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 --> 00:00:06.640 Your support will help MIT OpenCourseWare continue to 00:00:06.640 --> 00:00:09.980 offer high-quality educational resources for free. 00:00:09.980 --> 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. 00:00:26.555 --> 00:00:27.510 PROFESSOR: Hi. 00:00:27.510 --> 00:00:30.770 Today we begin our final block of material in this particular 00:00:30.770 --> 00:00:34.650 course, and it's the segment entitled Infinite Series. 00:00:34.650 --> 00:00:37.510 And perhaps the best way to motivate this rather difficult 00:00:37.510 --> 00:00:40.260 block of material is in terms of the concept 00:00:40.260 --> 00:00:42.080 of many versus infinite. 00:00:42.080 --> 00:00:45.190 In many respects, this particular block could've been 00:00:45.190 --> 00:00:46.860 given much earlier in the course. 00:00:46.860 --> 00:00:50.140 But somehow or other, until we have some sort of a feeling as 00:00:50.140 --> 00:00:53.760 to what infinity really means, we have a maturity problem in 00:00:53.760 --> 00:00:55.910 trying to really grasp the significance of 00:00:55.910 --> 00:00:56.850 what's going on. 00:00:56.850 --> 00:00:59.120 In fact, in a manner of speaking, with all of this 00:00:59.120 --> 00:01:02.130 experience, there may be a maturity problem in trying to 00:01:02.130 --> 00:01:03.770 grasp the fundamental ideas. 00:01:03.770 --> 00:01:07.000 What I shall do throughout the material on this block is to 00:01:07.000 --> 00:01:10.180 utilize the lectures again to make sure that the concepts 00:01:10.180 --> 00:01:13.720 become crystallized and use the learning exercises plus 00:01:13.720 --> 00:01:17.140 the text plus supplements notes to make sure that the 00:01:17.140 --> 00:01:20.750 details are taken care of in adequate fashion. 00:01:20.750 --> 00:01:23.040 At any rate, I've entitle today's lecture 00:01:23.040 --> 00:01:24.450 'Many Versus Infinite'. 00:01:24.450 --> 00:01:27.290 And I thought the best way to get started on this was to 00:01:27.290 --> 00:01:29.990 think of a number that's very easy to write in terms of 00:01:29.990 --> 00:01:31.790 exponential notation. 00:01:31.790 --> 00:01:35.560 Let capital 'N' be 10 to the 10 to the 10th power. 00:01:35.560 --> 00:01:37.910 10 to the 10, by the way, is 10 billion, a 1 00:01:37.910 --> 00:01:39.270 followed by 10 zeroes. 00:01:39.270 --> 00:01:42.030 That's 10 to the 10-billionth power. 00:01:42.030 --> 00:01:44.530 That, of course, means, if written in place value, that 00:01:44.530 --> 00:01:48.200 would be a 1 followed by 10 billion zeroes. 00:01:48.200 --> 00:01:50.190 And for those of you who would like an exercise in 00:01:50.190 --> 00:01:52.810 multiplication and long division and you want to 00:01:52.810 --> 00:01:55.790 compute the number of seconds in a year and what have you, 00:01:55.790 --> 00:01:57.970 it turns out without too much difficulty 00:01:57.970 --> 00:01:59.360 that it can be shown. 00:01:59.360 --> 00:02:02.990 That to write 1 billion zeroes at the rate of one per second 00:02:02.990 --> 00:02:06.380 would take in the order of magnitude of some 32 years. 00:02:06.380 --> 00:02:10.060 In other words, this number capital 'N', roughly speaking, 00:02:10.060 --> 00:02:12.960 writing it in place value notation at the rate of one 00:02:12.960 --> 00:02:16.640 digit per second would take 320 years to write. 00:02:16.640 --> 00:02:18.070 And you say so what? 00:02:18.070 --> 00:02:20.810 And the answer is, well, after you've got out that far-- and 00:02:20.810 --> 00:02:22.560 by the way, this is crucial. 00:02:22.560 --> 00:02:25.380 320 years is a long time. 00:02:25.380 --> 00:02:26.230 I was going to say it's a lifetime. 00:02:26.230 --> 00:02:27.610 It's more than the lifetime. 00:02:27.610 --> 00:02:29.860 It's a long time, but it's finite. 00:02:29.860 --> 00:02:32.190 Eventually, the job could be completed. 00:02:32.190 --> 00:02:34.710 But the interesting point is that once it's completed, the 00:02:34.710 --> 00:02:38.400 next number in our system is capital 'N plus 1', capital 'N 00:02:38.400 --> 00:02:42.920 plus 2', capital 'N plus 3', where in a sense then, with 00:02:42.920 --> 00:02:46.140 'N' as a new reference point, we're back to the beginning of 00:02:46.140 --> 00:02:47.030 our number system. 00:02:47.030 --> 00:02:50.740 In other words, granted that 'N' is a fantastically large 00:02:50.740 --> 00:02:54.020 number, if you wanted to become wealthy, to own 'N' 00:02:54.020 --> 00:02:56.810 dollars would more than realize your dream. 00:02:56.810 --> 00:03:00.850 But if your aim was to own infinitely much money, 'N' 00:03:00.850 --> 00:03:04.220 would be no closer than having no money at all. 'N' is no 00:03:04.220 --> 00:03:06.520 nearer the end of a number system than is 00:03:06.520 --> 00:03:07.990 the number 1 itself. 00:03:07.990 --> 00:03:11.020 There is the story that signifies the difference 00:03:11.020 --> 00:03:12.670 between many and infinite. 00:03:12.670 --> 00:03:14.590 And to hammer this point home, let me give 00:03:14.590 --> 00:03:16.600 you a few more examples. 00:03:16.600 --> 00:03:19.740 I cleverly call this additional examples. 00:03:19.740 --> 00:03:22.720 We all know that there are just as many odd numbers and 00:03:22.720 --> 00:03:23.640 even numbers, right? 00:03:23.640 --> 00:03:25.600 The odds and the evens match up. 00:03:25.600 --> 00:03:27.630 Now, watch the following little gimmick. 00:03:27.630 --> 00:03:30.450 Write the first two odd numbers, then the first even 00:03:30.450 --> 00:03:33.490 number, the next two odd numbers, then the next even 00:03:33.490 --> 00:03:35.570 number, the next two odd numbers, than 00:03:35.570 --> 00:03:36.780 the next even number. 00:03:36.780 --> 00:03:39.080 And go on like this as long as you want. 00:03:39.080 --> 00:03:43.510 And no matter where we stop, even if we go to the 10 to the 00:03:43.510 --> 00:03:47.390 10 to the 10th term, no matter what even number we stop at, 00:03:47.390 --> 00:03:50.470 there will always be twice as many odd numbers written on 00:03:50.470 --> 00:03:53.450 the board as there would be even numbers. 00:03:53.450 --> 00:03:57.090 In other words, even though in the long run in terms of the 00:03:57.090 --> 00:04:00.070 infinity of each there are as many odds and evens, if we 00:04:00.070 --> 00:04:03.650 stop this process at any finite time no matter how far 00:04:03.650 --> 00:04:06.130 out, there will always be twice as many odds 00:04:06.130 --> 00:04:07.130 as there are evens. 00:04:07.130 --> 00:04:10.290 In fact, if you want to compound this little dilemma, 00:04:10.290 --> 00:04:12.680 write the first two evens, then an odd, in other words, 00:04:12.680 --> 00:04:18.390 2, 4, 1, 6, 8, 3, 10, 12, 5, and you can get twice as many 00:04:18.390 --> 00:04:20.490 evens as there are odds, et cetera. 00:04:20.490 --> 00:04:22.890 And the whole argument again hinges on what? 00:04:22.890 --> 00:04:26.560 Confusing the concept of going out very far 00:04:26.560 --> 00:04:28.940 with going out endlessly. 00:04:28.940 --> 00:04:30.480 Oh, let me give you another example or two. 00:04:30.480 --> 00:04:32.940 I just want to throw these around so you at least get the 00:04:32.940 --> 00:04:36.850 mood created as to what we're really dealing with right now. 00:04:36.850 --> 00:04:40.460 Let's take the endless sequence of numbers, the sum, 00:04:40.460 --> 00:04:44.370 1 plus 'minus 1' plus 1 plus 'minus 1', and 00:04:44.370 --> 00:04:45.900 say let's go on forever. 00:04:45.900 --> 00:04:48.050 What will this sum be? 00:04:48.050 --> 00:04:49.760 Now, lookit, one way of grouping 00:04:49.760 --> 00:04:51.750 these terms is in twos. 00:04:51.750 --> 00:04:54.550 In other words, we'll start with the first two terms, the 00:04:54.550 --> 00:04:55.330 next two terms. 00:04:55.330 --> 00:04:59.460 In other words, we can write this as 1 plus minus 1 plus 1 00:04:59.460 --> 00:05:00.700 plus minus 1. 00:05:00.700 --> 00:05:04.200 And writing it this way, we can see that each term adds up 00:05:04.200 --> 00:05:07.210 to 0, and the infinite sum would be 0. 00:05:07.210 --> 00:05:10.430 On the other hand, if we now leave the first term alone and 00:05:10.430 --> 00:05:15.030 now start grouping the remaining terms in twos, we 00:05:15.030 --> 00:05:17.440 find that the infinite sum is 1. 00:05:17.440 --> 00:05:20.200 Now, we're not going to argue that something is fishy here. 00:05:20.200 --> 00:05:21.340 We're not going to say I wonder which 00:05:21.340 --> 00:05:22.170 is the right answer. 00:05:22.170 --> 00:05:26.790 What we have shown without fear of contradiction is that 00:05:26.790 --> 00:05:29.600 the answer that you get when you add infinitely many terms 00:05:29.600 --> 00:05:33.060 does depend on how you group them, unlike the situation of 00:05:33.060 --> 00:05:35.560 what happens when you add finitely many terms. 00:05:35.560 --> 00:05:38.730 In other words, notice the need for order as well as the 00:05:38.730 --> 00:05:41.530 terms themselves when you have a sum of 00:05:41.530 --> 00:05:42.960 infinitely many terms. 00:05:42.960 --> 00:05:47.000 And the key point is don't be upset when you find out that 00:05:47.000 --> 00:05:48.480 your intuition is defied here. 00:05:48.480 --> 00:05:50.910 You say this doesn't seem real to me. 00:05:50.910 --> 00:05:52.880 It seems intuitively false. 00:05:52.880 --> 00:05:56.110 The point is our intuition is defied. 00:05:56.110 --> 00:05:56.850 Why? 00:05:56.850 --> 00:05:58.710 Because it doesn't apply. 00:05:58.710 --> 00:06:00.310 And why doesn't it apply? 00:06:00.310 --> 00:06:03.810 It doesn't apply because our intuition is based on 00:06:03.810 --> 00:06:08.190 visualizing large but finite amounts, not based on 00:06:08.190 --> 00:06:10.380 visualizing infinity. 00:06:10.380 --> 00:06:13.250 You see, all of these paradoxes stem, because in our 00:06:13.250 --> 00:06:16.510 mind, we're trying to visualize infinity as meaning 00:06:16.510 --> 00:06:18.010 the same as very large. 00:06:18.010 --> 00:06:20.470 Well, you know, now we come to a very important crossroad. 00:06:20.470 --> 00:06:23.450 After all, if infinity is going to be this difficult a 00:06:23.450 --> 00:06:26.770 concept to handle, let's get rid of it the easy way. 00:06:26.770 --> 00:06:28.720 Let's refuse to study it. 00:06:28.720 --> 00:06:30.560 That's one way of solving problems. 00:06:30.560 --> 00:06:33.140 It's what I call the right wing conservative educational 00:06:33.140 --> 00:06:33.700 philosophy. 00:06:33.700 --> 00:06:36.060 If you don't like something, throw it out. 00:06:36.060 --> 00:06:37.870 The only trouble is we need it. 00:06:37.870 --> 00:06:40.160 For example, why do we need it? 00:06:40.160 --> 00:06:41.380 See, why deal with infinite sums? 00:06:41.380 --> 00:06:42.430 Well, because we need them. 00:06:42.430 --> 00:06:44.700 Among other places, we've already used them. 00:06:44.700 --> 00:06:46.430 For example, in computing areas. 00:06:46.430 --> 00:06:48.780 We've taken a limit as 'N' goes to infinity. 00:06:48.780 --> 00:06:53.240 Summation, 'k' goes from 1 to 'n', 'f of 'c sub k'' 'delta 00:06:53.240 --> 00:06:54.630 x', you see. 00:06:54.630 --> 00:06:55.840 And we need this limit. 00:06:55.840 --> 00:06:58.820 And so the question comes up, how shall we add infinitely 00:06:58.820 --> 00:06:59.390 many terms? 00:06:59.390 --> 00:07:00.560 We have a choice now. 00:07:00.560 --> 00:07:02.480 We can throw the thing out, but we don't 00:07:02.480 --> 00:07:03.180 want to throw it out. 00:07:03.180 --> 00:07:04.280 We need it. 00:07:04.280 --> 00:07:06.860 So the question is how shall we add infinitely many terms? 00:07:06.860 --> 00:07:09.690 And even though we know that our intuition can get us in 00:07:09.690 --> 00:07:13.260 trouble, we do have nothing else to begin with. 00:07:13.260 --> 00:07:16.880 So we say OK, let's mimic what happened in the finite case 00:07:16.880 --> 00:07:20.380 and see if we can't extend that in a plausible way to 00:07:20.380 --> 00:07:21.720 cover the infinite case. 00:07:21.720 --> 00:07:24.900 Let me pick a particularly straightforward example. 00:07:24.900 --> 00:07:27.840 Let's suppose I have the three numbers which I'll call 'a sub 00:07:27.840 --> 00:07:31.730 1', 'a sub 2', and 'a sub 3', where 'a sub 1' will be 1/2, 00:07:31.730 --> 00:07:35.310 'a sub 2' will be 1/4, and 'a sub 3' will be 1/8. 00:07:35.310 --> 00:07:37.640 In other words, just for reasons of identification 00:07:37.640 --> 00:07:41.270 later on in what I'm going to be doing, each term is half of 00:07:41.270 --> 00:07:43.190 the previous one. 00:07:43.190 --> 00:07:45.680 Now, I want to find the sum of these three terms. 00:07:45.680 --> 00:07:49.320 I want to find 'a1' plus 'a2' plus 'a3'. 00:07:49.320 --> 00:07:51.750 Now, colloquially, we just say, oh, that's 1/2 plus 1/4 00:07:51.750 --> 00:07:54.270 plus 1/8, and I'll just add them up. 00:07:54.270 --> 00:07:55.650 But do you remember how you learned to add? 00:07:55.650 --> 00:07:57.470 You may not have paid attention to it, but you 00:07:57.470 --> 00:07:58.780 learned to add as a sequence. 00:07:58.780 --> 00:08:00.750 You said I'll add the first one. 00:08:00.750 --> 00:08:04.030 Then the first plus the second gives me a number. 00:08:04.030 --> 00:08:05.850 That's my second partial sum. 00:08:05.850 --> 00:08:07.500 Then I'll add on the third number. 00:08:07.500 --> 00:08:09.830 That will give me my third partial sum. 00:08:09.830 --> 00:08:11.940 Then I have no more numbers to add. 00:08:11.940 --> 00:08:16.460 Consequently, my third partial sum is by definition the sum 00:08:16.460 --> 00:08:17.840 of these three numbers. 00:08:17.840 --> 00:08:21.160 Writing it more symbolically, we say lookit, the first 00:08:21.160 --> 00:08:23.720 partial sum, 's sub 1', is 1/2. 00:08:23.720 --> 00:08:26.430 The second partial sum as 1/2 plus 1/4. 00:08:26.430 --> 00:08:27.990 Another way of saying that is what? 00:08:27.990 --> 00:08:30.550 It's the first partial sums plus the next 00:08:30.550 --> 00:08:31.860 term, which is 1/4. 00:08:31.860 --> 00:08:34.549 1/2 plus 1/4 is 3/4. 00:08:34.549 --> 00:08:38.049 Then we said OK, the third partial sum is what we had 00:08:38.049 --> 00:08:42.820 before, namely, 3/4, plus the next term, which was 1/8, and 00:08:42.820 --> 00:08:45.070 that gives rise to 7/8. 00:08:45.070 --> 00:08:49.320 In other words, we said let's form 'a1', then 'a1 plus a2', 00:08:49.320 --> 00:08:51.640 'a1 plus a2 plus a3'. 00:08:51.640 --> 00:08:54.420 And when we finally finished with our sequence of partial 00:08:54.420 --> 00:08:57.760 sums, the last partial sum was the answer. 00:08:57.760 --> 00:09:01.330 And by the way, let me take time out here to hit home at 00:09:01.330 --> 00:09:04.220 the most important point, the point that I think is 00:09:04.220 --> 00:09:06.460 extremely crucial as a starting point if we're going 00:09:06.460 --> 00:09:09.200 to understand what this whole block is all about. 00:09:09.200 --> 00:09:12.135 It's to distinguish between a series and a sequence. 00:09:12.135 --> 00:09:14.370 And I'll have much more to say about this in the 00:09:14.370 --> 00:09:15.430 supplementary notes. 00:09:15.430 --> 00:09:17.500 But for now, think of it this way. 00:09:17.500 --> 00:09:20.530 A series is a sum of terms. 00:09:20.530 --> 00:09:22.960 A sequence is just a listing of terms. 00:09:22.960 --> 00:09:25.800 In other words, in this particular problem, do not 00:09:25.800 --> 00:09:29.960 confuse the role of the 'a's with the role of the 's's. 00:09:29.960 --> 00:09:33.240 Notice that the a's refer to the sequence of 00:09:33.240 --> 00:09:34.850 numbers being added. 00:09:34.850 --> 00:09:36.600 In other words, the 'a's were what? 00:09:36.600 --> 00:09:39.170 They were 1/2, 1/4 and 1/8. 00:09:39.170 --> 00:09:41.440 These were the three numbers being added. 00:09:41.440 --> 00:09:44.660 Notice that the 's's were the partial sums. 00:09:44.660 --> 00:09:48.690 In other words, the partial sums form the sequence 's1', 00:09:48.690 --> 00:09:50.430 's2', 's3'. 00:09:50.430 --> 00:09:52.780 And to refresh your memories on this, that would be the 00:09:52.780 --> 00:09:53.770 sequence what? 00:09:53.770 --> 00:09:57.150 1/2, 3/4, 7/8. 00:09:57.150 --> 00:09:59.260 In other words, this was the sum of the first number that 00:09:59.260 --> 00:10:00.060 you were adding. 00:10:00.060 --> 00:10:05.340 3/4 was the sum of first two, and 7/8 was the sum of all 00:10:05.340 --> 00:10:06.240 three of them. 00:10:06.240 --> 00:10:10.330 And notice, by the way, the last partial sum, 's sub 3', 00:10:10.330 --> 00:10:13.360 the sum was defined to be the last partial 00:10:13.360 --> 00:10:14.910 sum, and that is what? 00:10:14.910 --> 00:10:15.790 The number-- 00:10:15.790 --> 00:10:17.140 this is very, very crucial. 00:10:17.140 --> 00:10:21.500 1/2 plus 1/4 plus 1/8 is the sum of three numbers, but it's 00:10:21.500 --> 00:10:24.480 one number, and that number is called 7/8. 00:10:24.480 --> 00:10:26.280 OK, you see what we're talking about now? 00:10:26.280 --> 00:10:27.960 We're looking at a bunch of terms. 00:10:27.960 --> 00:10:30.390 We're adding them up, and we see how the sum 00:10:30.390 --> 00:10:32.110 changes with each term. 00:10:32.110 --> 00:10:35.150 In fact, in terms of a very trivial analogy, think of an 00:10:35.150 --> 00:10:37.450 adding machine. 00:10:37.450 --> 00:10:41.050 As you punch numbers in, the 's's are the sums that you see 00:10:41.050 --> 00:10:44.730 being read as your total sum, whereas the a's are the 00:10:44.730 --> 00:10:48.450 individual numbers being punched in to add up, OK? 00:10:48.450 --> 00:10:49.800 I hope that's a trivial example. 00:10:49.800 --> 00:10:52.840 As I listen to myself saying it, it sounds like I made it 00:10:52.840 --> 00:10:54.230 harder than it really is. 00:10:54.230 --> 00:10:57.230 At any rate, let's generalize this particular problem. 00:10:57.230 --> 00:11:00.190 Let's suppose now instead of wanting to add 1/2 plus 1/4 00:11:00.190 --> 00:11:04.320 plus 1/8, we want to add up the first 'n' terms of the 00:11:04.320 --> 00:11:06.710 form 1/2 plus 1/4 plus 1/8. 00:11:06.710 --> 00:11:09.060 In other words, let the n-th term that we're going to add, 00:11:09.060 --> 00:11:12.190 'a sub n', be '1 over '2 to the n''. 00:11:12.190 --> 00:11:16.280 Then the sum, the n-th partial sum here, the sum of these 'n' 00:11:16.280 --> 00:11:18.570 terms is, of course, what? 'a1' plus, et 00:11:18.570 --> 00:11:19.880 cetera, 'a sub n'. 00:11:19.880 --> 00:11:23.950 That turns out to be 1/2 plus 1/4 plus, et cetera, '1 over 00:11:23.950 --> 00:11:25.680 '2 to the n''. 00:11:25.680 --> 00:11:29.670 And by the way, rather than take time to develop this 00:11:29.670 --> 00:11:32.780 recipe over here, I thought you might like to see another 00:11:32.780 --> 00:11:35.690 place that might be interesting to review 00:11:35.690 --> 00:11:37.260 mathematical induction. 00:11:37.260 --> 00:11:40.280 If you'll bear with me and just come back over here where 00:11:40.280 --> 00:11:44.660 we were computing these partial sums, notice that in 00:11:44.660 --> 00:11:48.130 each of these partial sums, notice that your denominator 00:11:48.130 --> 00:11:52.160 is always 2 raised to the same power as this subscript. 00:11:52.160 --> 00:11:54.390 See, 2 the first power is 2. 00:11:54.390 --> 00:11:57.030 2 to the second power is 4. 00:11:57.030 --> 00:11:59.380 2 to the third power is 8. 00:11:59.380 --> 00:12:01.940 In other words, if your subscript is n, your 00:12:01.940 --> 00:12:04.240 denominator is '2 to the n'. 00:12:04.240 --> 00:12:07.190 Notice that your numerator is always one less than your 00:12:07.190 --> 00:12:08.080 denominator. 00:12:08.080 --> 00:12:11.180 In other words, if your denominator is '2 to the n', 00:12:11.180 --> 00:12:14.060 the numerator is '2 to the 'n minus 1''. 00:12:14.060 --> 00:12:17.570 And once we suspect this, this particular result can be 00:12:17.570 --> 00:12:19.040 proven by induction. 00:12:19.040 --> 00:12:20.780 I won't take the time to do this here. 00:12:20.780 --> 00:12:23.690 What I will take the time to do is to observe that this 00:12:23.690 --> 00:12:27.250 particular sum can be written more conveniently if we divide 00:12:27.250 --> 00:12:30.260 through by '2 to the n-th' to get 1 minus 00:12:30.260 --> 00:12:31.990 '1 over '2 the n''. 00:12:31.990 --> 00:12:35.440 For example, suppose we wanted to add up the 10 numbers. 00:12:35.440 --> 00:12:36.570 I say 10 numbers here. 00:12:36.570 --> 00:12:39.050 2 to the 10th is 1.024. 00:12:39.050 --> 00:12:43.190 But according to this recipe, 1/2 plus 1/4 plus 1/8 plus 00:12:43.190 --> 00:12:48.405 1/16 plus, et cetera, et cetera, plus 1/1,024 would add 00:12:48.405 --> 00:12:49.390 up to be what? 00:12:49.390 --> 00:12:53.820 1 minus 1/1,024. 00:12:53.820 --> 00:12:59.300 In other words, this would be 1,023/1,024, which seems to be 00:12:59.300 --> 00:13:00.700 pretty close to 1. 00:13:00.700 --> 00:13:03.390 In fact, you can begin to suspect that as 'n' gets 00:13:03.390 --> 00:13:06.660 arbitrarily large, 's sub n' gets arbitrarily 00:13:06.660 --> 00:13:08.100 close to 1 in value. 00:13:08.100 --> 00:13:09.750 I'm just talking fairly intuitively 00:13:09.750 --> 00:13:11.550 for the time being. 00:13:11.550 --> 00:13:14.460 But, you see, the major question now is suppose you 00:13:14.460 --> 00:13:16.410 elect not to stop at that. 00:13:16.410 --> 00:13:17.860 And you see, this is a very key point. 00:13:17.860 --> 00:13:21.810 We've already seen how the whole world seems to change as 00:13:21.810 --> 00:13:25.720 soon as you say let's never stop as opposed to saying 00:13:25.720 --> 00:13:27.650 let's go out as far as you want. 00:13:27.650 --> 00:13:29.810 See, if we now say what happens if you go on 00:13:29.810 --> 00:13:31.010 endlessly over here? 00:13:31.010 --> 00:13:34.190 Well, it becomes very natural to say lookit, the n-th 00:13:34.190 --> 00:13:37.920 partial sum was 1 minus '1 over '2 to the n-th'. 00:13:37.920 --> 00:13:40.360 In the case where you were adding up a finite number of 00:13:40.360 --> 00:13:44.040 terms, when you came to the last partial sum, that was by 00:13:44.040 --> 00:13:45.550 definition your answer. 00:13:45.550 --> 00:13:47.970 Now, what we're saying is lookit, because we have 00:13:47.970 --> 00:13:51.520 infinitely many terms to add, there is no last partial sum. 00:13:51.520 --> 00:13:54.450 And so what we say is lookit, instead of the last term, 00:13:54.450 --> 00:13:57.210 since there is no last term, why don't we just take the 00:13:57.210 --> 00:14:01.160 limit of the n-th partial sum as 'n' goes to infinity. 00:14:01.160 --> 00:14:03.605 In other words, in this particular case, notice that 00:14:03.605 --> 00:14:07.220 as 'n' approaches infinity, 1 minus '1 over '2 to the n'' 00:14:07.220 --> 00:14:10.910 approaches 1, and we then define the infinite sum, 00:14:10.910 --> 00:14:12.010 meaning what? 00:14:12.010 --> 00:14:14.600 I write it this way: as sigma 'n' goes from 1 to infinity, 00:14:14.600 --> 00:14:16.050 '1 over '2 to the n''. 00:14:16.050 --> 00:14:17.150 It really means what? 00:14:17.150 --> 00:14:20.850 The limit as 'n' goes to infinity: 1/2 plus 1/4 plus 00:14:20.850 --> 00:14:22.410 1/8 plus 1/16-- 00:14:22.410 --> 00:14:24.190 endlessly-- 00:14:24.190 --> 00:14:28.380 that that limit is 1, and we define that to be the sum. 00:14:28.380 --> 00:14:31.730 And again, as I say, I'm going to write that in greatly more 00:14:31.730 --> 00:14:34.160 detail in the notes, and also we'll have many 00:14:34.160 --> 00:14:35.360 exercises on this. 00:14:35.360 --> 00:14:38.830 I just wanted you to see how we get to infinite sums, which 00:14:38.830 --> 00:14:41.600 are called series by generalizing what happens in 00:14:41.600 --> 00:14:42.930 the finite case. 00:14:42.930 --> 00:14:45.350 And because this may seem a little vague to you, let me 00:14:45.350 --> 00:14:47.800 give you a pictorial representation 00:14:47.800 --> 00:14:49.930 of this same thing. 00:14:49.930 --> 00:14:52.380 You see, what's happening here is this. 00:14:52.380 --> 00:14:58.240 Draw a little circle around 1 of bandwidth epsilon. 00:14:58.240 --> 00:15:00.900 In other words, let's mark off an interval epsilon on 00:15:00.900 --> 00:15:02.080 either side of 1. 00:15:02.080 --> 00:15:05.270 And let's call this point here 1 minus epsilon. 00:15:05.270 --> 00:15:07.870 Let's call this point here 1 plus epsilon. 00:15:07.870 --> 00:15:11.140 And what we're saying about our partial sums is this. 00:15:11.140 --> 00:15:13.570 That when you start off and you're adding up terms here, 00:15:13.570 --> 00:15:14.770 you have 1/2. 00:15:14.770 --> 00:15:18.720 1/2 plus 1/4 brings you over to 3/4. 00:15:18.720 --> 00:15:21.420 The next possible sum is 7/8, et cetera. 00:15:21.420 --> 00:15:24.870 And all we're saying is that these terms get arbitrarily 00:15:24.870 --> 00:15:28.240 close to 1 in value, meaning that after a while-- 00:15:28.240 --> 00:15:30.450 and I'll define more rigorously what after a while 00:15:30.450 --> 00:15:31.490 means in a moment-- 00:15:31.490 --> 00:15:35.350 all of the 's sub n's are within epsilon of 1. 00:15:35.350 --> 00:15:38.590 After a while, all of your partial sums are in here. 00:15:38.590 --> 00:15:41.330 And what you mean by after a while certainly depends on how 00:15:41.330 --> 00:15:42.260 big epsilon is. 00:15:42.260 --> 00:15:45.590 In other words, the smaller the bandwidth you allow 00:15:45.590 --> 00:15:49.350 yourself, the more terms you may have to take before you 00:15:49.350 --> 00:15:50.730 get within the tolerance limits 00:15:50.730 --> 00:15:52.270 that you allow yourself. 00:15:52.270 --> 00:15:55.500 In any event, going back to something that we've been 00:15:55.500 --> 00:15:57.470 using for a long time, our basic 00:15:57.470 --> 00:15:58.980 definition is the following. 00:15:58.980 --> 00:16:01.890 If you have an infinite sequence, say, a collection of 00:16:01.890 --> 00:16:05.077 terms 'b sub n', in other words, 'b1', 'b2', 'b3', et 00:16:05.077 --> 00:16:09.900 cetera, without end, we say that that sequence converges 00:16:09.900 --> 00:16:13.390 to the limit 'L' written the limit of 'b sub n' as 'n' 00:16:13.390 --> 00:16:17.360 approaches infinity equals 'L', if and only if for every 00:16:17.360 --> 00:16:21.510 epsilon greater than 0 we can find a number 'N' which 00:16:21.510 --> 00:16:22.910 depends on epsilon-- 00:16:22.910 --> 00:16:25.480 notice the notation here: 'N' as a function of epsilon-- 00:16:25.480 --> 00:16:29.280 such that whenever little 'n' is greater than capital 'N', 00:16:29.280 --> 00:16:31.740 the absolute value of 'a sub n' minus 'L' 00:16:31.740 --> 00:16:33.050 is less than epsilon. 00:16:33.050 --> 00:16:35.755 And, you see, again, you may wonder how in the world that 00:16:35.755 --> 00:16:36.920 you're going to remember this. 00:16:36.920 --> 00:16:39.690 If you memorize this, I guarantee you, in two day's 00:16:39.690 --> 00:16:42.020 time, you'll have to memorize it again. 00:16:42.020 --> 00:16:44.790 I also hope you have enough faith in me to recognize I 00:16:44.790 --> 00:16:46.110 didn't memorize this. 00:16:46.110 --> 00:16:48.060 There is a feeling that one gets for this. 00:16:48.060 --> 00:16:50.090 And let me give you what that feeling is. 00:16:50.090 --> 00:16:53.140 Again, in terms of a picture, what it means-- well, I'll 00:16:53.140 --> 00:16:55.720 change these to 'a's now because that's the symbols 00:16:55.720 --> 00:16:58.170 that we've been using before in terms of 00:16:58.170 --> 00:16:59.005 the sequence of terms. 00:16:59.005 --> 00:17:01.220 What we really mean-- and I don't care what symbol you 00:17:01.220 --> 00:17:02.270 really use here-- 00:17:02.270 --> 00:17:06.060 is if you want to talk about the limit of 'a sub n' as 'n' 00:17:06.060 --> 00:17:08.930 approaches infinity, if that limit equals 'L', what the 00:17:08.930 --> 00:17:10.940 rigorous definition says is this. 00:17:10.940 --> 00:17:14.150 Draw yourself an interval around 'L' of bandwidth 00:17:14.150 --> 00:17:15.740 epsilon, in other words, from 'L minus 00:17:15.740 --> 00:17:18.089 epsilon' to 'L plus epsilon'. 00:17:18.089 --> 00:17:20.970 And what this thing says is that beyond a certain term, 00:17:20.970 --> 00:17:24.250 say, the capital N-th term, every term beyond this certain 00:17:24.250 --> 00:17:26.700 one is in here. 00:17:29.790 --> 00:17:35.240 Well, all 'a n's are in here if 'n' is sufficiently large. 00:17:35.240 --> 00:17:37.100 I don't know if you can read that very well, but just 00:17:37.100 --> 00:17:38.020 listen to what I'm saying. 00:17:38.020 --> 00:17:42.060 All of the terms are in here if 'n' is sufficiently large. 00:17:42.060 --> 00:17:45.820 What this means again is that to all intents and purposes, 00:17:45.820 --> 00:17:49.440 if you think of this bandwidth as giving you a dot, see, a 00:17:49.440 --> 00:17:53.160 thick dot here where the endpoints are 'L minus 00:17:53.160 --> 00:17:56.520 epsilon' and 'L plus epsilon', what we're saying is lookit, 00:17:56.520 --> 00:17:59.600 after a certain term, the way I've drawn here, after the 00:17:59.600 --> 00:18:02.290 fifth term, all the remaining terms of my 00:18:02.290 --> 00:18:04.310 sequence are in here. 00:18:04.310 --> 00:18:07.740 By the way, notice the role of the subscripts here. 00:18:07.740 --> 00:18:10.330 All the subscript tells you is where the term 00:18:10.330 --> 00:18:12.550 appears in your sequence. 00:18:12.550 --> 00:18:15.300 For example, the third term in your sequence could be a 00:18:15.300 --> 00:18:18.040 smaller number than the second term of your sequence. 00:18:18.040 --> 00:18:21.210 Do not confuse the size of the terms with the subscripts. 00:18:21.210 --> 00:18:24.170 The subscripts order the terms, but the third term in 00:18:24.170 --> 00:18:27.790 the sequence can be less than in size than the second term 00:18:27.790 --> 00:18:28.560 in the sequence. 00:18:28.560 --> 00:18:30.170 But again, I'll talk about that in more 00:18:30.170 --> 00:18:31.280 detail in the notes. 00:18:31.280 --> 00:18:36.020 The point that I want you to see is that in concept what 00:18:36.020 --> 00:18:37.840 limit does is the following. 00:18:37.840 --> 00:18:42.750 Limit is to an infinite sequence as last term is to a 00:18:42.750 --> 00:18:44.200 finite sequence. 00:18:44.200 --> 00:18:48.320 In other words, a limit replaces infinitely many 00:18:48.320 --> 00:18:52.720 points by a finite number of points plus a dot. 00:18:52.720 --> 00:18:55.780 You see, going back to this example here, how many 'a sub 00:18:55.780 --> 00:18:56.710 n's were there? 00:18:56.710 --> 00:18:58.760 Well, there were infinitely many. 00:18:58.760 --> 00:19:01.530 Well, to keep track of these infinitely many, what do I 00:19:01.530 --> 00:19:02.780 have to keep track of now? 00:19:02.780 --> 00:19:07.770 Well, in this diagram, the first five 'a's plus this dot, 00:19:07.770 --> 00:19:12.290 because you see, every one of my infinitely many 'a's past 00:19:12.290 --> 00:19:16.170 the fifth one is inside this dot, you see? 00:19:16.170 --> 00:19:18.450 So in other words then, what's happened? 00:19:18.450 --> 00:19:20.170 The thing that had to happen. 00:19:20.170 --> 00:19:22.670 We had to deal with infinite sequences. 00:19:22.670 --> 00:19:26.200 We saw the big philosophic difference between infinitely 00:19:26.200 --> 00:19:28.630 many and just large. 00:19:28.630 --> 00:19:33.550 And so our definition of limit had to be such that we could 00:19:33.550 --> 00:19:37.930 reduce in a way that was compatible with our intuition 00:19:37.930 --> 00:19:42.060 the concept of infinitely many points to a finite number. 00:19:42.060 --> 00:19:45.430 Because, you see, as I'll show you in the notes also, all of 00:19:45.430 --> 00:19:49.300 our arithmetic is geared for just a finite number of 00:19:49.300 --> 00:19:50.610 operations. 00:19:50.610 --> 00:19:52.280 See, this is why this definition 00:19:52.280 --> 00:19:53.920 of limit is so crucial. 00:19:53.920 --> 00:19:57.360 Again, you may notice, and I'll remind you of this also 00:19:57.360 --> 00:20:00.960 in the exercises, that structurally this definition 00:20:00.960 --> 00:20:04.040 of limit is the same as the limit that we use when we 00:20:04.040 --> 00:20:07.000 talked about the limit of 'f of x', as 'x' approaches 'a', 00:20:07.000 --> 00:20:07.880 equals 'L'. 00:20:07.880 --> 00:20:12.060 The absolute value signs have the same properties as before. 00:20:12.060 --> 00:20:14.470 And by the way, before I go on, let me just remind you 00:20:14.470 --> 00:20:17.810 again of one more thing while I'm talking that way. 00:20:17.810 --> 00:20:20.950 Instead of memorizing this, remember how you read this. 00:20:20.950 --> 00:20:24.180 The absolute value of 'a sub n' minus 'L' is less than 00:20:24.180 --> 00:20:25.660 epsilon means what? 00:20:25.660 --> 00:20:28.820 That 'a sub n' is within epsilon of 'L'. 00:20:28.820 --> 00:20:30.710 That's what we use in our diagram. 00:20:30.710 --> 00:20:32.740 But it seems to me I forgot to mention this. 00:20:32.740 --> 00:20:34.240 And I want you to see that what? 00:20:34.240 --> 00:20:37.790 The key building block analytically is the absolute 00:20:37.790 --> 00:20:40.690 value, and the meaning of absolute value is the same 00:20:40.690 --> 00:20:45.210 here as it was in blocks one and two of our course. 00:20:45.210 --> 00:20:49.010 So what I'm driving at is that the same limit theorems that 00:20:49.010 --> 00:20:52.260 we've been able to use up until now still apply. 00:20:52.260 --> 00:20:54.640 Oh, by means of an example. 00:20:54.640 --> 00:20:57.830 Suppose I have the limit as 'n' approaches infinity, '2n 00:20:57.830 --> 00:21:01.110 plus 3' over '5n plus 7'. 00:21:01.110 --> 00:21:03.130 Notice that I can divide numerator and denominator 00:21:03.130 --> 00:21:04.190 through by 'n'. 00:21:04.190 --> 00:21:07.100 If I do that, I have the limit as 'n' approaches infinity. 00:21:07.100 --> 00:21:11.060 '2 plus '3/n'' over '5 plus '7/n''. 00:21:11.060 --> 00:21:13.910 Now using the fact that the limit of a sum is the sum of 00:21:13.910 --> 00:21:16.380 the limits, the limit of a quotient is the quotient of 00:21:16.380 --> 00:21:21.320 the limits, the limit of '1/n' as 'n' goes to infinity is 0. 00:21:21.320 --> 00:21:24.190 Notice that I can use the limit theorems to conclude 00:21:24.190 --> 00:21:27.450 that the limit of this particular sequence is 2/5. 00:21:27.450 --> 00:21:31.980 If I wanted to, the same ways we did in block one, block 00:21:31.980 --> 00:21:35.710 two, where we're talking about limits, given an epsilon, I 00:21:35.710 --> 00:21:39.360 can actually exhibit how far out I have to go before each 00:21:39.360 --> 00:21:41.800 of the terms in this sequence is within that 00:21:41.800 --> 00:21:44.180 given epsilon of 2/5. 00:21:44.180 --> 00:21:48.310 By the way, again to emphasize once more, because this is so 00:21:48.310 --> 00:21:52.230 important, the difference between an infinite sum and an 00:21:52.230 --> 00:21:55.560 infinite sequence, observe that whereas the limit of the 00:21:55.560 --> 00:22:01.070 sequence of terms '2n plus 3' over '5n plus 7' is 2/5, the 00:22:01.070 --> 00:22:07.360 infinite sum composed of the terms of the form '2n plus 3' 00:22:07.360 --> 00:22:13.240 over '5n plus 7' is infinity since after a while each term 00:22:13.240 --> 00:22:16.080 that you're adding here behaves like 2/5. 00:22:16.080 --> 00:22:18.390 In other words, if you write this thing out to see what 00:22:18.390 --> 00:22:20.550 this means, pick 'n' to be 1. 00:22:20.550 --> 00:22:22.800 When 'n' is 1, this term is 5/12. 00:22:22.800 --> 00:22:24.570 When 'n' is 2, this is what? 00:22:24.570 --> 00:22:27.620 7 plus 17, 7/17. 00:22:27.620 --> 00:22:30.660 When 'n' is 3, this is 9/22. 00:22:30.660 --> 00:22:35.020 When 'n' is 4, this is 8 plus 3 is 11, over 27. 00:22:35.020 --> 00:22:37.070 In other words, what you're saying is the infinite sum 00:22:37.070 --> 00:22:40.060 means to add up all of these terms. 00:22:40.060 --> 00:22:44.080 The thing whose limit was 2/5 was the sequence of terms 00:22:44.080 --> 00:22:44.710 themselves. 00:22:44.710 --> 00:22:47.410 In other words, what we're saying is that after a certain 00:22:47.410 --> 00:22:50.810 point, every one of these terms behaves like 2/5. 00:22:50.810 --> 00:22:53.360 And what you're saying is lookit, after a point, what 00:22:53.360 --> 00:22:56.610 you're really doing is essentially adding on 2/5 00:22:56.610 --> 00:22:58.370 every time you add on another term. 00:22:58.370 --> 00:23:02.230 And therefore, this sum can get as large as you want, just 00:23:02.230 --> 00:23:03.840 by adding on enough terms. 00:23:03.840 --> 00:23:07.930 Again, observe the difference between the partial sums and 00:23:07.930 --> 00:23:09.260 the terms themselves. 00:23:09.260 --> 00:23:11.230 The terms that you're adding are 00:23:11.230 --> 00:23:13.430 approaching 2/5 as a limit. 00:23:13.430 --> 00:23:16.120 The thing that's becoming infinite is the sequence of 00:23:16.120 --> 00:23:17.210 partial sums. 00:23:17.210 --> 00:23:20.300 Because what you're saying is to get from one partial sum to 00:23:20.300 --> 00:23:22.150 the next, you're, roughly speaking, 00:23:22.150 --> 00:23:24.940 adding on 2/5 each time. 00:23:24.940 --> 00:23:28.380 To generalize this, what we're saying is if the sequence of 00:23:28.380 --> 00:23:32.220 partial sums converges, the individual terms that you're 00:23:32.220 --> 00:23:35.210 adding must approach 0 in the limit. 00:23:35.210 --> 00:23:38.450 For if the limit of the 'a sub n's as 'n' approaches infinity 00:23:38.450 --> 00:23:43.500 is 'L', where 'L' is not 0, then beyond a certain term, 00:23:43.500 --> 00:23:45.820 the sum of the 'a sub n's behaves like 00:23:45.820 --> 00:23:47.280 the sum of the 'L's. 00:23:47.280 --> 00:23:50.250 And what you're saying is if 'L' is non zero, by adding on 00:23:50.250 --> 00:23:53.330 enough of these fixed 'L's, you can make the sum as large 00:23:53.330 --> 00:23:55.380 as you wish. 00:23:55.380 --> 00:23:58.550 In other words, then, a sort of negative test is that if 00:23:58.550 --> 00:24:01.520 you know that the series converges, then the terms that 00:24:01.520 --> 00:24:04.520 you're adding on must approach 0 in the limit. 00:24:04.520 --> 00:24:07.650 Unfortunately, by the way, the converse is not true. 00:24:07.650 --> 00:24:10.790 Namely, if you know that the terms that you're adding on go 00:24:10.790 --> 00:24:15.680 to 0, you cannot conclude that their sum is finite. 00:24:15.680 --> 00:24:19.230 Again, it's our old friend of infinity times 0. 00:24:19.230 --> 00:24:25.140 You see, as these terms approach 0, when you start to 00:24:25.140 --> 00:24:27.840 add them up, it may be that they're not 00:24:27.840 --> 00:24:29.030 going to 0 fast enough. 00:24:29.030 --> 00:24:31.160 In other words, notice that the terms are getting small, 00:24:31.160 --> 00:24:33.500 but you're also adding more and more of them. 00:24:33.500 --> 00:24:34.870 You see, what I wrote here is what? 00:24:34.870 --> 00:24:37.630 On the other hand, the limit of 'a sub n' as 'n' approaches 00:24:37.630 --> 00:24:40.840 infinity equals 0 is not enough to guarantee the 00:24:40.840 --> 00:24:43.460 convergence of this particular sum. 00:24:43.460 --> 00:24:46.350 In fact, a trivial example to show this is look at the 00:24:46.350 --> 00:24:48.440 following contrived example. 00:24:48.440 --> 00:24:50.970 Start out with the first number being 1. 00:24:50.970 --> 00:24:57.720 Then take 1/2 twice, 1/3 three times, 1/4 four times, 1/5 00:24:57.720 --> 00:25:00.520 five times, 1/6 six times. 00:25:00.520 --> 00:25:02.830 Form the n-th partial sum. 00:25:02.830 --> 00:25:06.100 Lookit, is it clear that the terms that are going into your 00:25:06.100 --> 00:25:09.030 sum are approaching 0 in the limit? 00:25:09.030 --> 00:25:12.590 You see, you have a one, then there are halves, then thirds, 00:25:12.590 --> 00:25:15.060 then fourths, then fifths, then sixths, 00:25:15.060 --> 00:25:16.180 sevenths, et cetera. 00:25:16.180 --> 00:25:17.900 The terms themselves are getting 00:25:17.900 --> 00:25:19.780 arbitrarily close to 0. 00:25:19.780 --> 00:25:22.060 On the other hand, what is the sum becoming? 00:25:22.060 --> 00:25:23.820 Well, this adds up to 1. 00:25:23.820 --> 00:25:25.290 This adds up to 1. 00:25:25.290 --> 00:25:28.090 This adds up to 1, and this that up to 1. 00:25:28.090 --> 00:25:31.700 And in other words, by taking enough terms, I can tack on as 00:25:31.700 --> 00:25:34.540 many ones is I want, and ultimately, even though the 00:25:34.540 --> 00:25:37.390 terms become small, the sum becomes large. 00:25:37.390 --> 00:25:40.520 In fact, it's precisely because of this unpleasantness 00:25:40.520 --> 00:25:44.790 that we have to go into a rather difficult lecture next 00:25:44.790 --> 00:25:48.610 time, talking about OK, how then can you tell when an 00:25:48.610 --> 00:25:51.840 infinite sum converges to a finite limit and 00:25:51.840 --> 00:25:53.260 when doesn't it? 00:25:53.260 --> 00:25:54.430 At any rate, that's what I said we're going to 00:25:54.430 --> 00:25:56.050 talk about next time. 00:25:56.050 --> 00:25:58.660 As far as today's lesson is concerned, I hope that we've 00:25:58.660 --> 00:26:00.710 straightened out the difference between a sequence 00:26:00.710 --> 00:26:04.100 and the series, partial sums and the terms being added. 00:26:04.100 --> 00:26:06.590 And in the hopes that we've done that, let me say, until 00:26:06.590 --> 00:26:07.840 next time, goodbye. 00:26:09.970 --> 00:26:13.170 Funding for the publication of this video was provided by the 00:26:13.170 --> 00:26:17.220 Gabriella and Paul Rosenbaum Foundation. 00:26:17.220 --> 00:26:21.390 Help OCW continue to provide free and open access to MIT 00:26:21.390 --> 00:26:25.590 courses by making a donation at ocw.mit.edu/donate.