1 00:00:00,040 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,690 Commons license. 3 00:00:03,690 --> 00:00:06,640 Your support will help MIT OpenCourseWare continue to 4 00:00:06,640 --> 00:00:09,980 offer high-quality educational resources for free. 5 00:00:09,980 --> 00:00:12,830 To make a donation or to view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:26,555 --> 00:00:27,510 PROFESSOR: Hi. 9 00:00:27,510 --> 00:00:30,770 Today we begin our final block of material in this particular 10 00:00:30,770 --> 00:00:34,650 course, and it's the segment entitled Infinite Series. 11 00:00:34,650 --> 00:00:37,510 And perhaps the best way to motivate this rather difficult 12 00:00:37,510 --> 00:00:40,260 block of material is in terms of the concept 13 00:00:40,260 --> 00:00:42,080 of many versus infinite. 14 00:00:42,080 --> 00:00:45,190 In many respects, this particular block could've been 15 00:00:45,190 --> 00:00:46,860 given much earlier in the course. 16 00:00:46,860 --> 00:00:50,140 But somehow or other, until we have some sort of a feeling as 17 00:00:50,140 --> 00:00:53,760 to what infinity really means, we have a maturity problem in 18 00:00:53,760 --> 00:00:55,910 trying to really grasp the significance of 19 00:00:55,910 --> 00:00:56,850 what's going on. 20 00:00:56,850 --> 00:00:59,120 In fact, in a manner of speaking, with all of this 21 00:00:59,120 --> 00:01:02,130 experience, there may be a maturity problem in trying to 22 00:01:02,130 --> 00:01:03,770 grasp the fundamental ideas. 23 00:01:03,770 --> 00:01:07,000 What I shall do throughout the material on this block is to 24 00:01:07,000 --> 00:01:10,180 utilize the lectures again to make sure that the concepts 25 00:01:10,180 --> 00:01:13,720 become crystallized and use the learning exercises plus 26 00:01:13,720 --> 00:01:17,140 the text plus supplements notes to make sure that the 27 00:01:17,140 --> 00:01:20,750 details are taken care of in adequate fashion. 28 00:01:20,750 --> 00:01:23,040 At any rate, I've entitle today's lecture 29 00:01:23,040 --> 00:01:24,450 'Many Versus Infinite'. 30 00:01:24,450 --> 00:01:27,290 And I thought the best way to get started on this was to 31 00:01:27,290 --> 00:01:29,990 think of a number that's very easy to write in terms of 32 00:01:29,990 --> 00:01:31,790 exponential notation. 33 00:01:31,790 --> 00:01:35,560 Let capital 'N' be 10 to the 10 to the 10th power. 34 00:01:35,560 --> 00:01:37,910 10 to the 10, by the way, is 10 billion, a 1 35 00:01:37,910 --> 00:01:39,270 followed by 10 zeroes. 36 00:01:39,270 --> 00:01:42,030 That's 10 to the 10-billionth power. 37 00:01:42,030 --> 00:01:44,530 That, of course, means, if written in place value, that 38 00:01:44,530 --> 00:01:48,200 would be a 1 followed by 10 billion zeroes. 39 00:01:48,200 --> 00:01:50,190 And for those of you who would like an exercise in 40 00:01:50,190 --> 00:01:52,810 multiplication and long division and you want to 41 00:01:52,810 --> 00:01:55,790 compute the number of seconds in a year and what have you, 42 00:01:55,790 --> 00:01:57,970 it turns out without too much difficulty 43 00:01:57,970 --> 00:01:59,360 that it can be shown. 44 00:01:59,360 --> 00:02:02,990 That to write 1 billion zeroes at the rate of one per second 45 00:02:02,990 --> 00:02:06,380 would take in the order of magnitude of some 32 years. 46 00:02:06,380 --> 00:02:10,060 In other words, this number capital 'N', roughly speaking, 47 00:02:10,060 --> 00:02:12,960 writing it in place value notation at the rate of one 48 00:02:12,960 --> 00:02:16,640 digit per second would take 320 years to write. 49 00:02:16,640 --> 00:02:18,070 And you say so what? 50 00:02:18,070 --> 00:02:20,810 And the answer is, well, after you've got out that far-- and 51 00:02:20,810 --> 00:02:22,560 by the way, this is crucial. 52 00:02:22,560 --> 00:02:25,380 320 years is a long time. 53 00:02:25,380 --> 00:02:26,230 I was going to say it's a lifetime. 54 00:02:26,230 --> 00:02:27,610 It's more than the lifetime. 55 00:02:27,610 --> 00:02:29,860 It's a long time, but it's finite. 56 00:02:29,860 --> 00:02:32,190 Eventually, the job could be completed. 57 00:02:32,190 --> 00:02:34,710 But the interesting point is that once it's completed, the 58 00:02:34,710 --> 00:02:38,400 next number in our system is capital 'N plus 1', capital 'N 59 00:02:38,400 --> 00:02:42,920 plus 2', capital 'N plus 3', where in a sense then, with 60 00:02:42,920 --> 00:02:46,140 'N' as a new reference point, we're back to the beginning of 61 00:02:46,140 --> 00:02:47,030 our number system. 62 00:02:47,030 --> 00:02:50,740 In other words, granted that 'N' is a fantastically large 63 00:02:50,740 --> 00:02:54,020 number, if you wanted to become wealthy, to own 'N' 64 00:02:54,020 --> 00:02:56,810 dollars would more than realize your dream. 65 00:02:56,810 --> 00:03:00,850 But if your aim was to own infinitely much money, 'N' 66 00:03:00,850 --> 00:03:04,220 would be no closer than having no money at all. 'N' is no 67 00:03:04,220 --> 00:03:06,520 nearer the end of a number system than is 68 00:03:06,520 --> 00:03:07,990 the number 1 itself. 69 00:03:07,990 --> 00:03:11,020 There is the story that signifies the difference 70 00:03:11,020 --> 00:03:12,670 between many and infinite. 71 00:03:12,670 --> 00:03:14,590 And to hammer this point home, let me give 72 00:03:14,590 --> 00:03:16,600 you a few more examples. 73 00:03:16,600 --> 00:03:19,740 I cleverly call this additional examples. 74 00:03:19,740 --> 00:03:22,720 We all know that there are just as many odd numbers and 75 00:03:22,720 --> 00:03:23,640 even numbers, right? 76 00:03:23,640 --> 00:03:25,600 The odds and the evens match up. 77 00:03:25,600 --> 00:03:27,630 Now, watch the following little gimmick. 78 00:03:27,630 --> 00:03:30,450 Write the first two odd numbers, then the first even 79 00:03:30,450 --> 00:03:33,490 number, the next two odd numbers, then the next even 80 00:03:33,490 --> 00:03:35,570 number, the next two odd numbers, than 81 00:03:35,570 --> 00:03:36,780 the next even number. 82 00:03:36,780 --> 00:03:39,080 And go on like this as long as you want. 83 00:03:39,080 --> 00:03:43,510 And no matter where we stop, even if we go to the 10 to the 84 00:03:43,510 --> 00:03:47,390 10 to the 10th term, no matter what even number we stop at, 85 00:03:47,390 --> 00:03:50,470 there will always be twice as many odd numbers written on 86 00:03:50,470 --> 00:03:53,450 the board as there would be even numbers. 87 00:03:53,450 --> 00:03:57,090 In other words, even though in the long run in terms of the 88 00:03:57,090 --> 00:04:00,070 infinity of each there are as many odds and evens, if we 89 00:04:00,070 --> 00:04:03,650 stop this process at any finite time no matter how far 90 00:04:03,650 --> 00:04:06,130 out, there will always be twice as many odds 91 00:04:06,130 --> 00:04:07,130 as there are evens. 92 00:04:07,130 --> 00:04:10,290 In fact, if you want to compound this little dilemma, 93 00:04:10,290 --> 00:04:12,680 write the first two evens, then an odd, in other words, 94 00:04:12,680 --> 00:04:18,390 2, 4, 1, 6, 8, 3, 10, 12, 5, and you can get twice as many 95 00:04:18,390 --> 00:04:20,490 evens as there are odds, et cetera. 96 00:04:20,490 --> 00:04:22,890 And the whole argument again hinges on what? 97 00:04:22,890 --> 00:04:26,560 Confusing the concept of going out very far 98 00:04:26,560 --> 00:04:28,940 with going out endlessly. 99 00:04:28,940 --> 00:04:30,480 Oh, let me give you another example or two. 100 00:04:30,480 --> 00:04:32,940 I just want to throw these around so you at least get the 101 00:04:32,940 --> 00:04:36,850 mood created as to what we're really dealing with right now. 102 00:04:36,850 --> 00:04:40,460 Let's take the endless sequence of numbers, the sum, 103 00:04:40,460 --> 00:04:44,370 1 plus 'minus 1' plus 1 plus 'minus 1', and 104 00:04:44,370 --> 00:04:45,900 say let's go on forever. 105 00:04:45,900 --> 00:04:48,050 What will this sum be? 106 00:04:48,050 --> 00:04:49,760 Now, lookit, one way of grouping 107 00:04:49,760 --> 00:04:51,750 these terms is in twos. 108 00:04:51,750 --> 00:04:54,550 In other words, we'll start with the first two terms, the 109 00:04:54,550 --> 00:04:55,330 next two terms. 110 00:04:55,330 --> 00:04:59,460 In other words, we can write this as 1 plus minus 1 plus 1 111 00:04:59,460 --> 00:05:00,700 plus minus 1. 112 00:05:00,700 --> 00:05:04,200 And writing it this way, we can see that each term adds up 113 00:05:04,200 --> 00:05:07,210 to 0, and the infinite sum would be 0. 114 00:05:07,210 --> 00:05:10,430 On the other hand, if we now leave the first term alone and 115 00:05:10,430 --> 00:05:15,030 now start grouping the remaining terms in twos, we 116 00:05:15,030 --> 00:05:17,440 find that the infinite sum is 1. 117 00:05:17,440 --> 00:05:20,200 Now, we're not going to argue that something is fishy here. 118 00:05:20,200 --> 00:05:21,340 We're not going to say I wonder which 119 00:05:21,340 --> 00:05:22,170 is the right answer. 120 00:05:22,170 --> 00:05:26,790 What we have shown without fear of contradiction is that 121 00:05:26,790 --> 00:05:29,600 the answer that you get when you add infinitely many terms 122 00:05:29,600 --> 00:05:33,060 does depend on how you group them, unlike the situation of 123 00:05:33,060 --> 00:05:35,560 what happens when you add finitely many terms. 124 00:05:35,560 --> 00:05:38,730 In other words, notice the need for order as well as the 125 00:05:38,730 --> 00:05:41,530 terms themselves when you have a sum of 126 00:05:41,530 --> 00:05:42,960 infinitely many terms. 127 00:05:42,960 --> 00:05:47,000 And the key point is don't be upset when you find out that 128 00:05:47,000 --> 00:05:48,480 your intuition is defied here. 129 00:05:48,480 --> 00:05:50,910 You say this doesn't seem real to me. 130 00:05:50,910 --> 00:05:52,880 It seems intuitively false. 131 00:05:52,880 --> 00:05:56,110 The point is our intuition is defied. 132 00:05:56,110 --> 00:05:56,850 Why? 133 00:05:56,850 --> 00:05:58,710 Because it doesn't apply. 134 00:05:58,710 --> 00:06:00,310 And why doesn't it apply? 135 00:06:00,310 --> 00:06:03,810 It doesn't apply because our intuition is based on 136 00:06:03,810 --> 00:06:08,190 visualizing large but finite amounts, not based on 137 00:06:08,190 --> 00:06:10,380 visualizing infinity. 138 00:06:10,380 --> 00:06:13,250 You see, all of these paradoxes stem, because in our 139 00:06:13,250 --> 00:06:16,510 mind, we're trying to visualize infinity as meaning 140 00:06:16,510 --> 00:06:18,010 the same as very large. 141 00:06:18,010 --> 00:06:20,470 Well, you know, now we come to a very important crossroad. 142 00:06:20,470 --> 00:06:23,450 After all, if infinity is going to be this difficult a 143 00:06:23,450 --> 00:06:26,770 concept to handle, let's get rid of it the easy way. 144 00:06:26,770 --> 00:06:28,720 Let's refuse to study it. 145 00:06:28,720 --> 00:06:30,560 That's one way of solving problems. 146 00:06:30,560 --> 00:06:33,140 It's what I call the right wing conservative educational 147 00:06:33,140 --> 00:06:33,700 philosophy. 148 00:06:33,700 --> 00:06:36,060 If you don't like something, throw it out. 149 00:06:36,060 --> 00:06:37,870 The only trouble is we need it. 150 00:06:37,870 --> 00:06:40,160 For example, why do we need it? 151 00:06:40,160 --> 00:06:41,380 See, why deal with infinite sums? 152 00:06:41,380 --> 00:06:42,430 Well, because we need them. 153 00:06:42,430 --> 00:06:44,700 Among other places, we've already used them. 154 00:06:44,700 --> 00:06:46,430 For example, in computing areas. 155 00:06:46,430 --> 00:06:48,780 We've taken a limit as 'N' goes to infinity. 156 00:06:48,780 --> 00:06:53,240 Summation, 'k' goes from 1 to 'n', 'f of 'c sub k'' 'delta 157 00:06:53,240 --> 00:06:54,630 x', you see. 158 00:06:54,630 --> 00:06:55,840 And we need this limit. 159 00:06:55,840 --> 00:06:58,820 And so the question comes up, how shall we add infinitely 160 00:06:58,820 --> 00:06:59,390 many terms? 161 00:06:59,390 --> 00:07:00,560 We have a choice now. 162 00:07:00,560 --> 00:07:02,480 We can throw the thing out, but we don't 163 00:07:02,480 --> 00:07:03,180 want to throw it out. 164 00:07:03,180 --> 00:07:04,280 We need it. 165 00:07:04,280 --> 00:07:06,860 So the question is how shall we add infinitely many terms? 166 00:07:06,860 --> 00:07:09,690 And even though we know that our intuition can get us in 167 00:07:09,690 --> 00:07:13,260 trouble, we do have nothing else to begin with. 168 00:07:13,260 --> 00:07:16,880 So we say OK, let's mimic what happened in the finite case 169 00:07:16,880 --> 00:07:20,380 and see if we can't extend that in a plausible way to 170 00:07:20,380 --> 00:07:21,720 cover the infinite case. 171 00:07:21,720 --> 00:07:24,900 Let me pick a particularly straightforward example. 172 00:07:24,900 --> 00:07:27,840 Let's suppose I have the three numbers which I'll call 'a sub 173 00:07:27,840 --> 00:07:31,730 1', 'a sub 2', and 'a sub 3', where 'a sub 1' will be 1/2, 174 00:07:31,730 --> 00:07:35,310 'a sub 2' will be 1/4, and 'a sub 3' will be 1/8. 175 00:07:35,310 --> 00:07:37,640 In other words, just for reasons of identification 176 00:07:37,640 --> 00:07:41,270 later on in what I'm going to be doing, each term is half of 177 00:07:41,270 --> 00:07:43,190 the previous one. 178 00:07:43,190 --> 00:07:45,680 Now, I want to find the sum of these three terms. 179 00:07:45,680 --> 00:07:49,320 I want to find 'a1' plus 'a2' plus 'a3'. 180 00:07:49,320 --> 00:07:51,750 Now, colloquially, we just say, oh, that's 1/2 plus 1/4 181 00:07:51,750 --> 00:07:54,270 plus 1/8, and I'll just add them up. 182 00:07:54,270 --> 00:07:55,650 But do you remember how you learned to add? 183 00:07:55,650 --> 00:07:57,470 You may not have paid attention to it, but you 184 00:07:57,470 --> 00:07:58,780 learned to add as a sequence. 185 00:07:58,780 --> 00:08:00,750 You said I'll add the first one. 186 00:08:00,750 --> 00:08:04,030 Then the first plus the second gives me a number. 187 00:08:04,030 --> 00:08:05,850 That's my second partial sum. 188 00:08:05,850 --> 00:08:07,500 Then I'll add on the third number. 189 00:08:07,500 --> 00:08:09,830 That will give me my third partial sum. 190 00:08:09,830 --> 00:08:11,940 Then I have no more numbers to add. 191 00:08:11,940 --> 00:08:16,460 Consequently, my third partial sum is by definition the sum 192 00:08:16,460 --> 00:08:17,840 of these three numbers. 193 00:08:17,840 --> 00:08:21,160 Writing it more symbolically, we say lookit, the first 194 00:08:21,160 --> 00:08:23,720 partial sum, 's sub 1', is 1/2. 195 00:08:23,720 --> 00:08:26,430 The second partial sum as 1/2 plus 1/4. 196 00:08:26,430 --> 00:08:27,990 Another way of saying that is what? 197 00:08:27,990 --> 00:08:30,550 It's the first partial sums plus the next 198 00:08:30,550 --> 00:08:31,860 term, which is 1/4. 199 00:08:31,860 --> 00:08:34,549 1/2 plus 1/4 is 3/4. 200 00:08:34,549 --> 00:08:38,049 Then we said OK, the third partial sum is what we had 201 00:08:38,049 --> 00:08:42,820 before, namely, 3/4, plus the next term, which was 1/8, and 202 00:08:42,820 --> 00:08:45,070 that gives rise to 7/8. 203 00:08:45,070 --> 00:08:49,320 In other words, we said let's form 'a1', then 'a1 plus a2', 204 00:08:49,320 --> 00:08:51,640 'a1 plus a2 plus a3'. 205 00:08:51,640 --> 00:08:54,420 And when we finally finished with our sequence of partial 206 00:08:54,420 --> 00:08:57,760 sums, the last partial sum was the answer. 207 00:08:57,760 --> 00:09:01,330 And by the way, let me take time out here to hit home at 208 00:09:01,330 --> 00:09:04,220 the most important point, the point that I think is 209 00:09:04,220 --> 00:09:06,460 extremely crucial as a starting point if we're going 210 00:09:06,460 --> 00:09:09,200 to understand what this whole block is all about. 211 00:09:09,200 --> 00:09:12,135 It's to distinguish between a series and a sequence. 212 00:09:12,135 --> 00:09:14,370 And I'll have much more to say about this in the 213 00:09:14,370 --> 00:09:15,430 supplementary notes. 214 00:09:15,430 --> 00:09:17,500 But for now, think of it this way. 215 00:09:17,500 --> 00:09:20,530 A series is a sum of terms. 216 00:09:20,530 --> 00:09:22,960 A sequence is just a listing of terms. 217 00:09:22,960 --> 00:09:25,800 In other words, in this particular problem, do not 218 00:09:25,800 --> 00:09:29,960 confuse the role of the 'a's with the role of the 's's. 219 00:09:29,960 --> 00:09:33,240 Notice that the a's refer to the sequence of 220 00:09:33,240 --> 00:09:34,850 numbers being added. 221 00:09:34,850 --> 00:09:36,600 In other words, the 'a's were what? 222 00:09:36,600 --> 00:09:39,170 They were 1/2, 1/4 and 1/8. 223 00:09:39,170 --> 00:09:41,440 These were the three numbers being added. 224 00:09:41,440 --> 00:09:44,660 Notice that the 's's were the partial sums. 225 00:09:44,660 --> 00:09:48,690 In other words, the partial sums form the sequence 's1', 226 00:09:48,690 --> 00:09:50,430 's2', 's3'. 227 00:09:50,430 --> 00:09:52,780 And to refresh your memories on this, that would be the 228 00:09:52,780 --> 00:09:53,770 sequence what? 229 00:09:53,770 --> 00:09:57,150 1/2, 3/4, 7/8. 230 00:09:57,150 --> 00:09:59,260 In other words, this was the sum of the first number that 231 00:09:59,260 --> 00:10:00,060 you were adding. 232 00:10:00,060 --> 00:10:05,340 3/4 was the sum of first two, and 7/8 was the sum of all 233 00:10:05,340 --> 00:10:06,240 three of them. 234 00:10:06,240 --> 00:10:10,330 And notice, by the way, the last partial sum, 's sub 3', 235 00:10:10,330 --> 00:10:13,360 the sum was defined to be the last partial 236 00:10:13,360 --> 00:10:14,910 sum, and that is what? 237 00:10:14,910 --> 00:10:15,790 The number-- 238 00:10:15,790 --> 00:10:17,140 this is very, very crucial. 239 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 240 00:10:21,500 --> 00:10:24,480 one number, and that number is called 7/8. 241 00:10:24,480 --> 00:10:26,280 OK, you see what we're talking about now? 242 00:10:26,280 --> 00:10:27,960 We're looking at a bunch of terms. 243 00:10:27,960 --> 00:10:30,390 We're adding them up, and we see how the sum 244 00:10:30,390 --> 00:10:32,110 changes with each term. 245 00:10:32,110 --> 00:10:35,150 In fact, in terms of a very trivial analogy, think of an 246 00:10:35,150 --> 00:10:37,450 adding machine. 247 00:10:37,450 --> 00:10:41,050 As you punch numbers in, the 's's are the sums that you see 248 00:10:41,050 --> 00:10:44,730 being read as your total sum, whereas the a's are the 249 00:10:44,730 --> 00:10:48,450 individual numbers being punched in to add up, OK? 250 00:10:48,450 --> 00:10:49,800 I hope that's a trivial example. 251 00:10:49,800 --> 00:10:52,840 As I listen to myself saying it, it sounds like I made it 252 00:10:52,840 --> 00:10:54,230 harder than it really is. 253 00:10:54,230 --> 00:10:57,230 At any rate, let's generalize this particular problem. 254 00:10:57,230 --> 00:11:00,190 Let's suppose now instead of wanting to add 1/2 plus 1/4 255 00:11:00,190 --> 00:11:04,320 plus 1/8, we want to add up the first 'n' terms of the 256 00:11:04,320 --> 00:11:06,710 form 1/2 plus 1/4 plus 1/8. 257 00:11:06,710 --> 00:11:09,060 In other words, let the n-th term that we're going to add, 258 00:11:09,060 --> 00:11:12,190 'a sub n', be '1 over '2 to the n''. 259 00:11:12,190 --> 00:11:16,280 Then the sum, the n-th partial sum here, the sum of these 'n' 260 00:11:16,280 --> 00:11:18,570 terms is, of course, what? 'a1' plus, et 261 00:11:18,570 --> 00:11:19,880 cetera, 'a sub n'. 262 00:11:19,880 --> 00:11:23,950 That turns out to be 1/2 plus 1/4 plus, et cetera, '1 over 263 00:11:23,950 --> 00:11:25,680 '2 to the n''. 264 00:11:25,680 --> 00:11:29,670 And by the way, rather than take time to develop this 265 00:11:29,670 --> 00:11:32,780 recipe over here, I thought you might like to see another 266 00:11:32,780 --> 00:11:35,690 place that might be interesting to review 267 00:11:35,690 --> 00:11:37,260 mathematical induction. 268 00:11:37,260 --> 00:11:40,280 If you'll bear with me and just come back over here where 269 00:11:40,280 --> 00:11:44,660 we were computing these partial sums, notice that in 270 00:11:44,660 --> 00:11:48,130 each of these partial sums, notice that your denominator 271 00:11:48,130 --> 00:11:52,160 is always 2 raised to the same power as this subscript. 272 00:11:52,160 --> 00:11:54,390 See, 2 the first power is 2. 273 00:11:54,390 --> 00:11:57,030 2 to the second power is 4. 274 00:11:57,030 --> 00:11:59,380 2 to the third power is 8. 275 00:11:59,380 --> 00:12:01,940 In other words, if your subscript is n, your 276 00:12:01,940 --> 00:12:04,240 denominator is '2 to the n'. 277 00:12:04,240 --> 00:12:07,190 Notice that your numerator is always one less than your 278 00:12:07,190 --> 00:12:08,080 denominator. 279 00:12:08,080 --> 00:12:11,180 In other words, if your denominator is '2 to the n', 280 00:12:11,180 --> 00:12:14,060 the numerator is '2 to the 'n minus 1''. 281 00:12:14,060 --> 00:12:17,570 And once we suspect this, this particular result can be 282 00:12:17,570 --> 00:12:19,040 proven by induction. 283 00:12:19,040 --> 00:12:20,780 I won't take the time to do this here. 284 00:12:20,780 --> 00:12:23,690 What I will take the time to do is to observe that this 285 00:12:23,690 --> 00:12:27,250 particular sum can be written more conveniently if we divide 286 00:12:27,250 --> 00:12:30,260 through by '2 to the n-th' to get 1 minus 287 00:12:30,260 --> 00:12:31,990 '1 over '2 the n''. 288 00:12:31,990 --> 00:12:35,440 For example, suppose we wanted to add up the 10 numbers. 289 00:12:35,440 --> 00:12:36,570 I say 10 numbers here. 290 00:12:36,570 --> 00:12:39,050 2 to the 10th is 1.024. 291 00:12:39,050 --> 00:12:43,190 But according to this recipe, 1/2 plus 1/4 plus 1/8 plus 292 00:12:43,190 --> 00:12:48,405 1/16 plus, et cetera, et cetera, plus 1/1,024 would add 293 00:12:48,405 --> 00:12:49,390 up to be what? 294 00:12:49,390 --> 00:12:53,820 1 minus 1/1,024. 295 00:12:53,820 --> 00:12:59,300 In other words, this would be 1,023/1,024, which seems to be 296 00:12:59,300 --> 00:13:00,700 pretty close to 1. 297 00:13:00,700 --> 00:13:03,390 In fact, you can begin to suspect that as 'n' gets 298 00:13:03,390 --> 00:13:06,660 arbitrarily large, 's sub n' gets arbitrarily 299 00:13:06,660 --> 00:13:08,100 close to 1 in value. 300 00:13:08,100 --> 00:13:09,750 I'm just talking fairly intuitively 301 00:13:09,750 --> 00:13:11,550 for the time being. 302 00:13:11,550 --> 00:13:14,460 But, you see, the major question now is suppose you 303 00:13:14,460 --> 00:13:16,410 elect not to stop at that. 304 00:13:16,410 --> 00:13:17,860 And you see, this is a very key point. 305 00:13:17,860 --> 00:13:21,810 We've already seen how the whole world seems to change as 306 00:13:21,810 --> 00:13:25,720 soon as you say let's never stop as opposed to saying 307 00:13:25,720 --> 00:13:27,650 let's go out as far as you want. 308 00:13:27,650 --> 00:13:29,810 See, if we now say what happens if you go on 309 00:13:29,810 --> 00:13:31,010 endlessly over here? 310 00:13:31,010 --> 00:13:34,190 Well, it becomes very natural to say lookit, the n-th 311 00:13:34,190 --> 00:13:37,920 partial sum was 1 minus '1 over '2 to the n-th'. 312 00:13:37,920 --> 00:13:40,360 In the case where you were adding up a finite number of 313 00:13:40,360 --> 00:13:44,040 terms, when you came to the last partial sum, that was by 314 00:13:44,040 --> 00:13:45,550 definition your answer. 315 00:13:45,550 --> 00:13:47,970 Now, what we're saying is lookit, because we have 316 00:13:47,970 --> 00:13:51,520 infinitely many terms to add, there is no last partial sum. 317 00:13:51,520 --> 00:13:54,450 And so what we say is lookit, instead of the last term, 318 00:13:54,450 --> 00:13:57,210 since there is no last term, why don't we just take the 319 00:13:57,210 --> 00:14:01,160 limit of the n-th partial sum as 'n' goes to infinity. 320 00:14:01,160 --> 00:14:03,605 In other words, in this particular case, notice that 321 00:14:03,605 --> 00:14:07,220 as 'n' approaches infinity, 1 minus '1 over '2 to the n'' 322 00:14:07,220 --> 00:14:10,910 approaches 1, and we then define the infinite sum, 323 00:14:10,910 --> 00:14:12,010 meaning what? 324 00:14:12,010 --> 00:14:14,600 I write it this way: as sigma 'n' goes from 1 to infinity, 325 00:14:14,600 --> 00:14:16,050 '1 over '2 to the n''. 326 00:14:16,050 --> 00:14:17,150 It really means what? 327 00:14:17,150 --> 00:14:20,850 The limit as 'n' goes to infinity: 1/2 plus 1/4 plus 328 00:14:20,850 --> 00:14:22,410 1/8 plus 1/16-- 329 00:14:22,410 --> 00:14:24,190 endlessly-- 330 00:14:24,190 --> 00:14:28,380 that that limit is 1, and we define that to be the sum. 331 00:14:28,380 --> 00:14:31,730 And again, as I say, I'm going to write that in greatly more 332 00:14:31,730 --> 00:14:34,160 detail in the notes, and also we'll have many 333 00:14:34,160 --> 00:14:35,360 exercises on this. 334 00:14:35,360 --> 00:14:38,830 I just wanted you to see how we get to infinite sums, which 335 00:14:38,830 --> 00:14:41,600 are called series by generalizing what happens in 336 00:14:41,600 --> 00:14:42,930 the finite case. 337 00:14:42,930 --> 00:14:45,350 And because this may seem a little vague to you, let me 338 00:14:45,350 --> 00:14:47,800 give you a pictorial representation 339 00:14:47,800 --> 00:14:49,930 of this same thing. 340 00:14:49,930 --> 00:14:52,380 You see, what's happening here is this. 341 00:14:52,380 --> 00:14:58,240 Draw a little circle around 1 of bandwidth epsilon. 342 00:14:58,240 --> 00:15:00,900 In other words, let's mark off an interval epsilon on 343 00:15:00,900 --> 00:15:02,080 either side of 1. 344 00:15:02,080 --> 00:15:05,270 And let's call this point here 1 minus epsilon. 345 00:15:05,270 --> 00:15:07,870 Let's call this point here 1 plus epsilon. 346 00:15:07,870 --> 00:15:11,140 And what we're saying about our partial sums is this. 347 00:15:11,140 --> 00:15:13,570 That when you start off and you're adding up terms here, 348 00:15:13,570 --> 00:15:14,770 you have 1/2. 349 00:15:14,770 --> 00:15:18,720 1/2 plus 1/4 brings you over to 3/4. 350 00:15:18,720 --> 00:15:21,420 The next possible sum is 7/8, et cetera. 351 00:15:21,420 --> 00:15:24,870 And all we're saying is that these terms get arbitrarily 352 00:15:24,870 --> 00:15:28,240 close to 1 in value, meaning that after a while-- 353 00:15:28,240 --> 00:15:30,450 and I'll define more rigorously what after a while 354 00:15:30,450 --> 00:15:31,490 means in a moment-- 355 00:15:31,490 --> 00:15:35,350 all of the 's sub n's are within epsilon of 1. 356 00:15:35,350 --> 00:15:38,590 After a while, all of your partial sums are in here. 357 00:15:38,590 --> 00:15:41,330 And what you mean by after a while certainly depends on how 358 00:15:41,330 --> 00:15:42,260 big epsilon is. 359 00:15:42,260 --> 00:15:45,590 In other words, the smaller the bandwidth you allow 360 00:15:45,590 --> 00:15:49,350 yourself, the more terms you may have to take before you 361 00:15:49,350 --> 00:15:50,730 get within the tolerance limits 362 00:15:50,730 --> 00:15:52,270 that you allow yourself. 363 00:15:52,270 --> 00:15:55,500 In any event, going back to something that we've been 364 00:15:55,500 --> 00:15:57,470 using for a long time, our basic 365 00:15:57,470 --> 00:15:58,980 definition is the following. 366 00:15:58,980 --> 00:16:01,890 If you have an infinite sequence, say, a collection of 367 00:16:01,890 --> 00:16:05,077 terms 'b sub n', in other words, 'b1', 'b2', 'b3', et 368 00:16:05,077 --> 00:16:09,900 cetera, without end, we say that that sequence converges 369 00:16:09,900 --> 00:16:13,390 to the limit 'L' written the limit of 'b sub n' as 'n' 370 00:16:13,390 --> 00:16:17,360 approaches infinity equals 'L', if and only if for every 371 00:16:17,360 --> 00:16:21,510 epsilon greater than 0 we can find a number 'N' which 372 00:16:21,510 --> 00:16:22,910 depends on epsilon-- 373 00:16:22,910 --> 00:16:25,480 notice the notation here: 'N' as a function of epsilon-- 374 00:16:25,480 --> 00:16:29,280 such that whenever little 'n' is greater than capital 'N', 375 00:16:29,280 --> 00:16:31,740 the absolute value of 'a sub n' minus 'L' 376 00:16:31,740 --> 00:16:33,050 is less than epsilon. 377 00:16:33,050 --> 00:16:35,755 And, you see, again, you may wonder how in the world that 378 00:16:35,755 --> 00:16:36,920 you're going to remember this. 379 00:16:36,920 --> 00:16:39,690 If you memorize this, I guarantee you, in two day's 380 00:16:39,690 --> 00:16:42,020 time, you'll have to memorize it again. 381 00:16:42,020 --> 00:16:44,790 I also hope you have enough faith in me to recognize I 382 00:16:44,790 --> 00:16:46,110 didn't memorize this. 383 00:16:46,110 --> 00:16:48,060 There is a feeling that one gets for this. 384 00:16:48,060 --> 00:16:50,090 And let me give you what that feeling is. 385 00:16:50,090 --> 00:16:53,140 Again, in terms of a picture, what it means-- well, I'll 386 00:16:53,140 --> 00:16:55,720 change these to 'a's now because that's the symbols 387 00:16:55,720 --> 00:16:58,170 that we've been using before in terms of 388 00:16:58,170 --> 00:16:59,005 the sequence of terms. 389 00:16:59,005 --> 00:17:01,220 What we really mean-- and I don't care what symbol you 390 00:17:01,220 --> 00:17:02,270 really use here-- 391 00:17:02,270 --> 00:17:06,060 is if you want to talk about the limit of 'a sub n' as 'n' 392 00:17:06,060 --> 00:17:08,930 approaches infinity, if that limit equals 'L', what the 393 00:17:08,930 --> 00:17:10,940 rigorous definition says is this. 394 00:17:10,940 --> 00:17:14,150 Draw yourself an interval around 'L' of bandwidth 395 00:17:14,150 --> 00:17:15,740 epsilon, in other words, from 'L minus 396 00:17:15,740 --> 00:17:18,089 epsilon' to 'L plus epsilon'. 397 00:17:18,089 --> 00:17:20,970 And what this thing says is that beyond a certain term, 398 00:17:20,970 --> 00:17:24,250 say, the capital N-th term, every term beyond this certain 399 00:17:24,250 --> 00:17:26,700 one is in here. 400 00:17:29,790 --> 00:17:35,240 Well, all 'a n's are in here if 'n' is sufficiently large. 401 00:17:35,240 --> 00:17:37,100 I don't know if you can read that very well, but just 402 00:17:37,100 --> 00:17:38,020 listen to what I'm saying. 403 00:17:38,020 --> 00:17:42,060 All of the terms are in here if 'n' is sufficiently large. 404 00:17:42,060 --> 00:17:45,820 What this means again is that to all intents and purposes, 405 00:17:45,820 --> 00:17:49,440 if you think of this bandwidth as giving you a dot, see, a 406 00:17:49,440 --> 00:17:53,160 thick dot here where the endpoints are 'L minus 407 00:17:53,160 --> 00:17:56,520 epsilon' and 'L plus epsilon', what we're saying is lookit, 408 00:17:56,520 --> 00:17:59,600 after a certain term, the way I've drawn here, after the 409 00:17:59,600 --> 00:18:02,290 fifth term, all the remaining terms of my 410 00:18:02,290 --> 00:18:04,310 sequence are in here. 411 00:18:04,310 --> 00:18:07,740 By the way, notice the role of the subscripts here. 412 00:18:07,740 --> 00:18:10,330 All the subscript tells you is where the term 413 00:18:10,330 --> 00:18:12,550 appears in your sequence. 414 00:18:12,550 --> 00:18:15,300 For example, the third term in your sequence could be a 415 00:18:15,300 --> 00:18:18,040 smaller number than the second term of your sequence. 416 00:18:18,040 --> 00:18:21,210 Do not confuse the size of the terms with the subscripts. 417 00:18:21,210 --> 00:18:24,170 The subscripts order the terms, but the third term in 418 00:18:24,170 --> 00:18:27,790 the sequence can be less than in size than the second term 419 00:18:27,790 --> 00:18:28,560 in the sequence. 420 00:18:28,560 --> 00:18:30,170 But again, I'll talk about that in more 421 00:18:30,170 --> 00:18:31,280 detail in the notes. 422 00:18:31,280 --> 00:18:36,020 The point that I want you to see is that in concept what 423 00:18:36,020 --> 00:18:37,840 limit does is the following. 424 00:18:37,840 --> 00:18:42,750 Limit is to an infinite sequence as last term is to a 425 00:18:42,750 --> 00:18:44,200 finite sequence. 426 00:18:44,200 --> 00:18:48,320 In other words, a limit replaces infinitely many 427 00:18:48,320 --> 00:18:52,720 points by a finite number of points plus a dot. 428 00:18:52,720 --> 00:18:55,780 You see, going back to this example here, how many 'a sub 429 00:18:55,780 --> 00:18:56,710 n's were there? 430 00:18:56,710 --> 00:18:58,760 Well, there were infinitely many. 431 00:18:58,760 --> 00:19:01,530 Well, to keep track of these infinitely many, what do I 432 00:19:01,530 --> 00:19:02,780 have to keep track of now? 433 00:19:02,780 --> 00:19:07,770 Well, in this diagram, the first five 'a's plus this dot, 434 00:19:07,770 --> 00:19:12,290 because you see, every one of my infinitely many 'a's past 435 00:19:12,290 --> 00:19:16,170 the fifth one is inside this dot, you see? 436 00:19:16,170 --> 00:19:18,450 So in other words then, what's happened? 437 00:19:18,450 --> 00:19:20,170 The thing that had to happen. 438 00:19:20,170 --> 00:19:22,670 We had to deal with infinite sequences. 439 00:19:22,670 --> 00:19:26,200 We saw the big philosophic difference between infinitely 440 00:19:26,200 --> 00:19:28,630 many and just large. 441 00:19:28,630 --> 00:19:33,550 And so our definition of limit had to be such that we could 442 00:19:33,550 --> 00:19:37,930 reduce in a way that was compatible with our intuition 443 00:19:37,930 --> 00:19:42,060 the concept of infinitely many points to a finite number. 444 00:19:42,060 --> 00:19:45,430 Because, you see, as I'll show you in the notes also, all of 445 00:19:45,430 --> 00:19:49,300 our arithmetic is geared for just a finite number of 446 00:19:49,300 --> 00:19:50,610 operations. 447 00:19:50,610 --> 00:19:52,280 See, this is why this definition 448 00:19:52,280 --> 00:19:53,920 of limit is so crucial. 449 00:19:53,920 --> 00:19:57,360 Again, you may notice, and I'll remind you of this also 450 00:19:57,360 --> 00:20:00,960 in the exercises, that structurally this definition 451 00:20:00,960 --> 00:20:04,040 of limit is the same as the limit that we use when we 452 00:20:04,040 --> 00:20:07,000 talked about the limit of 'f of x', as 'x' approaches 'a', 453 00:20:07,000 --> 00:20:07,880 equals 'L'. 454 00:20:07,880 --> 00:20:12,060 The absolute value signs have the same properties as before. 455 00:20:12,060 --> 00:20:14,470 And by the way, before I go on, let me just remind you 456 00:20:14,470 --> 00:20:17,810 again of one more thing while I'm talking that way. 457 00:20:17,810 --> 00:20:20,950 Instead of memorizing this, remember how you read this. 458 00:20:20,950 --> 00:20:24,180 The absolute value of 'a sub n' minus 'L' is less than 459 00:20:24,180 --> 00:20:25,660 epsilon means what? 460 00:20:25,660 --> 00:20:28,820 That 'a sub n' is within epsilon of 'L'. 461 00:20:28,820 --> 00:20:30,710 That's what we use in our diagram. 462 00:20:30,710 --> 00:20:32,740 But it seems to me I forgot to mention this. 463 00:20:32,740 --> 00:20:34,240 And I want you to see that what? 464 00:20:34,240 --> 00:20:37,790 The key building block analytically is the absolute 465 00:20:37,790 --> 00:20:40,690 value, and the meaning of absolute value is the same 466 00:20:40,690 --> 00:20:45,210 here as it was in blocks one and two of our course. 467 00:20:45,210 --> 00:20:49,010 So what I'm driving at is that the same limit theorems that 468 00:20:49,010 --> 00:20:52,260 we've been able to use up until now still apply. 469 00:20:52,260 --> 00:20:54,640 Oh, by means of an example. 470 00:20:54,640 --> 00:20:57,830 Suppose I have the limit as 'n' approaches infinity, '2n 471 00:20:57,830 --> 00:21:01,110 plus 3' over '5n plus 7'. 472 00:21:01,110 --> 00:21:03,130 Notice that I can divide numerator and denominator 473 00:21:03,130 --> 00:21:04,190 through by 'n'. 474 00:21:04,190 --> 00:21:07,100 If I do that, I have the limit as 'n' approaches infinity. 475 00:21:07,100 --> 00:21:11,060 '2 plus '3/n'' over '5 plus '7/n''. 476 00:21:11,060 --> 00:21:13,910 Now using the fact that the limit of a sum is the sum of 477 00:21:13,910 --> 00:21:16,380 the limits, the limit of a quotient is the quotient of 478 00:21:16,380 --> 00:21:21,320 the limits, the limit of '1/n' as 'n' goes to infinity is 0. 479 00:21:21,320 --> 00:21:24,190 Notice that I can use the limit theorems to conclude 480 00:21:24,190 --> 00:21:27,450 that the limit of this particular sequence is 2/5. 481 00:21:27,450 --> 00:21:31,980 If I wanted to, the same ways we did in block one, block 482 00:21:31,980 --> 00:21:35,710 two, where we're talking about limits, given an epsilon, I 483 00:21:35,710 --> 00:21:39,360 can actually exhibit how far out I have to go before each 484 00:21:39,360 --> 00:21:41,800 of the terms in this sequence is within that 485 00:21:41,800 --> 00:21:44,180 given epsilon of 2/5. 486 00:21:44,180 --> 00:21:48,310 By the way, again to emphasize once more, because this is so 487 00:21:48,310 --> 00:21:52,230 important, the difference between an infinite sum and an 488 00:21:52,230 --> 00:21:55,560 infinite sequence, observe that whereas the limit of the 489 00:21:55,560 --> 00:22:01,070 sequence of terms '2n plus 3' over '5n plus 7' is 2/5, the 490 00:22:01,070 --> 00:22:07,360 infinite sum composed of the terms of the form '2n plus 3' 491 00:22:07,360 --> 00:22:13,240 over '5n plus 7' is infinity since after a while each term 492 00:22:13,240 --> 00:22:16,080 that you're adding here behaves like 2/5. 493 00:22:16,080 --> 00:22:18,390 In other words, if you write this thing out to see what 494 00:22:18,390 --> 00:22:20,550 this means, pick 'n' to be 1. 495 00:22:20,550 --> 00:22:22,800 When 'n' is 1, this term is 5/12. 496 00:22:22,800 --> 00:22:24,570 When 'n' is 2, this is what? 497 00:22:24,570 --> 00:22:27,620 7 plus 17, 7/17. 498 00:22:27,620 --> 00:22:30,660 When 'n' is 3, this is 9/22. 499 00:22:30,660 --> 00:22:35,020 When 'n' is 4, this is 8 plus 3 is 11, over 27. 500 00:22:35,020 --> 00:22:37,070 In other words, what you're saying is the infinite sum 501 00:22:37,070 --> 00:22:40,060 means to add up all of these terms. 502 00:22:40,060 --> 00:22:44,080 The thing whose limit was 2/5 was the sequence of terms 503 00:22:44,080 --> 00:22:44,710 themselves. 504 00:22:44,710 --> 00:22:47,410 In other words, what we're saying is that after a certain 505 00:22:47,410 --> 00:22:50,810 point, every one of these terms behaves like 2/5. 506 00:22:50,810 --> 00:22:53,360 And what you're saying is lookit, after a point, what 507 00:22:53,360 --> 00:22:56,610 you're really doing is essentially adding on 2/5 508 00:22:56,610 --> 00:22:58,370 every time you add on another term. 509 00:22:58,370 --> 00:23:02,230 And therefore, this sum can get as large as you want, just 510 00:23:02,230 --> 00:23:03,840 by adding on enough terms. 511 00:23:03,840 --> 00:23:07,930 Again, observe the difference between the partial sums and 512 00:23:07,930 --> 00:23:09,260 the terms themselves. 513 00:23:09,260 --> 00:23:11,230 The terms that you're adding are 514 00:23:11,230 --> 00:23:13,430 approaching 2/5 as a limit. 515 00:23:13,430 --> 00:23:16,120 The thing that's becoming infinite is the sequence of 516 00:23:16,120 --> 00:23:17,210 partial sums. 517 00:23:17,210 --> 00:23:20,300 Because what you're saying is to get from one partial sum to 518 00:23:20,300 --> 00:23:22,150 the next, you're, roughly speaking, 519 00:23:22,150 --> 00:23:24,940 adding on 2/5 each time. 520 00:23:24,940 --> 00:23:28,380 To generalize this, what we're saying is if the sequence of 521 00:23:28,380 --> 00:23:32,220 partial sums converges, the individual terms that you're 522 00:23:32,220 --> 00:23:35,210 adding must approach 0 in the limit. 523 00:23:35,210 --> 00:23:38,450 For if the limit of the 'a sub n's as 'n' approaches infinity 524 00:23:38,450 --> 00:23:43,500 is 'L', where 'L' is not 0, then beyond a certain term, 525 00:23:43,500 --> 00:23:45,820 the sum of the 'a sub n's behaves like 526 00:23:45,820 --> 00:23:47,280 the sum of the 'L's. 527 00:23:47,280 --> 00:23:50,250 And what you're saying is if 'L' is non zero, by adding on 528 00:23:50,250 --> 00:23:53,330 enough of these fixed 'L's, you can make the sum as large 529 00:23:53,330 --> 00:23:55,380 as you wish. 530 00:23:55,380 --> 00:23:58,550 In other words, then, a sort of negative test is that if 531 00:23:58,550 --> 00:24:01,520 you know that the series converges, then the terms that 532 00:24:01,520 --> 00:24:04,520 you're adding on must approach 0 in the limit. 533 00:24:04,520 --> 00:24:07,650 Unfortunately, by the way, the converse is not true. 534 00:24:07,650 --> 00:24:10,790 Namely, if you know that the terms that you're adding on go 535 00:24:10,790 --> 00:24:15,680 to 0, you cannot conclude that their sum is finite. 536 00:24:15,680 --> 00:24:19,230 Again, it's our old friend of infinity times 0. 537 00:24:19,230 --> 00:24:25,140 You see, as these terms approach 0, when you start to 538 00:24:25,140 --> 00:24:27,840 add them up, it may be that they're not 539 00:24:27,840 --> 00:24:29,030 going to 0 fast enough. 540 00:24:29,030 --> 00:24:31,160 In other words, notice that the terms are getting small, 541 00:24:31,160 --> 00:24:33,500 but you're also adding more and more of them. 542 00:24:33,500 --> 00:24:34,870 You see, what I wrote here is what? 543 00:24:34,870 --> 00:24:37,630 On the other hand, the limit of 'a sub n' as 'n' approaches 544 00:24:37,630 --> 00:24:40,840 infinity equals 0 is not enough to guarantee the 545 00:24:40,840 --> 00:24:43,460 convergence of this particular sum. 546 00:24:43,460 --> 00:24:46,350 In fact, a trivial example to show this is look at the 547 00:24:46,350 --> 00:24:48,440 following contrived example. 548 00:24:48,440 --> 00:24:50,970 Start out with the first number being 1. 549 00:24:50,970 --> 00:24:57,720 Then take 1/2 twice, 1/3 three times, 1/4 four times, 1/5 550 00:24:57,720 --> 00:25:00,520 five times, 1/6 six times. 551 00:25:00,520 --> 00:25:02,830 Form the n-th partial sum. 552 00:25:02,830 --> 00:25:06,100 Lookit, is it clear that the terms that are going into your 553 00:25:06,100 --> 00:25:09,030 sum are approaching 0 in the limit? 554 00:25:09,030 --> 00:25:12,590 You see, you have a one, then there are halves, then thirds, 555 00:25:12,590 --> 00:25:15,060 then fourths, then fifths, then sixths, 556 00:25:15,060 --> 00:25:16,180 sevenths, et cetera. 557 00:25:16,180 --> 00:25:17,900 The terms themselves are getting 558 00:25:17,900 --> 00:25:19,780 arbitrarily close to 0. 559 00:25:19,780 --> 00:25:22,060 On the other hand, what is the sum becoming? 560 00:25:22,060 --> 00:25:23,820 Well, this adds up to 1. 561 00:25:23,820 --> 00:25:25,290 This adds up to 1. 562 00:25:25,290 --> 00:25:28,090 This adds up to 1, and this that up to 1. 563 00:25:28,090 --> 00:25:31,700 And in other words, by taking enough terms, I can tack on as 564 00:25:31,700 --> 00:25:34,540 many ones is I want, and ultimately, even though the 565 00:25:34,540 --> 00:25:37,390 terms become small, the sum becomes large. 566 00:25:37,390 --> 00:25:40,520 In fact, it's precisely because of this unpleasantness 567 00:25:40,520 --> 00:25:44,790 that we have to go into a rather difficult lecture next 568 00:25:44,790 --> 00:25:48,610 time, talking about OK, how then can you tell when an 569 00:25:48,610 --> 00:25:51,840 infinite sum converges to a finite limit and 570 00:25:51,840 --> 00:25:53,260 when doesn't it? 571 00:25:53,260 --> 00:25:54,430 At any rate, that's what I said we're going to 572 00:25:54,430 --> 00:25:56,050 talk about next time. 573 00:25:56,050 --> 00:25:58,660 As far as today's lesson is concerned, I hope that we've 574 00:25:58,660 --> 00:26:00,710 straightened out the difference between a sequence 575 00:26:00,710 --> 00:26:04,100 and the series, partial sums and the terms being added. 576 00:26:04,100 --> 00:26:06,590 And in the hopes that we've done that, let me say, until 577 00:26:06,590 --> 00:26:07,840 next time, goodbye. 578 00:26:09,970 --> 00:26:13,170 Funding for the publication of this video was provided by the 579 00:26:13,170 --> 00:26:17,220 Gabriella and Paul Rosenbaum Foundation. 580 00:26:17,220 --> 00:26:21,390 Help OCW continue to provide free and open access to MIT 581 00:26:21,390 --> 00:26:25,590 courses by making a donation at ocw.mit.edu/donate.