 00:30:03.270 --> 00:30:04.520 And this series is convergent. 00:30:04.520 --> 00:30:11.070 Now it turns out that evaluating this is very easy. 00:30:11.070 --> 00:30:12.820 This is 1. 00:30:12.820 --> 00:30:15.310 It's easy to calculate. 00:30:15.310 --> 00:30:19.500 Evaluating this is very tricky. 00:30:19.500 --> 00:30:21.610 And Euler did it. 00:30:21.610 --> 00:30:26.440 And the answer is pi^2 / 6. 00:30:26.440 --> 00:30:29.050 That's an amazing calculation. 00:30:29.050 --> 00:30:33.570 And it was done very early in the history of mathematics. 00:30:33.570 --> 00:30:38.110 If you look at another example-- so maybe example two here, 00:30:38.110 --> 00:30:45.790 if you look at 1/n^3, n equals-- well you can't start here at 0 00:30:45.790 --> 00:30:46.580 by the way. 00:30:46.580 --> 00:30:48.930 I get to start wherever I want in these series. 00:30:48.930 --> 00:30:49.920 Here I start with 0. 00:30:49.920 --> 00:30:51.220 Here I started with 1. 00:30:51.220 --> 00:30:52.880 And notice the reason why I started-- 00:30:52.880 --> 00:30:56.650 it was a bad idea to start with 0 was that 1 over 0 00:30:56.650 --> 00:30:57.860 is undefined. 00:30:57.860 --> 00:30:58.360 Right? 00:30:58.360 --> 00:31:00.443 So I'm just starting where it's convenient for me. 00:31:00.443 --> 00:31:03.560 And since I'm interested mostly in the tail behavior 00:31:03.560 --> 00:31:06.150 it doesn't matter to me so much where I start. 00:31:06.150 --> 00:31:07.800 Although if I want an exact answer 00:31:07.800 --> 00:31:10.165 I need to start exactly at n = 1. 00:31:10.165 --> 00:31:10.890 All right. 00:31:10.890 --> 00:31:17.932 This one is similar to this integral here. 00:31:17.932 --> 00:31:18.700 All right. 00:31:18.700 --> 00:31:20.440 Which is convergent again. 00:31:20.440 --> 00:31:22.390 So there's a number that you get. 00:31:22.390 --> 00:31:27.180 And let's see what is it something like 2/3 or something 00:31:27.180 --> 00:31:30.190 like that, all right, for this for this number. 00:31:30.190 --> 00:31:32.440 Or 1/3. 00:31:32.440 --> 00:31:33.110 What is it? 00:31:33.110 --> 00:31:33.690 No 1/2. 00:31:33.690 --> 00:31:35.000 I guess it's 1/2. 00:31:35.000 --> 00:31:37.110 This one is 1/2. 00:31:37.110 --> 00:31:38.780 You check that, I'm not positive, 00:31:38.780 --> 00:31:40.600 but anyway just doing it in my head quickly 00:31:40.600 --> 00:31:42.150 it seems to be 1/2. 00:31:42.150 --> 00:31:44.100 Anyway it's an easy number to calculate. 00:31:44.100 --> 00:31:49.710 This one over here stumped mathematicians basically 00:31:49.710 --> 00:31:51.870 for all time. 00:31:51.870 --> 00:31:56.250 It doesn't have any kind of elementary form like this. 00:31:56.250 --> 00:31:59.842 And it was only very recently proved to be rational. 00:31:59.842 --> 00:32:01.550 People couldn't even couldn't even decide 00:32:01.550 --> 00:32:05.260 whether this was a rational number or not. 00:32:05.260 --> 00:32:07.830 But anyway that's been resolved; it is an irrational number 00:32:07.830 --> 00:32:09.540 which is what people suspected. 00:32:09.540 --> 00:32:10.650 Yeah question. 00:32:10.650 --> 00:32:11.816 AUDIENCE: [INAUDIBLE PHRASE] 00:32:14.360 --> 00:32:16.220 PROFESSOR: Yeah sorry. 00:32:16.220 --> 00:32:16.720 OK. 00:32:19.820 --> 00:32:25.070 I violated a rule of mathematics-- 00:32:25.070 --> 00:32:26.720 you said why is this similar? 00:32:26.720 --> 00:32:29.140 I thought that similar was something else. 00:32:29.140 --> 00:32:30.380 And you're absolutely right. 00:32:30.380 --> 00:32:33.520 And I violated a rule of mathematics. 00:32:33.520 --> 00:32:37.825 Which is that I used this symbol for two different things. 00:32:41.196 --> 00:32:42.820 I should have written this symbol here. 00:32:42.820 --> 00:32:43.530 All right. 00:32:43.530 --> 00:32:45.260 I'll create a new symbol here. 00:32:45.260 --> 00:32:48.920 The question of whether this converges or this converges. 00:32:48.920 --> 00:32:51.920 These are the the same type of question. 00:32:51.920 --> 00:32:54.090 And we'll see why they're the same question it 00:32:54.090 --> 00:32:55.000 in a few minutes. 00:32:55.000 --> 00:32:58.190 But in fact the wiggle I used, "similar", 00:32:58.190 --> 00:33:02.360 I used for the connection between functions. 00:33:02.360 --> 00:33:09.070 The things that are really similar are that 1/n resembles 00:33:09.070 --> 00:33:10.230 1/x^2. 00:33:10.230 --> 00:33:12.066 So I apologize I didn't-- 00:33:12.066 --> 00:33:13.232 AUDIENCE: [INAUDIBLE PHRASE] 00:33:15.890 --> 00:33:17.890 PROFESSOR: Oh you thought that this 00:33:17.890 --> 00:33:19.210 was the definition of that. 00:33:19.210 --> 00:33:21.460 That's actually the reason why these things correspond 00:33:21.460 --> 00:33:21.990 so closely. 00:33:21.990 --> 00:33:25.560 That is that the Riemann sum is close to this. 00:33:25.560 --> 00:33:27.520 But that doesn't mean they're equal. 00:33:27.520 --> 00:33:31.245 The Riemann sum only works when the delta x goes to 0. 00:33:31.245 --> 00:33:33.870 The way that we're going to get a connection between these two, 00:33:33.870 --> 00:33:38.710 as we will just a second, is with a Riemann sum with-- What 00:33:38.710 --> 00:33:47.180 we're going to use is a Riemann sum with delta x = 1. 00:33:47.180 --> 00:33:49.617 All right and then that will be the connection between. 00:33:49.617 --> 00:33:53.240 All right that's absolutely right. 00:33:53.240 --> 00:33:53.740 All right. 00:33:57.020 --> 00:34:00.280 So in order to illustrate exactly this idea 00:34:00.280 --> 00:34:02.110 that you've just come up with, and in fact 00:34:02.110 --> 00:34:04.490 that we're going to use, we'll do the same thing 00:34:04.490 --> 00:34:07.740 but we're going to do it on the example sum 1/n. 00:34:13.380 --> 00:34:20.170 So here's Example 3 and it's going to be sum 1/n, 00:34:20.170 --> 00:34:21.369 n equals 1 to infinity. 00:34:24.090 --> 00:34:27.040 And what we're now going to see is 00:34:27.040 --> 00:34:29.340 that it corresponds to this integral here. 00:34:32.620 --> 00:34:34.850 And we're going to show therefore 00:34:34.850 --> 00:34:37.870 that this thing diverges. 00:34:37.870 --> 00:34:40.300 But we're going to do this more carefully. 00:34:40.300 --> 00:34:42.750 We're going to do this in some detail 00:34:42.750 --> 00:34:45.880 so that you see what it is, that the correspondence is 00:34:45.880 --> 00:34:47.390 between these quantities. 00:34:47.390 --> 00:34:49.890 And the same sort of reasoning applies 00:34:49.890 --> 00:34:51.275 to these other examples. 00:34:55.820 --> 00:34:59.180 So here we go. 00:34:59.180 --> 00:35:04.290 I'm going to take the integral and draw 00:35:04.290 --> 00:35:07.220 the picture of the Riemann sum. 00:35:07.220 --> 00:35:13.840 So here's the level 1 and here's the function y = 1/x. 00:35:13.840 --> 00:35:15.730 And I'm going to take the Riemann sum. 00:35:21.980 --> 00:35:25.200 With delta x = 1. 00:35:25.200 --> 00:35:28.820 And that's going to be closely connected to the series 00:35:28.820 --> 00:35:29.570 that I have. 00:35:32.160 --> 00:35:35.670 But now I have to decide whether I want a lower Riemann 00:35:35.670 --> 00:35:37.797 sum or an upper Riemann sum. 00:35:37.797 --> 00:35:39.630 And actually I'm going to check both of them 00:35:39.630 --> 00:35:41.213 because both of them are illuminating. 00:35:44.620 --> 00:35:47.000 First we'll do the upper Riemann's sum. 00:35:47.000 --> 00:35:48.700 Now that's this staircase here. 00:35:51.780 --> 00:35:54.600 So we'll call this the upper Riemann's sum. 00:35:58.130 --> 00:35:59.710 And let's check what its levels are. 00:35:59.710 --> 00:36:01.560 This is not to scale. 00:36:01.560 --> 00:36:03.250 This level should be 1/2. 00:36:03.250 --> 00:36:05.245 So if this is 1 and this is 2 and that level 00:36:05.245 --> 00:36:10.340 was supposed to be 1/2 and this next level should be 1/3. 00:36:10.340 --> 00:36:12.830 That's how the Riemann sums are working out. 00:36:17.040 --> 00:36:21.770 And now I have the following phenomenon. 00:36:21.770 --> 00:36:24.640 Let's cut it off at the nth stage. 00:36:24.640 --> 00:36:27.095 So that means that I'm going, the integral 00:36:27.095 --> 00:36:30.760 is from 1 to n, dx / x. 00:36:30.760 --> 00:36:33.690 And the Riemann sum is something that's bigger than it. 00:36:33.690 --> 00:36:40.300 Because the areas are enclosing the area of the curved region. 00:36:40.300 --> 00:36:43.510 And that's going to be the area of the first box which 00:36:43.510 --> 00:36:50.330 is 1, plus the area of the second box which is 1/2, 00:36:50.330 --> 00:36:54.150 plus the area of the third box which is 1/3. 00:36:54.150 --> 00:37:00.840 All the way up the last one, but the last one starts at N-1. 00:37:00.840 --> 00:37:03.450 So it has 1/(N-1). 00:37:03.450 --> 00:37:05.780 There are not N boxes here. 00:37:05.780 --> 00:37:07.960 There are only N-1 boxes. 00:37:07.960 --> 00:37:11.830 Because the distance between 1 and N is N-1. 00:37:11.830 --> 00:37:13.555 Right so this is N-1 terms. 00:37:17.330 --> 00:37:25.140 However, if I use the notation for partial sum. 00:37:25.140 --> 00:37:33.367 Which is 1 + 1/2 plus all the way up to 1/(n-1) 1 + 1/n. 00:37:33.367 --> 00:37:35.200 In other words I go out to the Nth one which 00:37:35.200 --> 00:37:37.370 is what I would ordinarily do. 00:37:37.370 --> 00:37:42.900 Then this sum that I have here certainly is less than S_N. 00:37:42.900 --> 00:37:47.850 Because there's one more term there. 00:37:47.850 --> 00:37:50.430 And so here I have an integral which 00:37:50.430 --> 00:37:53.570 is underneath this sum S_N. 00:38:01.700 --> 00:38:19.031 Now this is going to allow us to prove conclusively 00:38:19.031 --> 00:38:21.156 that the-- So I'm just going to rewrite this, prove 00:38:21.156 --> 00:38:22.614 conclusively that the sum diverges. 00:38:22.614 --> 00:38:23.340 Why is that? 00:38:23.340 --> 00:38:26.000 Because this term here we can calculate. 00:38:26.000 --> 00:38:29.470 This is log x evaluated at 1 and n. 00:38:29.470 --> 00:38:34.850 Which is the same thing as log N minus 0. 00:38:34.850 --> 00:38:39.270 All right, the quantity log N - log 1 which is 0. 00:38:39.270 --> 00:38:46.892 And so what we have here is that log N is less than S_N. 00:38:46.892 --> 00:38:51.455 All right and clearly this goes to infinity right. 00:38:51.455 --> 00:38:57.830 As N goes to infinity this thing goes to infinity. 00:38:57.830 --> 00:38:58.435 So we're done. 00:38:58.435 --> 00:39:00.330 All right we've shown divergence. 00:39:08.730 --> 00:39:15.040 Now the way I'm going to use the lower Riemann's sum 00:39:15.040 --> 00:39:20.010 is to recognize that we've captured 00:39:20.010 --> 00:39:21.960 the rate appropriately. 00:39:21.960 --> 00:39:24.400 That is not only do I have a lower bound like this 00:39:24.400 --> 00:39:27.860 but I have an upper bound which is very similar. 00:39:27.860 --> 00:39:30.440 So if I use the upper Riemann-- oh sorry, 00:39:30.440 --> 00:39:39.540 lower Riemann sum again with delta x = 1. 00:39:43.760 --> 00:39:53.946 Then I have that the integral from 1 to n of dx 00:39:53.946 --> 00:39:57.970 / x is bigger than-- Well what are the terms going 00:39:57.970 --> 00:40:00.640 to be if fit them underneath? 00:40:00.640 --> 00:40:03.210 If I fit them underneath I'm missing the first term. 00:40:03.210 --> 00:40:05.600 That is the box is going to be half height. 00:40:05.600 --> 00:40:08.300 It's going to be this lower piece. 00:40:08.300 --> 00:40:10.480 So I'm missing this first term. 00:40:10.480 --> 00:40:16.765 So it'll be a 1/2 + 1/3 plus... 00:40:16.765 --> 00:40:18.150 All right, it will keep on going. 00:40:18.150 --> 00:40:22.150 But now the last one instead of being 1/(N-1), 00:40:22.150 --> 00:40:25.850 it's going to be 1 over N. This is again a total of the N-1 00:40:25.850 --> 00:40:27.050 terms. 00:40:27.050 --> 00:40:28.430 This is the lower Riemann sum. 00:40:31.190 --> 00:40:38.808 And now we can recognize that this is exactly equal to-- well 00:40:38.808 --> 00:40:40.558 so I'll put it over here-- this is exactly 00:40:40.558 --> 00:40:43.520 equal to S_N minus 1, minus the first term. 00:40:43.520 --> 00:40:47.110 So we missed the first term but we got all the rest of them. 00:40:47.110 --> 00:40:49.360 So if I put this to the other side 00:40:49.360 --> 00:40:53.746 remember this is log N. All right. 00:40:53.746 --> 00:40:55.120 If I put this to the other side I 00:40:55.120 --> 00:40:57.130 have the other side of this bound. 00:40:57.130 --> 00:41:07.010 I have that S S_N is less than, if I reverse it, log N + 1. 00:41:07.010 --> 00:41:09.040 And so I've trapped it on the other side. 00:41:09.040 --> 00:41:10.950 And here I have the lower bound. 00:41:10.950 --> 00:41:13.170 So I'm going to combine those together. 00:41:13.170 --> 00:41:18.030 So all told I have this correspondence here. 00:41:18.030 --> 00:41:23.150 It is the size of log N is trapped between the-- sorry, 00:41:23.150 --> 00:41:26.150 the size of S_N, which is relatively hard to calculate 00:41:26.150 --> 00:41:30.780 and understand exactly, is trapped between log N 00:41:30.780 --> 00:41:34.260 and log N + 1. 00:41:34.260 --> 00:41:35.160 Yeah question. 00:41:35.160 --> 00:41:36.326 AUDIENCE: [INAUDIBLE PHRASE] 00:41:46.039 --> 00:41:47.580 PROFESSOR: This step here is the step 00:41:47.580 --> 00:41:49.550 that you're concerned about. 00:41:49.550 --> 00:41:54.160 So this step is a geometric argument which 00:41:54.160 --> 00:41:57.070 is analogous to this step. 00:41:57.070 --> 00:42:01.320 All right it's the same type of argument. 00:42:01.320 --> 00:42:04.720 And in this case it's that the rectangles are on top 00:42:04.720 --> 00:42:07.270 and so the area represented on the right hand side 00:42:07.270 --> 00:42:09.840 is less than the area represented on this side. 00:42:09.840 --> 00:42:11.560 And this is the same type of thing 00:42:11.560 --> 00:42:14.400 except that the rectangles are underneath. 00:42:14.400 --> 00:42:17.310 So the sum of the areas of the rectangles 00:42:17.310 --> 00:42:19.190 is less than the area under the curve. 00:42:23.741 --> 00:42:24.240 All right. 00:42:24.240 --> 00:42:27.380 So I've now trapped this quantity. 00:42:27.380 --> 00:42:34.440 And I'm now going to state the sort of general results. 00:42:38.420 --> 00:42:42.230 So here's what's known as integral comparison. 00:42:42.230 --> 00:42:44.900 It's this double arrow correspondence 00:42:44.900 --> 00:42:54.580 in the general case, for a very general case. 00:42:54.580 --> 00:42:57.130 There are actually even more cases where it works. 00:42:57.130 --> 00:43:01.220 But this is a good case and convenient. 00:43:01.220 --> 00:43:02.845 Now this is called integral comparison. 00:43:07.060 --> 00:43:12.030 And it comes with hypotheses but it follows the same argument 00:43:12.030 --> 00:43:13.495 that I just gave. 00:43:13.495 --> 00:43:27.540 If f(x) is decreasing and it's positive, 00:43:27.540 --> 00:43:37.520 then the sum f(n), n equals 1 to infinity, 00:43:37.520 --> 00:43:44.570 minus the integral from 1 to infinity of f(x) dx 00:43:44.570 --> 00:43:45.420 is less than f(1). 00:43:50.060 --> 00:43:51.590 That's basically what we showed. 00:43:51.590 --> 00:43:54.570 We showed that the difference between S_N and log N 00:43:54.570 --> 00:43:55.930 was at most 1. 00:43:59.321 --> 00:43:59.820 All right. 00:43:59.820 --> 00:44:11.590 Now if both of them are-- And the sum 00:44:11.590 --> 00:44:25.380 and the integral converge or diverge together. 00:44:25.380 --> 00:44:27.750 That is they either both converge or both diverge. 00:44:27.750 --> 00:44:30.780 This is the type of test that we like because then we can just 00:44:30.780 --> 00:44:33.000 convert the question of convergence over here 00:44:33.000 --> 00:44:37.770 to this question of convergence over on the other side. 00:44:37.770 --> 00:44:41.590 Now I remind you that it's incredibly hard 00:44:41.590 --> 00:44:44.870 to calculate these numbers. 00:44:44.870 --> 00:44:47.220 Whereas these numbers are easier to calculate. 00:44:47.220 --> 00:44:50.390 Our goal is to reduce things to simpler things. 00:44:50.390 --> 00:44:52.910 And in this case sums, infinite sums 00:44:52.910 --> 00:44:54.720 are much harder than infinite integrals. 00:45:00.080 --> 00:45:03.150 All right so that's the integral comparison. 00:45:03.150 --> 00:45:12.140 And now I have one last bit on comparisons 00:45:12.140 --> 00:45:13.840 that I need to tell you about. 00:45:13.840 --> 00:45:16.310 And this is very much like what we did with integrals. 00:45:16.310 --> 00:45:18.240 Which is a so called limit comparison. 00:45:29.200 --> 00:45:31.190 The limit comparison says the following: 00:45:31.190 --> 00:45:45.210 if f(n) is similar to g(n) -- recall that means f(n) / g(n) 00:45:45.210 --> 00:45:51.760 tends to 1 as n goes to infinity -- 00:45:51.760 --> 00:45:55.000 and we're in the positive case. 00:45:55.000 --> 00:45:57.535 So let's just say g(n) is positive. 00:46:03.920 --> 00:46:05.490 Then-- that doesn't even, well-- then 00:46:05.490 --> 00:46:15.055 sum f(n), sum g(n) either both-- same thing 00:46:15.055 --> 00:46:21.884 as above, either both converge or both diverge. 00:46:27.488 --> 00:46:28.770 All right. 00:46:28.770 --> 00:46:30.730 This is just saying that if they behave 00:46:30.730 --> 00:46:33.920 the same way in the tail, which is all we really care about, 00:46:33.920 --> 00:46:40.820 then they have similar behavior, similar convergence properties. 00:46:44.107 --> 00:46:45.690 And let me give you a couple examples. 00:46:50.370 --> 00:46:56.580 So here's one example: if you take the sum 1 over n^2 + 1, 00:46:56.580 --> 00:46:57.080 square root. 00:47:01.890 --> 00:47:05.350 This is going to be replaced by something simpler. 00:47:05.350 --> 00:47:07.240 Which is the main term here. 00:47:07.240 --> 00:47:10.460 Which is 1 over square root of n^2, 00:47:10.460 --> 00:47:15.100 which we recognize as sum 1/n, which diverges. 00:47:17.920 --> 00:47:20.440 So this guy is one of the red guys. 00:47:24.300 --> 00:47:26.410 On the red team. 00:47:26.410 --> 00:47:30.330 Now we have another example. 00:47:33.370 --> 00:47:38.960 Which is let's say the square root of n, I don't know, 00:47:38.960 --> 00:47:43.080 to the fifth minus n^2. 00:47:43.080 --> 00:47:44.665 Now if you have something where it's 00:47:44.665 --> 00:47:46.290 negative in the denominator you kind of 00:47:46.290 --> 00:47:49.540 do have to watch out that denominator makes sense. 00:47:49.540 --> 00:47:50.570 It isn't 0. 00:47:50.570 --> 00:47:53.042 So we're going to be careful and start this at n = 2. 00:47:53.042 --> 00:48:00.370 In which case, the first term, I don't like 1/0 00:48:00.370 --> 00:48:01.720 as a term in my series. 00:48:01.720 --> 00:48:04.130 So I'm just going to be a little careful about how-- 00:48:04.130 --> 00:48:05.710 as I said I was kind of lazy here. 00:48:05.710 --> 00:48:09.930 I could have started this one at 0 for instance. 00:48:09.930 --> 00:48:10.810 All right. 00:48:10.810 --> 00:48:14.110 So here's the picture. 00:48:14.110 --> 00:48:19.070 Now this I just replace by its main term which is 1 over n^5, 00:48:19.070 --> 00:48:20.340 square root. 00:48:20.340 --> 00:48:25.975 Which is sum 1/n^(5/2), which converges. 00:48:28.705 --> 00:48:29.520 All right. 00:48:29.520 --> 00:48:30.800 The power is bigger than 1. 00:48:30.800 --> 00:48:33.960 1 is the divider for these things and it just misses. 00:48:33.960 --> 00:48:38.752 This one converges. 00:48:38.752 --> 00:48:41.800 All right so these are the typical ways 00:48:41.800 --> 00:48:47.355 in which these convergence processes are used. 00:48:47.355 --> 00:48:47.870 All right. 00:48:47.870 --> 00:48:49.930 So I have one more thing for you. 00:48:49.930 --> 00:48:52.600 Which is an advertisement for next time. 00:48:52.600 --> 00:48:56.360 And I have this demo here which I will grab. 00:48:56.360 --> 00:48:58.300 But you will see this next time. 00:48:58.300 --> 00:49:00.880 So here's a question for you to think about overnight 00:49:00.880 --> 00:49:04.240 but don't ask friends, you have to think about it yourself. 00:49:04.240 --> 00:49:05.492 So here's the problem. 00:49:05.492 --> 00:49:08.000 Here are some blocks which I acquired 00:49:08.000 --> 00:49:09.390 when my kids left home. 00:49:12.190 --> 00:49:20.550 Anyway yeah that'll happen to you too in about four years. 00:49:20.550 --> 00:49:25.929 So now here you are, these are blocks. 00:49:25.929 --> 00:49:27.470 So now here's the question that we're 00:49:27.470 --> 00:49:29.546 going to deal with next time. 00:49:29.546 --> 00:49:30.920 I'm going to build it, maybe I'll 00:49:30.920 --> 00:49:32.961 put it over here because I want to have some room 00:49:32.961 --> 00:49:34.890 to head this way. 00:49:34.890 --> 00:49:41.320 I want to stack them up so that-- oh didn't work. 00:49:41.320 --> 00:49:44.430 Going to stack them up in the following way. 00:49:44.430 --> 00:49:48.280 I want to do it so that the top one is completely 00:49:48.280 --> 00:49:51.540 to the right of the bottom one. 00:49:51.540 --> 00:49:53.170 That's the question can I do that? 00:49:53.170 --> 00:49:57.870 Can I get-- Can I build this up? 00:49:57.870 --> 00:50:01.980 So let's see here. 00:50:01.980 --> 00:50:04.590 I just seem to be missing-- but anyway what I'm going to do 00:50:04.590 --> 00:50:05.965 is I'm going to try to build this 00:50:05.965 --> 00:50:10.880 and we're going to see how far we can get with this next time.