WEBVTT 00:00:00.040 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.690 Commons license. 00:00:03.690 --> 00:00:06.630 Your support will help MIT OpenCourseWare continue to 00:00:06.630 --> 00:00:09.990 offer high-quality educational resources for free. 00:00:09.990 --> 00:00:12.830 To make a donation or to view additional materials from 00:00:12.830 --> 00:00:16.760 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:16.760 --> 00:00:18.010 ocw.mit.edu. 00:00:32.404 --> 00:00:33.210 HERBERT GROSS: Hi. 00:00:33.210 --> 00:00:35.210 Well, I guess we're in the home stretch of the course. 00:00:35.210 --> 00:00:39.210 We're down to our last concept, and it will take us 00:00:39.210 --> 00:00:43.150 essentially two lectures to cover this new concept. 00:00:43.150 --> 00:00:46.430 Today we'll do the concept in general and next time we'll 00:00:46.430 --> 00:00:50.640 apply it specifically to the concept of series. 00:00:50.640 --> 00:00:53.710 But the concept we want to talk about is something called 00:00:53.710 --> 00:00:55.420 'Uniform Convergence'. 00:00:55.420 --> 00:00:59.050 Let me say at the outset that this is a very subtle topic. 00:00:59.050 --> 00:01:00.510 It is difficult. 00:01:00.510 --> 00:01:03.620 It seems to be beyond the scope of our textbook, because 00:01:03.620 --> 00:01:04.900 it's not mentioned there. 00:01:04.900 --> 00:01:08.840 Consequently, I will try to give the highlights as I speak 00:01:08.840 --> 00:01:11.430 with you, but the supplementary notes will 00:01:11.430 --> 00:01:15.210 contain a more detailed explanation of the things that 00:01:15.210 --> 00:01:16.770 we're going to talk about. 00:01:16.770 --> 00:01:20.690 Now, to set the stage properly for our discussion of uniform 00:01:20.690 --> 00:01:24.980 convergence, I think it's wise that we at least review the 00:01:24.980 --> 00:01:27.670 concept of convergence in general. 00:01:27.670 --> 00:01:30.180 Well, let's take a look over here. 00:01:30.180 --> 00:01:34.320 Recall that when we write that the limit of ''f sub n' of x' 00:01:34.320 --> 00:01:39.360 as 'n' approaches infinity equals 'f of x' for all 'x' in 00:01:39.360 --> 00:01:40.980 the closed interval from 'a' to 'b'-- 00:01:40.980 --> 00:01:44.910 when this happens, another way of saying this is that the 00:01:44.910 --> 00:01:48.110 sequence 'f sub n', the sequence of functions 'f sub 00:01:48.110 --> 00:01:52.710 n', converges to the function 'f' on the closed interval 00:01:52.710 --> 00:01:54.060 from 'a' to 'b'. 00:01:54.060 --> 00:01:57.350 Now, again, these tend to be words unless you look at a 00:01:57.350 --> 00:01:58.520 specific example. 00:01:58.520 --> 00:02:00.210 Let's just pick one over here. 00:02:00.210 --> 00:02:03.630 Let ''f sub n' of x' be ''n' over '2n plus 00:02:03.630 --> 00:02:05.940 1'' times 'x squared'. 00:02:05.940 --> 00:02:10.880 Notice, of course, that the value of this particular 00:02:10.880 --> 00:02:14.880 number depends both on 'x' and 'n'. 00:02:14.880 --> 00:02:19.770 At any rate, let's pick a fixed 'x', hold it that way, 00:02:19.770 --> 00:02:22.240 and take the limit of ''f sub n' of x' as 00:02:22.240 --> 00:02:23.770 'n' approaches infinity. 00:02:23.770 --> 00:02:26.300 In other words, as we let 'n' approach infinity in this 00:02:26.300 --> 00:02:29.020 case, notice that the limit of 'n' over '2n 00:02:29.020 --> 00:02:31.260 plus 1' becomes 1/2. 00:02:31.260 --> 00:02:33.050 'x' has been chosen independently of 00:02:33.050 --> 00:02:34.150 the choice of 'n'. 00:02:34.150 --> 00:02:36.510 Consequently, the limit function in this case, 'f of 00:02:36.510 --> 00:02:39.880 x', is 1/2 'x squared'. 00:02:39.880 --> 00:02:43.150 And in this case, what we say is that the sequence of 00:02:43.150 --> 00:02:48.440 functions 'n 'x squared'' over '2n plus 1' converges to 1/2 00:02:48.440 --> 00:02:49.720 'x squared'. 00:02:49.720 --> 00:02:52.240 Now what does this mean more specifically? 00:02:52.240 --> 00:02:54.250 In other words, let's see if we can look at a few specific 00:02:54.250 --> 00:02:55.450 values of 'x'. 00:02:55.450 --> 00:02:59.390 For example, if I choose 'x' to be 2-- 00:02:59.390 --> 00:03:02.360 in other words, if I choose 'x' to be 2, if we look over 00:03:02.360 --> 00:03:06.430 here, this says that ''f sub n' of x' is '4n' 00:03:06.430 --> 00:03:08.490 over '2n plus 1'. 00:03:08.490 --> 00:03:11.550 If I now take the limit as 'n' approaches infinity, I'm going 00:03:11.550 --> 00:03:13.800 to wind up with what? 00:03:13.800 --> 00:03:18.560 'x' is replaced by 2, 2 squared is 4, 1/2 of 4 is 2. 00:03:18.560 --> 00:03:22.500 In other words, the limit of ''f sub n' of 2', as 'n' goes 00:03:22.500 --> 00:03:24.270 to infinity, is 2. 00:03:24.270 --> 00:03:29.270 In a similar way, if I replace 'x' by 4, 1/2 'x 00:03:29.270 --> 00:03:30.560 squared' becomes 8. 00:03:30.560 --> 00:03:33.870 And so what we have is, at the limit, as 'n' approaches 00:03:33.870 --> 00:03:36.790 infinity, ''f sub n' of 4' is 8. 00:03:36.790 --> 00:03:39.980 The key thing being that once you choose 'x', notice that 00:03:39.980 --> 00:03:43.870 for a fixed 'x', ''f sub n' of x' is a constant and you're 00:03:43.870 --> 00:03:47.690 now taking the limit of a sequence of numbers. 00:03:47.690 --> 00:03:49.990 At any rate, here's what the key point is. 00:03:49.990 --> 00:03:53.120 What does this mean by our basic definition? 00:03:53.120 --> 00:03:55.910 By our basic definition, it means that we can find the 00:03:55.910 --> 00:03:59.540 number, capital 'N sub 1', such that when 'n' is greater 00:03:59.540 --> 00:04:03.020 than capital 'N sub 1', the absolute value of ''f sub n' 00:04:03.020 --> 00:04:06.260 of 2', minus 2, is less than epsilon. 00:04:06.260 --> 00:04:10.170 In a similar way, this means that we can find the number 00:04:10.170 --> 00:04:13.400 capital 'N sub 2' such that for any 'n' greater than 00:04:13.400 --> 00:04:16.426 capital 'N sub 2', the absolute value of ''f sub n' 00:04:16.426 --> 00:04:19.680 of 4' minus 8 is also less than epsilon. 00:04:19.680 --> 00:04:24.060 The key point is that 'N1' and 'N2' can be different. 00:04:24.060 --> 00:04:27.590 In other words, you may have to go out further to make this 00:04:27.590 --> 00:04:30.500 difference less than epsilon than you do to make this 00:04:30.500 --> 00:04:32.190 difference less than epsilon. 00:04:32.190 --> 00:04:34.900 In other words, you see what's happening here-- and we're 00:04:34.900 --> 00:04:36.830 going to review this in writing in a few minutes, so 00:04:36.830 --> 00:04:38.310 that you'll see it in front you. 00:04:38.310 --> 00:04:42.350 What happens here is that you see that for different values 00:04:42.350 --> 00:04:45.460 of 'x', we get different values of 'n'. 00:04:45.460 --> 00:04:48.910 And since there are infinitely many values of 'x', it means 00:04:48.910 --> 00:04:51.150 that in general, we're going to be in a little bit of 00:04:51.150 --> 00:04:55.430 trouble trying to find one 'n' that works for everything. 00:04:55.430 --> 00:04:56.460 And let me show you what that means 00:04:56.460 --> 00:04:59.230 again, going more slowly. 00:04:59.230 --> 00:05:01.720 I simply call this two basic definitions. 00:05:01.720 --> 00:05:04.720 In other words, if we have a sequence of functions 'f sub 00:05:04.720 --> 00:05:07.670 n', each of which is defined on the closed interval from 00:05:07.670 --> 00:05:11.680 'a' to 'b', we say that that sequence of functions 00:05:11.680 --> 00:05:13.450 converges point-wise-- 00:05:13.450 --> 00:05:14.910 that means number by number-- 00:05:14.910 --> 00:05:17.370 to 'f' on [a, b] 00:05:17.370 --> 00:05:21.170 if this limit, ''f sub n' of x' as 'n' approaches infinity, 00:05:21.170 --> 00:05:25.340 equals 'f of x' for each 'x' in [a,b]. 00:05:25.340 --> 00:05:27.930 In other words, given epsilon greater than 0, we can find 00:05:27.930 --> 00:05:31.550 'N1' such that 'n' greater than 'N1' implies that the 00:05:31.550 --> 00:05:35.900 absolute value of ''f sub n' of x1' minus 'f of x1' is less 00:05:35.900 --> 00:05:39.720 than epsilon for a given 'x1' in [a,b]. 00:05:39.720 --> 00:05:43.860 In general, the choice of 'N1' depends on the choice of 'x1', 00:05:43.860 --> 00:05:47.450 and there are infinitely many such choices to make in [a,b]. 00:05:47.450 --> 00:05:50.080 Now the key point is this, and this is where uniform 00:05:50.080 --> 00:05:51.740 convergence comes in. 00:05:51.740 --> 00:05:56.130 If we can find one 'N' such that whenever little 'n' is 00:05:56.130 --> 00:06:00.090 greater than that capital 'N', ''f sub n' of x' minus 'f of 00:06:00.090 --> 00:06:07.320 x' in absolute value is less than epsilon for every 'x' in 00:06:07.320 --> 00:06:10.000 the closed interval, then we say that the 00:06:10.000 --> 00:06:12.530 convergence is uniform. 00:06:12.530 --> 00:06:16.220 In other words, if we can find one 'N' that makes that 00:06:16.220 --> 00:06:19.960 difference less than epsilon for the entire interval, then 00:06:19.960 --> 00:06:22.970 we call the convergence uniform. 00:06:22.970 --> 00:06:26.070 Now, you see, convergence in general is a tough topic. 00:06:26.070 --> 00:06:30.240 In particular, uniform convergence may seem even more 00:06:30.240 --> 00:06:33.620 remote, and therefore what I'd like to do now is-- saving the 00:06:33.620 --> 00:06:36.490 formal proofs for the supplementary notes, let me 00:06:36.490 --> 00:06:40.860 show you pictorially just what the concept of uniform 00:06:40.860 --> 00:06:42.200 convergence really is. 00:06:45.570 --> 00:06:48.810 So let me give you a pictorial representation. 00:06:48.810 --> 00:06:53.020 Let's suppose I have the curve 'y' equals 'f of x'. 00:06:53.020 --> 00:06:57.580 Now, to be within epsilon of 'f of x' means I retrace this 00:06:57.580 --> 00:07:01.650 curve displaced epsilon units above the original position, 00:07:01.650 --> 00:07:03.350 and epsilon units below. 00:07:03.350 --> 00:07:06.430 In other words, for a given an epsilon, I now draw the curve 00:07:06.430 --> 00:07:09.710 'y' equals ''f of x' plus epsilon' and 'y' equals ''f of 00:07:09.710 --> 00:07:11.730 x' minus epsilon'. 00:07:11.730 --> 00:07:15.390 Now, what uniform convergence means is this, that for this 00:07:15.390 --> 00:07:19.630 given epsilon, I can find a capital 'N' such that whenever 00:07:19.630 --> 00:07:23.290 'n' is greater than capital 'N', the curve 'y' equals ''f 00:07:23.290 --> 00:07:26.190 sub n' of x' lies in this shaded region. 00:07:26.190 --> 00:07:29.960 In other words, it can bounce around all over, but it can't 00:07:29.960 --> 00:07:31.180 get outside of this region. 00:07:31.180 --> 00:07:35.270 In other words, once I'm far enough out of my sequence, all 00:07:35.270 --> 00:07:41.120 of the curves lie in this particular region. 00:07:41.120 --> 00:07:43.590 Now, of course, the question is what does this all mean? 00:07:43.590 --> 00:07:44.820 And the answer is-- well, look. 00:07:44.820 --> 00:07:48.510 Let's take epsilon to be, and I put this in quotation marks, 00:07:48.510 --> 00:07:52.020 "very, very small." Let's take epsilon, for example, to be so 00:07:52.020 --> 00:07:55.290 small that it's within the thickness of our chalk. 00:07:55.290 --> 00:07:56.540 If I now do this-- 00:07:58.900 --> 00:08:01.540 see, I draw the curve 'y' equals 'f of x'. 00:08:01.540 --> 00:08:04.930 Notice now that the thickness of my curve itself is the band 00:08:04.930 --> 00:08:06.710 width 2 epsilon. 00:08:06.710 --> 00:08:10.410 All I'm saying is that for this very, very small epsilon, 00:08:10.410 --> 00:08:14.140 when 'n' is sufficiently large, the curve 'y' equals 00:08:14.140 --> 00:08:18.720 ''f sub n' of x' appears to lie inside of this curve. 00:08:18.720 --> 00:08:21.590 You see, in other words, what you're saying is that for a 00:08:21.590 --> 00:08:24.930 large enough 'n' and small enough epsilon-- 00:08:24.930 --> 00:08:28.330 loosely speaking what you saying is that 'y' equals ''f 00:08:28.330 --> 00:08:31.620 sub n' of x' looks like 'y' equals 'f of x', for a 00:08:31.620 --> 00:08:33.510 sufficiently large values of 'n'. 00:08:33.510 --> 00:08:38.220 In other words, it appears that we can't really tell the 00:08:38.220 --> 00:08:42.750 n-th curve in the sequence from the limit function. 00:08:42.750 --> 00:08:44.430 And I want to make a few key observations 00:08:44.430 --> 00:08:45.850 about what that means. 00:08:45.850 --> 00:08:48.370 I've written the whole thing out on the blackboard so that 00:08:48.370 --> 00:08:51.860 you can see this after I say it, but what I want you to see 00:08:51.860 --> 00:08:55.100 is, can you begin to get the feeling that with this kind of 00:08:55.100 --> 00:08:56.970 a condition, for example-- 00:08:56.970 --> 00:09:01.260 if each 'f sub n' happens to be continuous, in other words, 00:09:01.260 --> 00:09:05.200 if each member of my sequence is unbroken, then the limit 00:09:05.200 --> 00:09:07.430 function itself must also be unbroken. 00:09:07.430 --> 00:09:10.970 Because you see I can squeeze this thing down to such a 00:09:10.970 --> 00:09:14.950 narrow width that there's no room for a break in here. 00:09:14.950 --> 00:09:19.700 Also notice that if the curve 'y' equals ''f sub n' of x' is 00:09:19.700 --> 00:09:21.490 caught inside this curve-- 00:09:21.490 --> 00:09:24.840 if I, for example, were computing the area of the 00:09:24.840 --> 00:09:26.010 region 'R'-- 00:09:26.010 --> 00:09:28.970 for a large enough 'n', I couldn't tell the difference 00:09:28.970 --> 00:09:33.470 in area if I use 'y' equals 'f of x' for my top curve, or 00:09:33.470 --> 00:09:36.730 whether I use 'y' equals ''f sub n' of x'. 00:09:36.730 --> 00:09:40.560 Well, keep that in mind, and all I'm saying is this, that 00:09:40.560 --> 00:09:43.810 from our picture, it should seem clear that if the 00:09:43.810 --> 00:09:47.670 sequence 'f sub n' converges uniformly to 'f' on [a, b], 00:09:47.670 --> 00:09:51.420 and if each 'f sub n' is continuous on [a,b], then-- 00:09:51.420 --> 00:09:53.140 and this is a fundamental result. 00:09:53.140 --> 00:09:57.380 Fundamental result one, 'f' is also continuous on [a, b]. 00:10:01.230 --> 00:10:02.240 Now you say, look. 00:10:02.240 --> 00:10:04.300 That's what you'd expect, isn't it, if every member of 00:10:04.300 --> 00:10:05.620 the sequence is continuous? 00:10:05.620 --> 00:10:08.570 Why shouldn't the limit function also be continuous? 00:10:08.570 --> 00:10:12.030 The point is, well, maybe that's what you expect, but 00:10:12.030 --> 00:10:13.500 note this-- 00:10:13.500 --> 00:10:17.070 in one of our earlier lectures, we already saw that 00:10:17.070 --> 00:10:20.640 if we only had point-wise convergence, this did not need 00:10:20.640 --> 00:10:21.640 to be true. 00:10:21.640 --> 00:10:25.240 In particular, recall our example in which we defined 00:10:25.240 --> 00:10:29.480 ''f sub n' of x' to be 'x to the n', where the domain of 00:10:29.480 --> 00:10:32.650 'f' was the closed interval from 0 to 1. 00:10:32.650 --> 00:10:34.330 Remember what happened in that case? 00:10:34.330 --> 00:10:36.960 Each of these 'f sub n's is continuous. 00:10:36.960 --> 00:10:42.060 But the limit function, you may recall, is what? 00:10:42.060 --> 00:10:47.990 It's 0 if 'x' is less than 1, and 1 if 'x' equals 1. 00:10:47.990 --> 00:10:51.540 In other words, the limit function was discontinuous at 00:10:51.540 --> 00:10:52.890 'x' equals 1. 00:10:52.890 --> 00:10:55.360 By the way, you might like to see what this means from 00:10:55.360 --> 00:10:56.870 another point of view. 00:10:56.870 --> 00:10:58.750 And let me show you what it does mean from 00:10:58.750 --> 00:10:59.910 another point of view. 00:10:59.910 --> 00:11:04.190 Since 'f' continuous at 'x' equals 'x sub 1' means that 00:11:04.190 --> 00:11:08.110 the limit of 'f of x' as 'x' approaches 'x1' is 'f of x1', 00:11:08.110 --> 00:11:13.620 and since 'f of x' itself, by definition, is the limit of 00:11:13.620 --> 00:11:22.360 ''f sub n' of x' as 'n' approaches infinity, we may 00:11:22.360 --> 00:11:26.550 rewrite our first condition in the form-- what? 00:11:26.550 --> 00:11:32.100 We may rewrite our first equation, this equation here, 00:11:32.100 --> 00:11:33.320 in what form? 00:11:33.320 --> 00:11:38.820 Limit as 'x' approaches 'x1' of limit 'n' approaches 00:11:38.820 --> 00:11:41.270 infinity, ''f sub n' of x'. 00:11:41.270 --> 00:11:44.470 And that equals the limit as 'n' approaches infinity, ''f 00:11:44.470 --> 00:11:46.660 sub n' of x1'. 00:11:46.660 --> 00:11:50.330 Now keep in mind that since each 'f sub n' is given to be 00:11:50.330 --> 00:11:55.510 continuous, by definition of continuity ''f sub n' of x1' 00:11:55.510 --> 00:11:59.280 is the same as saying the limit as 'x' approaches 'x1', 00:11:59.280 --> 00:12:00.690 ''f sub n' of x'. 00:12:00.690 --> 00:12:05.060 The point I'm making is, if you now put this together with 00:12:05.060 --> 00:12:09.410 this, it says the rather remarkable thing that unless 00:12:09.410 --> 00:12:13.860 you have uniform convergence when you interchange the order 00:12:13.860 --> 00:12:16.770 in which you take the limits over here, you may very well 00:12:16.770 --> 00:12:18.200 get a different answer. 00:12:18.200 --> 00:12:20.920 In other words, when you're dealing with non-uniform 00:12:20.920 --> 00:12:25.550 convergence, you must be very, very careful to perform every 00:12:25.550 --> 00:12:27.650 operation in the given order. 00:12:27.650 --> 00:12:29.210 What we're saying is what? 00:12:29.210 --> 00:12:31.090 This will give you an answer. 00:12:31.090 --> 00:12:32.710 This will give you an answer. 00:12:32.710 --> 00:12:36.140 But if the convergence is not uniform, the answers may be 00:12:36.140 --> 00:12:39.730 different, and consequently by changing the order you destroy 00:12:39.730 --> 00:12:42.000 the whole physical meaning of the problem. 00:12:42.000 --> 00:12:44.590 Well, again, that's reemphasized in the 00:12:44.590 --> 00:12:45.890 supplementary notes. 00:12:45.890 --> 00:12:47.940 Let me continue on here. 00:12:47.940 --> 00:12:51.510 Let me tell you another interesting property of 00:12:51.510 --> 00:12:54.510 uniform convergence. 00:12:54.510 --> 00:12:58.420 Suppose the sequence ''f sub n' of x' converges uniformly 00:12:58.420 --> 00:13:00.590 to 'f of x' on [a, b]. 00:13:00.590 --> 00:13:03.220 The point that's rather interesting is that you can 00:13:03.220 --> 00:13:06.530 reverse the order of integration and taking the 00:13:06.530 --> 00:13:08.210 limit in this particular case. 00:13:08.210 --> 00:13:12.060 In other words, suppose you want to compute the integral 00:13:12.060 --> 00:13:14.800 of the limit function from 'a' to 'b'. 00:13:14.800 --> 00:13:19.790 What you can do instead is compute the integral of the 00:13:19.790 --> 00:13:23.440 n-th number of your sequence, and then take the limit as 'n' 00:13:23.440 --> 00:13:24.700 goes to infinity. 00:13:24.700 --> 00:13:27.820 In other words, rewriting this, it says that if you have 00:13:27.820 --> 00:13:32.080 uniform convergence, you can take the limit inside the 00:13:32.080 --> 00:13:33.470 integral sign. 00:13:33.470 --> 00:13:37.740 And again, these results are proven in our 00:13:37.740 --> 00:13:39.400 supplementary notes. 00:13:39.400 --> 00:13:43.040 We also say a few words about corresponding results for 00:13:43.040 --> 00:13:45.750 differentiation in our supplementary notes. 00:13:45.750 --> 00:13:50.100 And I should point out that differentiation is a far more 00:13:50.100 --> 00:13:52.000 subtle thing than integration. 00:13:52.000 --> 00:13:55.080 See, remember that for integration, all you need is 00:13:55.080 --> 00:13:56.520 continuity. 00:13:56.520 --> 00:13:59.340 For differentiation, you need smoothness. 00:13:59.340 --> 00:14:04.420 The point is that as you put a thin band around the function 00:14:04.420 --> 00:14:07.760 'y' equals 'f of x' when you have uniform convergence, 00:14:07.760 --> 00:14:10.410 that's enough to make sure that the limit function must 00:14:10.410 --> 00:14:13.200 be continuous if each of the members in the sequence is 00:14:13.200 --> 00:14:13.960 continuous. 00:14:13.960 --> 00:14:16.790 But without going into the details of this thing, it does 00:14:16.790 --> 00:14:20.000 turn out that for the degree of smoothness that you need, 00:14:20.000 --> 00:14:22.510 these things can jump around enough so that for 00:14:22.510 --> 00:14:25.860 differentiation, we do have to be a little bit more careful. 00:14:25.860 --> 00:14:28.650 Rather than to becloud the issue, I will stick with 00:14:28.650 --> 00:14:31.810 integration topics for our lecture today. 00:14:31.810 --> 00:14:34.880 Now, the other thing that I want to mention is, again, in 00:14:34.880 --> 00:14:38.830 terms of doing what comes naturally, I think we are 00:14:38.830 --> 00:14:41.850 tempted to look at something like this and say, look, all I 00:14:41.850 --> 00:14:43.270 did was bring the limit inside. 00:14:43.270 --> 00:14:45.360 Why can't I do that? 00:14:45.360 --> 00:14:48.120 And instead of saying, look, you can't, I think the best 00:14:48.120 --> 00:14:50.290 thing to show is that when you don't have uniform 00:14:50.290 --> 00:14:52.570 convergence, you get two different answers. 00:14:52.570 --> 00:14:53.980 Again, the idea being what? 00:14:53.980 --> 00:14:56.270 We are not saying that you can't do this. 00:14:56.270 --> 00:14:58.920 We are not saying that you can't compute this. 00:14:58.920 --> 00:15:01.530 All we're saying is that if the convergence is not 00:15:01.530 --> 00:15:08.050 uniform, these two expressions may very well 00:15:08.050 --> 00:15:10.880 name different numbers. 00:15:10.880 --> 00:15:13.180 Now, to show you what I have in mind here, let me give you 00:15:13.180 --> 00:15:15.110 an example. 00:15:15.110 --> 00:15:18.180 See, to show you that 2 need not be true if the convergence 00:15:18.180 --> 00:15:21.630 is not uniform, consider the following example. 00:15:21.630 --> 00:15:25.290 Now, in the supplementary notes, I repeat this example 00:15:25.290 --> 00:15:28.270 both the way I have it on the board and also from an 00:15:28.270 --> 00:15:30.880 algebraic point of view, without using the pictures. 00:15:30.880 --> 00:15:33.700 But in terms of the picture, here's what we do. 00:15:33.700 --> 00:15:37.020 We define a function on the closed interval from 0 to 2, 00:15:37.020 --> 00:15:40.160 which I'll call 'f sub n', as follows. 00:15:40.160 --> 00:15:45.460 For a given 'n', I will locate the points '1/n' and '2/n'. 00:15:45.460 --> 00:15:49.900 For example, if 'n' happened to be 50, this would be 1/50 00:15:49.900 --> 00:15:52.030 and this would be 2/50. 00:15:52.030 --> 00:15:57.570 Now what I do, is at the 'x' value '1/n', I take as the 00:15:57.570 --> 00:16:01.510 corresponding y-value, 'n squared'. 00:16:01.510 --> 00:16:05.060 And I draw the straight line that goes from the origin to 00:16:05.060 --> 00:16:08.110 this point, '1/n' comma 'n squared'. 00:16:08.110 --> 00:16:12.290 Then I draw the straight line that comes right back to the 00:16:12.290 --> 00:16:17.800 x-intercept, '2/n', and I finish off the curve by just 00:16:17.800 --> 00:16:22.180 letting it hug the x-axis till we get over to 'x' equals 2. 00:16:22.180 --> 00:16:24.660 I'll come back to this on the next board, to show you why I 00:16:24.660 --> 00:16:27.590 chose this, but let's make a few observations just to make 00:16:27.590 --> 00:16:30.860 sure that you understand what this function looks like. 00:16:30.860 --> 00:16:33.630 I'll just make a few arbitrary remarks about it. 00:16:33.630 --> 00:16:36.310 First of all, for each 'n', ''f sub n' of 00:16:36.310 --> 00:16:38.640 '1/n'' is 'n squared'. 00:16:38.640 --> 00:16:41.010 That's just another way of indicating a 00:16:41.010 --> 00:16:42.260 label for this point. 00:16:44.610 --> 00:16:50.990 Secondly, my claim is that for any number 'x sub 0', if 'x 00:16:50.990 --> 00:16:56.580 sub 0' is greater than '2/n', ''f sub n' of x0' must be 0. 00:16:56.580 --> 00:16:58.810 And the reason for that is quite simple. 00:16:58.810 --> 00:17:00.760 I'm just trying to show you how to read this picture. 00:17:00.760 --> 00:17:04.410 Namely, notice that as soon as 'x' gets to be as great as 00:17:04.410 --> 00:17:10.140 '2/n', the 'f' value is 0, because the function is 00:17:10.140 --> 00:17:11.960 hugging the x-axis. 00:17:11.960 --> 00:17:14.579 And just as a final observation, notice that when 00:17:14.579 --> 00:17:19.420 'x sub 0' is 0, ''f sub n' of 0' is 0 for every 'n', meaning 00:17:19.420 --> 00:17:23.660 that every member of my family of functions goes through this 00:17:23.660 --> 00:17:24.520 particular point. 00:17:24.520 --> 00:17:26.020 In other words, let me just label this. 00:17:26.020 --> 00:17:31.850 This is 'y' equals ''f sub n' of x'. 00:17:31.850 --> 00:17:35.110 Well, by the way if ''f sub n' of 0' is 0 for each 'n', in 00:17:35.110 --> 00:17:37.810 particular the limit of ''f sub n' of 0' as 'n' approaches 00:17:37.810 --> 00:17:39.920 infinity is 0. 00:17:39.920 --> 00:17:43.080 What happens if we pick a non-zero value? 00:17:43.080 --> 00:17:47.490 For example, suppose I pick 'x0' to be greater than 0 but 00:17:47.490 --> 00:17:49.190 less than or equal to 2? 00:17:49.190 --> 00:17:54.030 The key point is this, that since the limit of '2/n' as 00:17:54.030 --> 00:17:58.490 'n' approaches infinity is 0, given a value of 'x0' which is 00:17:58.490 --> 00:18:02.720 not 0, I can find the capital 'N' such that when 'n' is 00:18:02.720 --> 00:18:06.660 greater than capital 'N', '2/n' is less than 'x0'. 00:18:06.660 --> 00:18:10.020 In other words, if 'x0' is greater than 0, and '2/n' 00:18:10.020 --> 00:18:13.860 approaches 0, for large enough values of 'n', '2/n' 00:18:13.860 --> 00:18:15.340 be less than 'x0'. 00:18:15.340 --> 00:18:18.210 In particular, when that happens, if we couple this 00:18:18.210 --> 00:18:20.010 with our earlier observation-- 00:18:20.010 --> 00:18:21.510 what earlier observation? 00:18:21.510 --> 00:18:24.780 Well, this one. 00:18:24.780 --> 00:18:27.610 If we couple that with our earlier observation, we see 00:18:27.610 --> 00:18:30.150 that when 'n' is greater than capital 'N', ''f sub 00:18:30.150 --> 00:18:32.110 n' of x0' is 0. 00:18:32.110 --> 00:18:36.530 Correspondingly, then, the limit of ''f sub n' of x0' as 00:18:36.530 --> 00:18:39.870 'n' approaches infinity, by definition, is 0. 00:18:39.870 --> 00:18:40.260 In other words-- 00:18:40.260 --> 00:18:42.650 I'm going to reinforce this later, but notice that the 00:18:42.650 --> 00:18:45.650 limit function here is the function which 00:18:45.650 --> 00:18:47.480 is identically 0. 00:18:47.480 --> 00:18:51.600 Now, since this may look very abstract to you, let me take a 00:18:51.600 --> 00:18:52.450 few minutes-- 00:18:52.450 --> 00:18:54.960 and I hope this doesn't insult your intelligence, but let me 00:18:54.960 --> 00:18:58.490 just take a few minutes and redraw this for a couple of 00:18:58.490 --> 00:19:01.200 different values of 'n', just so that you can see what's 00:19:01.200 --> 00:19:02.450 starting to happen here. 00:19:04.870 --> 00:19:08.730 Keep that picture in mind, and now look what this means. 00:19:08.730 --> 00:19:15.460 For example, when 'n' is 1, '1/n' is 1, '2/n' is 2, 'n 00:19:15.460 --> 00:19:16.760 squared' is 1. 00:19:16.760 --> 00:19:20.920 In other words, the graph 'y' equals 'f1 of x' is just this 00:19:20.920 --> 00:19:23.876 triangular-- 00:19:23.876 --> 00:19:25.480 just this. 00:19:25.480 --> 00:19:27.300 Why give it a name? 00:19:27.300 --> 00:19:30.030 Well, let's try a tougher one. 00:19:30.030 --> 00:19:34.100 Let's see what the member 'f sub 20' looks like. 00:19:34.100 --> 00:19:35.590 Recall how you draw this, now. 00:19:35.590 --> 00:19:38.000 With the subscript 20, what do you do? 00:19:38.000 --> 00:19:43.670 You come in to the point 1/20, and at that 00:19:43.670 --> 00:19:44.930 point, you do what? 00:19:44.930 --> 00:19:49.900 You locate the point 1/20 comma 'n squared'. 00:19:49.900 --> 00:19:52.200 In this case, it's 400. 00:19:52.200 --> 00:19:54.700 And I have obviously haven't drawn this to scale, but you 00:19:54.700 --> 00:19:55.440 now do what? 00:19:55.440 --> 00:19:57.060 Draw the straight line that goes from the 00:19:57.060 --> 00:19:59.120 origin to this point. 00:19:59.120 --> 00:20:01.870 Then from this point, you draw the straight line that comes 00:20:01.870 --> 00:20:06.190 back to the x-axis, hitting it at 'x' equals 1/10. 00:20:06.190 --> 00:20:08.640 And then you come across the x-axis all the way 00:20:08.640 --> 00:20:10.050 to 'x' equals 2. 00:20:10.050 --> 00:20:13.810 This would be the graph of 'y' equals ''f sub 20' of x'. 00:20:13.810 --> 00:20:17.010 And by the way, do you sense what's happening over here? 00:20:17.010 --> 00:20:23.040 See, notice that as 'n' gets very, very large, the curve 00:20:23.040 --> 00:20:26.650 hugs the x-axis, starting in closer and 00:20:26.650 --> 00:20:28.220 closer to the y-axis. 00:20:28.220 --> 00:20:31.580 But what happens is someplace in here, no matter how close 00:20:31.580 --> 00:20:35.560 'x sub 0' is to 0, there comes a very high peak. 00:20:35.560 --> 00:20:37.220 In fact, what is that high peak? 00:20:37.220 --> 00:20:38.830 It's 'n squared'. 00:20:38.830 --> 00:20:41.470 In other words, when this number is very close to the 00:20:41.470 --> 00:20:45.220 y-axis, the peak is very, very high. 00:20:45.220 --> 00:20:47.580 In other words, no matter how you put the squeeze on over 00:20:47.580 --> 00:20:52.800 here, this particular peak jumps out. 00:20:52.800 --> 00:20:55.830 This is why this particular sequence of functions is not 00:20:55.830 --> 00:20:57.070 uniformly convergent. 00:20:57.070 --> 00:20:59.820 Again, this is done more slowly in the notes. 00:20:59.820 --> 00:21:01.880 But at any rate, let me show you an interesting thing that 00:21:01.880 --> 00:21:03.760 happens over here. 00:21:03.760 --> 00:21:07.290 Let me redraw this now for a general 'n'. 00:21:07.290 --> 00:21:09.930 In other words, let me draw 'y' equals ''f sub n' of x' 00:21:09.930 --> 00:21:11.430 for any old 'n'. 00:21:11.430 --> 00:21:14.140 Recall what our definition was, now, especially 00:21:14.140 --> 00:21:16.000 with this as review. 00:21:16.000 --> 00:21:18.960 We locate the point '1/n' comma 'n squared'. 00:21:18.960 --> 00:21:20.710 We then draw the line that goes from the 00:21:20.710 --> 00:21:22.180 origin to that point. 00:21:22.180 --> 00:21:26.220 Then we draw the line that goes from that point back to 00:21:26.220 --> 00:21:29.410 the x-axis at the point '2/n'. 00:21:29.410 --> 00:21:32.300 And then we come across to 'x' equals 2. 00:21:32.300 --> 00:21:37.770 Let's try to visualize what the integral from 0 to 2, ''f 00:21:37.770 --> 00:21:39.790 sub n' of x', 'dx', means. 00:21:39.790 --> 00:21:43.090 After all, in a case of a continuous curve, which this 00:21:43.090 --> 00:21:46.250 is, isn't the definite integral interpreted just as 00:21:46.250 --> 00:21:47.790 the area under the curve? 00:21:47.790 --> 00:21:51.370 Well, you see, the curve coincides with the x-axis from 00:21:51.370 --> 00:21:53.380 '2/n' on to 2. 00:21:53.380 --> 00:21:57.860 Consequently, this triangular region which I call 'R' is the 00:21:57.860 --> 00:21:59.210 area under the curve. 00:21:59.210 --> 00:22:02.440 In other words, the integral from 0 to 2, ''f sub n' of x', 00:22:02.440 --> 00:22:05.720 'dx', is the area of the region 'R'. 00:22:05.720 --> 00:22:06.750 But here's the point. 00:22:06.750 --> 00:22:09.870 We can compute the area of the region 'R' very easily. 00:22:09.870 --> 00:22:11.780 It's a triangle, right? 00:22:11.780 --> 00:22:13.300 What is the area of a triangle? 00:22:13.300 --> 00:22:17.040 Well, it's 1/2 times the base-- 00:22:17.040 --> 00:22:19.410 but the base is just '2/n'-- 00:22:19.410 --> 00:22:20.790 times the height. 00:22:20.790 --> 00:22:22.900 The height is 'n squared'. 00:22:22.900 --> 00:22:27.150 In other words, the area of the region 'R' simply is 'n'. 00:22:27.150 --> 00:22:28.530 And that's rather interesting. 00:22:28.530 --> 00:22:32.730 In other words, for each 'n', this particular integral just 00:22:32.730 --> 00:22:34.770 turns out to be 'n' itself. 00:22:34.770 --> 00:22:37.220 That's what's interesting about this particular diagram. 00:22:37.220 --> 00:22:40.620 In other words, this thing rises so high that even though 00:22:40.620 --> 00:22:44.800 the base gets very, very small as 'n' gets large, the height 00:22:44.800 --> 00:22:47.980 increases so rapidly that the area under this curve, 00:22:47.980 --> 00:22:52.270 numerically, is always equal to 'n'. 00:22:52.270 --> 00:22:54.300 In fact, we can check that if you'd like. 00:22:54.300 --> 00:22:56.160 Come back to this particular case. 00:22:56.160 --> 00:22:58.280 Look at this particular triangle. 00:22:58.280 --> 00:23:00.510 The base is 2, the height is 1. 00:23:00.510 --> 00:23:04.560 The area is 1 unit. 00:23:04.560 --> 00:23:07.490 Look at this particular triangle. 00:23:07.490 --> 00:23:12.520 The base is 1/10, the height is 400. 00:23:12.520 --> 00:23:17.540 400 times 1/10 is 40, and half of that is 20. 00:23:17.540 --> 00:23:20.470 The area of this triangle is 20, which exactly 00:23:20.470 --> 00:23:21.590 matches this subscript. 00:23:21.590 --> 00:23:23.870 That's what's going to happen here all the time. 00:23:23.870 --> 00:23:27.640 In particular, then, if we compute the integral from 0 to 00:23:27.640 --> 00:23:33.810 2, 'f of n', 'x dx', and then let the limit as 'n' goes to 00:23:33.810 --> 00:23:36.430 infinity be computed, what do we get for an answer? 00:23:36.430 --> 00:23:40.450 We get that this limit is the limit of 'n' as 'n' approaches 00:23:40.450 --> 00:23:43.690 infinity, and that of course is infinity. 00:23:43.690 --> 00:23:47.050 On the other hand, suppose we bring the limit inside? 00:23:47.050 --> 00:23:49.960 In other words, suppose we compute this. 00:23:49.960 --> 00:23:53.620 Well, the point is that we have already shown that this 00:23:53.620 --> 00:23:57.370 is identically 0 for all 'x'. 00:23:57.370 --> 00:24:01.800 Consequently, this integral is the integral from 0 to 2, 0 00:24:01.800 --> 00:24:03.990 'dx', which is 0. 00:24:03.990 --> 00:24:06.760 In other words, if you first take the limit and then 00:24:06.760 --> 00:24:09.530 integrate, you get 0. 00:24:09.530 --> 00:24:12.490 On the other hand, if you first integrate and then take 00:24:12.490 --> 00:24:14.490 the limit, you get infinity. 00:24:14.490 --> 00:24:17.820 And this should be a glaring example to show you that the 00:24:17.820 --> 00:24:20.990 answer that you get indeed does depend on the order in 00:24:20.990 --> 00:24:22.620 which you do the operation. 00:24:22.620 --> 00:24:24.260 Again, let me emphasize-- 00:24:24.260 --> 00:24:26.600 which of these two is wrong? 00:24:26.600 --> 00:24:29.210 The answer is, neither is wrong. 00:24:29.210 --> 00:24:32.150 All we're saying is that if you were supposed to solve 00:24:32.150 --> 00:24:35.710 this problem and by mistake you solve this one, you are 00:24:35.710 --> 00:24:39.150 going to get a drastically different answer. 00:24:39.150 --> 00:24:40.230 OK. 00:24:40.230 --> 00:24:41.670 Let's not beat this to death. 00:24:41.670 --> 00:24:43.170 So far, so good. 00:24:43.170 --> 00:24:47.340 Let me make one more remark, namely, what does all of this 00:24:47.340 --> 00:24:49.550 have to do with the study of series? 00:24:49.550 --> 00:24:53.710 See, now we're just talking about sequences of functions. 00:24:53.710 --> 00:24:56.630 And you see, the answer to this question is essentially 00:24:56.630 --> 00:25:02.020 going to be our last lecture of the course. 00:25:02.020 --> 00:25:04.140 But for now, what I'd like to do is to give you 00:25:04.140 --> 00:25:05.420 a preview of that. 00:25:05.420 --> 00:25:08.580 Namely, the application of uniform convergence to series 00:25:08.580 --> 00:25:09.750 is the following. 00:25:09.750 --> 00:25:13.300 Recall that when we write summation 'n' goes from 0 to 00:25:13.300 --> 00:25:16.310 infinity, 'a sub n', 'x to the n', that's an 00:25:16.310 --> 00:25:18.070 abbreviation for what? 00:25:18.070 --> 00:25:23.860 A polynomial, 'k' goes from 0 to 'n', 'a sub k', 'x sub k', 00:25:23.860 --> 00:25:25.550 as 'n' goes to infinity. 00:25:25.550 --> 00:25:30.390 In other words, recall that the sum of the series is a 00:25:30.390 --> 00:25:34.240 limit of a sequence of partial sums, and this is the n-th 00:25:34.240 --> 00:25:37.020 member of that sequence of partial sums. 00:25:37.020 --> 00:25:40.560 Again, if the sigma notation is throwing you off, all I'm 00:25:40.560 --> 00:25:46.800 saying is to observe that 'a0' plus 'a1 x' plus 'a2 'x 00:25:46.800 --> 00:25:50.160 squared'' plus-- et cetera, et cetera, et cetera, forever, 00:25:50.160 --> 00:25:53.500 just represents the limit of the following sequence. 00:25:53.500 --> 00:25:57.940 'a0', next member is 'a0 plus a1 x'. 00:25:57.940 --> 00:26:02.090 Next member is 'a0' plus 'a1 x' plus 'a2 'x squared''. 00:26:02.090 --> 00:26:06.310 The next member is 'a0' plus 'a1 x' plus 'a2 'x squared'' 00:26:06.310 --> 00:26:07.960 plus 'a3 'x cubed''. 00:26:07.960 --> 00:26:10.860 By the way, what is each member of the sequence? 00:26:10.860 --> 00:26:13.290 It's a polynomial. 00:26:13.290 --> 00:26:16.570 And polynomials have very nice properties, 00:26:16.570 --> 00:26:17.960 among which are what? 00:26:17.960 --> 00:26:20.780 Well, a polynomial is a continuous function. 00:26:20.780 --> 00:26:23.850 A polynomial is an integral function, et cetera. 00:26:23.850 --> 00:26:27.670 The idea, therefore, is that if this sequence of partial 00:26:27.670 --> 00:26:35.840 sums converges uniformly to the limit function, then, for 00:26:35.840 --> 00:26:39.800 example, the limit function, namely the power series, must 00:26:39.800 --> 00:26:44.640 be continuous since each partial sum that makes up the 00:26:44.640 --> 00:26:47.570 sequence of partial sums is also continuous. 00:26:47.570 --> 00:26:50.100 Namely, every polynomial is continuous. 00:26:50.100 --> 00:26:53.670 Also, if, for some reason or other, you want to integrate 00:26:53.670 --> 00:26:57.260 that power series from 'a' to 'b', if the convergence is 00:26:57.260 --> 00:26:59.800 uniform, I can then do what? 00:26:59.800 --> 00:27:03.880 I can then take the summation sign outside, integrate the 00:27:03.880 --> 00:27:08.100 n-th partial sum, and add these all up. 00:27:08.100 --> 00:27:09.610 You see, the idea being what? 00:27:09.610 --> 00:27:13.350 That the n-th partial sum is a polynomial, and a polynomial 00:27:13.350 --> 00:27:16.560 is a particularly simple thing to integrate. 00:27:16.560 --> 00:27:18.400 That's one of the easiest functions to 00:27:18.400 --> 00:27:20.150 integrate, in fact. 00:27:20.150 --> 00:27:21.380 OK. 00:27:21.380 --> 00:27:23.120 Now, here's the wrap up, then. 00:27:23.120 --> 00:27:26.760 What we shall show next time is that within the interval of 00:27:26.760 --> 00:27:30.780 absolute convergence, the sequence of partial sums, 00:27:30.780 --> 00:27:36.620 which we already know converges absolutely to the 00:27:36.620 --> 00:27:39.840 limit function, also converges uniformly. 00:27:39.840 --> 00:27:43.040 In other words, within the radius of convergence, the 00:27:43.040 --> 00:27:43.670 power series-- 00:27:43.670 --> 00:27:46.530 and I don't know how to say this other than to say, it 00:27:46.530 --> 00:27:50.750 enjoys the usual polynomial properties associated with a 00:27:50.750 --> 00:27:54.610 polynomial such as summation 'k' goes from 0 to 'n', 'a sub 00:27:54.610 --> 00:27:57.090 k', 'x to the k'. 00:27:57.090 --> 00:28:00.820 In other words, then, this about wraps up what our 00:28:00.820 --> 00:28:03.780 introductory lecture for today wanted to be, namely the 00:28:03.780 --> 00:28:06.250 concept of uniform convergence. 00:28:06.250 --> 00:28:09.320 What I would like you to do now is to study this material 00:28:09.320 --> 00:28:12.040 very carefully in the supplementary notes, go over 00:28:12.040 --> 00:28:15.260 the learning exercises so that you become familiar with this. 00:28:15.260 --> 00:28:18.080 Then we will wrap up our course in our next lecture, 00:28:18.080 --> 00:28:23.480 when we show what a very, very powerful tool this particular 00:28:23.480 --> 00:28:28.480 concept of absolute convergence is in the study of 00:28:28.480 --> 00:28:31.110 the mathematical concept of convergence. 00:28:31.110 --> 00:28:33.970 At any rate, until next time, then, goodbye. 00:28:36.550 --> 00:28:39.750 Funding for the publication of this video was provided by the 00:28:39.750 --> 00:28:43.810 Gabriella and Paul Rosenbaum Foundation. 00:28:43.810 --> 00:28:47.970 Help OCW continue to provide free and open access to MIT 00:28:47.970 --> 00:28:52.170 courses by making a donation at ocw.mit.edu/donate.