1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:04,076 Commons license. 3 00:00:04,076 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:09,930 continue to offer high quality educational resources for free. 5 00:00:09,930 --> 00:00:12,590 To make a donation, or to view additional materials 6 00:00:12,590 --> 00:00:15,850 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,850 --> 00:00:21,770 at ocw.mit.edu. 8 00:00:21,770 --> 00:00:30,090 PROFESSOR: Now, today we are continuing with this last unit. 9 00:00:30,090 --> 00:00:36,650 Unit 5, continued. 10 00:00:36,650 --> 00:00:48,490 The informal title of this unit is Dealing With Infinity. 11 00:00:48,490 --> 00:00:51,210 That's really the extra little piece 12 00:00:51,210 --> 00:00:55,150 that we're putting in to our discussions of things 13 00:00:55,150 --> 00:01:01,080 like limits and integrals. 14 00:01:01,080 --> 00:01:09,680 To start out with today, I'd like to recall for you, 15 00:01:09,680 --> 00:01:20,290 L'Hôpital's Rule. 16 00:01:20,290 --> 00:01:22,260 And in keeping with the spirit here, 17 00:01:22,260 --> 00:01:32,670 we're just going to do the infinity / infinity case. 18 00:01:32,670 --> 00:01:35,010 I stated this a little differently last time, 19 00:01:35,010 --> 00:01:37,750 and I want to state it again today. 20 00:01:37,750 --> 00:01:39,870 Just to make clear what the hypotheses are 21 00:01:39,870 --> 00:01:43,880 and what the conclusion is. 22 00:01:43,880 --> 00:01:47,350 We start out with, really, three hypotheses. 23 00:01:47,350 --> 00:01:50,160 Two of them are kind of obvious. 24 00:01:50,160 --> 00:01:55,660 The three hypotheses are that f(x) tends to infinity, 25 00:01:55,660 --> 00:01:58,660 g(x) tends to infinity, that's what 26 00:01:58,660 --> 00:02:02,380 it means to be in this infinity / infinity case. 27 00:02:02,380 --> 00:02:08,590 And then the last assumption is that f'(x) / g'(x) tends 28 00:02:08,590 --> 00:02:15,660 to a limit, L. And this is all as x tends to some a. 29 00:02:15,660 --> 00:02:18,180 Some limit a. 30 00:02:18,180 --> 00:02:27,090 And then the conclusion is that f(x) / g(x) also 31 00:02:27,090 --> 00:02:37,260 tends to L, as x goes to a. 32 00:02:37,260 --> 00:02:38,900 Now, so that's the way it is. 33 00:02:38,900 --> 00:02:40,750 So it's three limits. 34 00:02:40,750 --> 00:02:43,030 But presumably these are obvious, 35 00:02:43,030 --> 00:02:49,180 and this one is exactly what we were going to check anyway. 36 00:02:49,180 --> 00:02:52,790 Gives us this one limit. 37 00:02:52,790 --> 00:02:54,390 So that's the statement. 38 00:02:54,390 --> 00:02:59,340 And then the other little interesting point here, 39 00:02:59,340 --> 00:03:02,900 which is consistent with this idea of dealing with infinity, 40 00:03:02,900 --> 00:03:07,200 is that a equals plus or minus infinity 41 00:03:07,200 --> 00:03:11,590 and L equals plus or minus infinity are OK. 42 00:03:11,590 --> 00:03:15,780 That is, the numbers capital L, the limit capital L, 43 00:03:15,780 --> 00:03:21,090 and the number a can also be infinite. 44 00:03:21,090 --> 00:03:26,770 Now in recitation yesterday, you should have discussed something 45 00:03:26,770 --> 00:03:31,830 about rates of growth, which follow from what 46 00:03:31,830 --> 00:03:34,680 I said in lecture last time and also maybe 47 00:03:34,680 --> 00:03:37,260 from some more detailed discussions that you 48 00:03:37,260 --> 00:03:40,440 had in recitation. 49 00:03:40,440 --> 00:03:44,130 And I'm going to introduce a notation to compare functions. 50 00:03:44,130 --> 00:03:49,580 Namely, we say that f(x) is a lot less than g(x) 51 00:03:49,580 --> 00:03:58,060 if-- So this means that the limit, as it goes to infinity, 52 00:03:58,060 --> 00:03:59,280 this tends to 0. 53 00:03:59,280 --> 00:04:04,700 As x goes to infinity, this would be. 54 00:04:04,700 --> 00:04:07,490 So this is a notation, a new notation for us. 55 00:04:07,490 --> 00:04:10,160 f is a lot less than g. 56 00:04:10,160 --> 00:04:13,150 And it's meant to be read only asymptotically. 57 00:04:13,150 --> 00:04:17,890 It's only in the limit as x goes to infinity that this happens. 58 00:04:17,890 --> 00:04:20,120 And implicitly here, I'm always assuming 59 00:04:20,120 --> 00:04:23,570 that these are positive quantities. f and g are 60 00:04:23,570 --> 00:04:28,130 positive. 61 00:04:28,130 --> 00:04:31,050 What you saw in recitation was that you 62 00:04:31,050 --> 00:04:34,159 can make a systematic comparison of all the standard functions 63 00:04:34,159 --> 00:04:34,950 that we know about. 64 00:04:34,950 --> 00:04:38,050 For example, the log function goes to infinity. 65 00:04:38,050 --> 00:04:41,500 But a lot more slowly than x to a power. 66 00:04:41,500 --> 00:04:44,920 A lot more slowly then e^x. 67 00:04:44,920 --> 00:04:48,220 A lot more slowly than, say, e^(x^2). 68 00:04:48,220 --> 00:04:50,850 So this one is slow. 69 00:04:50,850 --> 00:04:54,580 This one is moderate. 70 00:04:54,580 --> 00:04:56,810 And this one is fast. 71 00:04:56,810 --> 00:05:00,740 And this one is very fast. 72 00:05:00,740 --> 00:05:02,900 Going to infinity. 73 00:05:02,900 --> 00:05:04,650 Tends to infinity, and this is of course 74 00:05:04,650 --> 00:05:07,570 as x goes to infinity. 75 00:05:07,570 --> 00:05:12,270 All of them go to infinity, but at quite different rates. 76 00:05:12,270 --> 00:05:14,860 And, analogous to this, and today 77 00:05:14,860 --> 00:05:20,430 we're going to be doing this, needing to do this quite a bit, 78 00:05:20,430 --> 00:05:24,850 is rates of decay, which are more or less the opposite 79 00:05:24,850 --> 00:05:26,460 of rates of growth. 80 00:05:26,460 --> 00:05:30,270 So rates of decay are rates at which things tend to 0. 81 00:05:30,270 --> 00:05:39,820 So the rate of decay, and for that I'm just going to take 82 00:05:39,820 --> 00:05:41,890 reciprocals of these numbers. 83 00:05:41,890 --> 00:05:45,820 So 1 / ln x tends to 0. 84 00:05:45,820 --> 00:05:47,260 But rather slowly. 85 00:05:47,260 --> 00:05:51,220 It's much bigger than 1 / x^p. 86 00:05:51,220 --> 00:05:53,490 Oh, I didn't mention that this exponent 87 00:05:53,490 --> 00:05:55,750 p is meant to be positive. 88 00:05:55,750 --> 00:05:58,360 That's a convention that I'm using without saying. 89 00:05:58,360 --> 00:06:00,310 I should've told you that. 90 00:06:00,310 --> 00:06:06,660 So think x^(1/2), x^1, x^2, they're all in this sort 91 00:06:06,660 --> 00:06:09,000 of moderate intermediate range. 92 00:06:09,000 --> 00:06:15,350 And then that, in turn, goes to 0 but much more slowly then 1 / 93 00:06:15,350 --> 00:06:19,220 e^x, also known as e^(-x). 94 00:06:19,220 --> 00:06:27,830 And that, in turn, this guy here goes to 0 incredibly fast. 95 00:06:27,830 --> 00:06:28,780 e^(-x^2). 96 00:06:28,780 --> 00:06:32,950 vanishes really, really fast. 97 00:06:32,950 --> 00:06:37,130 So this is a review of L'Hôpital's Rule. 98 00:06:37,130 --> 00:06:39,860 What we said last time, and the application of it, 99 00:06:39,860 --> 00:06:42,920 which is to rates of growth and tells us 100 00:06:42,920 --> 00:06:50,860 what these rates of growth are. 101 00:06:50,860 --> 00:07:01,830 Today, I want to talk about improper integrals. 102 00:07:01,830 --> 00:07:05,160 And improper integrals, we've already 103 00:07:05,160 --> 00:07:09,050 really seen one or two of them on your exercises. 104 00:07:09,050 --> 00:07:11,805 And we mention them a little bit, briefly. 105 00:07:11,805 --> 00:07:13,180 I'm just going to go through them 106 00:07:13,180 --> 00:07:15,110 more carefully and more systematically now. 107 00:07:15,110 --> 00:07:17,530 And we want to get just exactly what's 108 00:07:17,530 --> 00:07:19,930 going on with these rates of decay 109 00:07:19,930 --> 00:07:21,940 and their relationship with improper integrals. 110 00:07:21,940 --> 00:07:26,350 So I need for you to understand, on the spectrum 111 00:07:26,350 --> 00:07:28,470 of the range of functions like this, 112 00:07:28,470 --> 00:07:32,940 which ones are suitable for integration 113 00:07:32,940 --> 00:07:38,320 as x goes to infinity. 114 00:07:38,320 --> 00:07:43,130 Well, let's start out with the definition. 115 00:07:43,130 --> 00:07:47,360 The integral from a to infinity of f(x) dx 116 00:07:47,360 --> 00:07:50,170 is, by definition, the limit as N 117 00:07:50,170 --> 00:07:56,620 goes to infinity of the ordinary definite integral 118 00:07:56,620 --> 00:08:00,430 up to some fixed, finite level. 119 00:08:00,430 --> 00:08:01,760 That's the definition. 120 00:08:01,760 --> 00:08:05,990 And there's a word that we use here, 121 00:08:05,990 --> 00:08:09,250 which is that we say the integral, 122 00:08:09,250 --> 00:08:20,560 so this is terminology for it, converges if the limit exists. 123 00:08:20,560 --> 00:08:28,630 And diverges if not. 124 00:08:28,630 --> 00:08:34,340 Well, these are the key words for today. 125 00:08:34,340 --> 00:08:39,870 So here's the issue that we're going to be addressing. 126 00:08:39,870 --> 00:08:42,430 Which is whether the limit exists or not. 127 00:08:42,430 --> 00:08:50,820 In other words, whether the integral converges or diverges. 128 00:08:50,820 --> 00:08:53,800 These notions have a geometric analog, 129 00:08:53,800 --> 00:08:56,190 which you should always be thinking of at the same time 130 00:08:56,190 --> 00:08:57,730 in the back of your head. 131 00:08:57,730 --> 00:09:00,390 I'll draw a picture of the function. 132 00:09:00,390 --> 00:09:02,510 Here it's starting out at a. 133 00:09:02,510 --> 00:09:05,790 And maybe it's going down like this. 134 00:09:05,790 --> 00:09:09,840 And it's interpreting it geometrically. 135 00:09:09,840 --> 00:09:15,520 This would only work if f is positive. 136 00:09:15,520 --> 00:09:25,900 Then the convergent case is the case where the area is finite. 137 00:09:25,900 --> 00:09:29,660 So the total area is finite under this curve. 138 00:09:29,660 --> 00:09:43,750 And the other case is the total area is infinite. 139 00:09:43,750 --> 00:09:46,920 I claim that both of these things are possible. 140 00:09:46,920 --> 00:09:50,460 Although this thing goes on forever, 141 00:09:50,460 --> 00:09:53,070 if you stop it at one stage, N, then of course 142 00:09:53,070 --> 00:09:54,310 it's a finite number. 143 00:09:54,310 --> 00:09:56,270 But as you go further and further and further, 144 00:09:56,270 --> 00:09:58,290 there's more and more and more area. 145 00:09:58,290 --> 00:10:00,420 And there are two possibilities. 146 00:10:00,420 --> 00:10:04,700 Either as you go all the way out here to infinity, 147 00:10:04,700 --> 00:10:08,900 the total that you get adds up to a finite total. 148 00:10:08,900 --> 00:10:11,280 Or else, maybe there's infinitely much. 149 00:10:11,280 --> 00:10:13,700 For instance, if it's a straight line going across, 150 00:10:13,700 --> 00:10:23,340 there's clearly infinitely much area underneath. 151 00:10:23,340 --> 00:10:25,500 So we need to do a bunch of examples. 152 00:10:25,500 --> 00:10:28,660 And that's really our main job for the day, 153 00:10:28,660 --> 00:10:30,700 and to make sure that we know exactly 154 00:10:30,700 --> 00:10:34,810 what to expect in all cases. 155 00:10:34,810 --> 00:10:42,200 The first example is the integral from 0 to infinity 156 00:10:42,200 --> 00:10:45,410 of e^(-kx) dx. 157 00:10:45,410 --> 00:10:48,550 Where k is going to be some positive number. 158 00:10:48,550 --> 00:10:54,080 Some positive constant. 159 00:10:54,080 --> 00:11:02,919 This is the most fundamental, by far, of the definite integrals. 160 00:11:02,919 --> 00:11:03,710 Improper integrals. 161 00:11:03,710 --> 00:11:07,370 And in order to handle this, the thing that I need to do is 162 00:11:07,370 --> 00:11:13,000 to check the integral from 0 up to N, e^(-kx) dx. 163 00:11:13,000 --> 00:11:15,960 And since this is an easy integral to evaluate, 164 00:11:15,960 --> 00:11:17,650 we're going to do it. 165 00:11:17,650 --> 00:11:22,590 It's -1/k e^(-kx), that's the antiderivative. 166 00:11:22,590 --> 00:11:30,370 Evaluated at 0 and N. And that, if I plug in these values, 167 00:11:30,370 --> 00:11:39,730 is -1/k e^(-kN), minus-- and if I evaluate it at 0, 168 00:11:39,730 --> 00:11:42,600 I get a -1/k e^0. 169 00:11:46,860 --> 00:11:48,690 So there's the answer. 170 00:11:48,690 --> 00:11:50,180 And now we have to think about what 171 00:11:50,180 --> 00:11:54,040 happens as N goes to infinity. 172 00:11:54,040 --> 00:11:59,050 So as N goes to infinity, what's happening 173 00:11:59,050 --> 00:12:03,400 is the second term here stays unchanged. 174 00:12:03,400 --> 00:12:06,784 But the first term is e to some negative power. 175 00:12:06,784 --> 00:12:08,700 And the exponent is getting larger and larger. 176 00:12:08,700 --> 00:12:10,830 That's because k is positive here. 177 00:12:10,830 --> 00:12:12,590 You've definitely got to pay attention. 178 00:12:12,590 --> 00:12:15,020 Even though I'm doing this with general variables here, 179 00:12:15,020 --> 00:12:17,922 you've got to pay attention to signs of things. 180 00:12:17,922 --> 00:12:20,130 Because otherwise you'll always get the wrong answer. 181 00:12:20,130 --> 00:12:22,710 So you have to pay very close attention here. 182 00:12:22,710 --> 00:12:24,990 So this is, if you like, e to the minus infinity 183 00:12:24,990 --> 00:12:26,720 in the limit, which is 0. 184 00:12:26,720 --> 00:12:30,720 And so in the limit, this thing tends to 0. 185 00:12:30,720 --> 00:12:33,870 And this thing is just equal to 1/k. 186 00:12:33,870 --> 00:12:36,100 And so all told, the answer is 1/k. 187 00:12:36,100 --> 00:12:43,610 And that's it. 188 00:12:43,610 --> 00:12:46,074 Now we're going to abbreviate this a little bit. 189 00:12:46,074 --> 00:12:47,490 This thought process, you're going 190 00:12:47,490 --> 00:12:50,070 to have to go through every single time you do this. 191 00:12:50,070 --> 00:12:52,830 But after a while you also get good enough at it 192 00:12:52,830 --> 00:12:55,770 that you can make it a little bit less cluttered. 193 00:12:55,770 --> 00:13:09,350 So let me show you a shorthand for this same calculation. 194 00:13:09,350 --> 00:13:14,650 Namely, I write 0 to infinity e^(-kx) dx. 195 00:13:14,650 --> 00:13:23,890 And that's equal to -1/k e^(-kx) 0 to infinity. 196 00:13:23,890 --> 00:13:27,710 That was cute. 197 00:13:27,710 --> 00:13:35,600 Not small enough, however. 198 00:13:35,600 --> 00:13:36,374 So, here we are. 199 00:13:36,374 --> 00:13:38,290 We have the same calculation as we had before. 200 00:13:38,290 --> 00:13:40,130 But now we're thinking, really, in our minds 201 00:13:40,130 --> 00:13:43,100 that this infinity is some very, very enormous number. 202 00:13:43,100 --> 00:13:44,790 And we're going to plug it in. 203 00:13:44,790 --> 00:13:47,060 And you can either do this in your head or not. 204 00:13:47,060 --> 00:13:50,760 You say -1/k e^(-infinity). 205 00:13:50,760 --> 00:13:53,510 Here's where I've used the fact that k is positive. 206 00:13:53,510 --> 00:13:57,300 Because e to the -k times a large number is minus infinity. 207 00:13:57,300 --> 00:14:01,570 And then here plus 1/k-- minus -1/k, let write it 208 00:14:01,570 --> 00:14:05,780 the same way I did before. 209 00:14:05,780 --> 00:14:11,840 And that's just equal to 0 + 1/k, which is what we want. 210 00:14:11,840 --> 00:14:17,240 So this is the same calculation, just slightly abbreviated. 211 00:14:17,240 --> 00:14:17,740 Yeah. 212 00:14:17,740 --> 00:14:18,240 Question. 213 00:14:18,240 --> 00:14:29,050 STUDENT: [INAUDIBLE] 214 00:14:29,050 --> 00:14:30,385 PROFESSOR: Good question. 215 00:14:30,385 --> 00:14:31,885 The question is, what about the case 216 00:14:31,885 --> 00:14:34,830 when the limit is infinity? 217 00:14:34,830 --> 00:14:37,710 I'm distinguishing between something existing 218 00:14:37,710 --> 00:14:39,690 and its limit being infinity here. 219 00:14:39,690 --> 00:14:45,660 Whenever I make a discussion of limits, I say a finite limit, 220 00:14:45,660 --> 00:14:49,100 or in this case, it works for infinite limits. 221 00:14:49,100 --> 00:14:50,920 So in other words, when I say exists, 222 00:14:50,920 --> 00:14:54,970 I mean exists and is finite. 223 00:14:54,970 --> 00:14:57,535 So here, when I say that it converges and I 224 00:14:57,535 --> 00:14:58,910 say the limit exists, what I mean 225 00:14:58,910 --> 00:15:00,500 is that it's a finite number. 226 00:15:00,500 --> 00:15:02,630 And so that's indeed what I said here. 227 00:15:02,630 --> 00:15:04,570 The total area is finite. 228 00:15:04,570 --> 00:15:06,100 And, similarly, over here. 229 00:15:06,100 --> 00:15:07,720 I might add, however, that there is 230 00:15:07,720 --> 00:15:09,770 another part of this subject. 231 00:15:09,770 --> 00:15:11,410 Which I'm skipping entirely. 232 00:15:11,410 --> 00:15:13,020 Which is a little bit subtle. 233 00:15:13,020 --> 00:15:14,330 Which is the following. 234 00:15:14,330 --> 00:15:17,860 If f changes sign, there can be some cancellation 235 00:15:17,860 --> 00:15:19,040 and oscillation. 236 00:15:19,040 --> 00:15:21,040 And then sometimes the limit exists, 237 00:15:21,040 --> 00:15:23,760 but the total area, if you counted it all positively, 238 00:15:23,760 --> 00:15:25,680 is actually still infinite. 239 00:15:25,680 --> 00:15:29,120 And we're going to avoid that case. 240 00:15:29,120 --> 00:15:31,820 We're just going to treat these positive cases. 241 00:15:31,820 --> 00:15:33,460 So don't worry about that for now. 242 00:15:33,460 --> 00:15:35,600 That's the next layer of complexity 243 00:15:35,600 --> 00:15:38,432 which we're not addressing in this class. 244 00:15:38,432 --> 00:15:39,140 Another question. 245 00:15:39,140 --> 00:15:45,610 STUDENT: [INAUDIBLE] 246 00:15:45,610 --> 00:15:48,260 PROFESSOR: The question is, would this be OK on tests. 247 00:15:48,260 --> 00:15:49,870 The answer is, absolutely yes. 248 00:15:49,870 --> 00:15:51,550 I want to encourage you to do this. 249 00:15:51,550 --> 00:15:53,660 If you can think about it correctly. 250 00:15:53,660 --> 00:15:55,170 The subtle point is just, you have 251 00:15:55,170 --> 00:15:57,270 to plug in infinity correctly. 252 00:15:57,270 --> 00:15:59,830 Namely, you have to realize that this only 253 00:15:59,830 --> 00:16:01,115 works if k is positive. 254 00:16:01,115 --> 00:16:03,240 This is the step where you're plugging in infinity. 255 00:16:03,240 --> 00:16:06,630 And I'm letting you put this infinity up here 256 00:16:06,630 --> 00:16:08,390 as an endpoint value. 257 00:16:08,390 --> 00:16:12,470 So in fact that's exactly the theme. 258 00:16:12,470 --> 00:16:16,800 The theme is dealing with infinity here. 259 00:16:16,800 --> 00:16:18,550 And I want you to be able to deal with it. 260 00:16:18,550 --> 00:16:20,440 That's my goal. 261 00:16:20,440 --> 00:16:32,660 STUDENT: [INAUDIBLE] 262 00:16:32,660 --> 00:16:35,060 PROFESSOR: OK, so another question is, 263 00:16:35,060 --> 00:16:37,940 so let's be sure here when the limit exists, 264 00:16:37,940 --> 00:16:40,640 I say it has to be finite. 265 00:16:40,640 --> 00:16:46,910 That means it's finite, not infinite. 266 00:16:46,910 --> 00:16:48,270 The limit can be 0. 267 00:16:48,270 --> 00:16:50,790 It can also be -1. 268 00:16:50,790 --> 00:16:51,880 It can be anything. 269 00:16:51,880 --> 00:16:58,590 Doesn't have to be a positive number. 270 00:16:58,590 --> 00:17:04,840 Other questions. 271 00:17:04,840 --> 00:17:07,180 So we've had our first example. 272 00:17:07,180 --> 00:17:23,650 And now I just want to add one physical interpretation here. 273 00:17:23,650 --> 00:17:29,430 This is Example 1, if you like. 274 00:17:29,430 --> 00:17:32,930 And this is something that was on your problem set, remember. 275 00:17:32,930 --> 00:17:36,680 That we talked about the probability, or the number, 276 00:17:36,680 --> 00:17:42,540 if you like, the number of particles 277 00:17:42,540 --> 00:18:02,540 on average that decay in some radioactive substance. 278 00:18:02,540 --> 00:18:09,870 Say, in time between 0 and some capital 279 00:18:09,870 --> 00:18:15,990 T. And then that would be this integral, 0 to capital 280 00:18:15,990 --> 00:18:22,560 T, some total quantity times this integral here. 281 00:18:22,560 --> 00:18:28,000 This is the typical kind of radioactive decay number 282 00:18:28,000 --> 00:18:29,800 that one gets. 283 00:18:29,800 --> 00:18:38,620 Now, in the limit, so this is some number of particles. 284 00:18:38,620 --> 00:18:42,720 If the substance is radioactive, then in the limit, 285 00:18:42,720 --> 00:18:47,610 we have this. 286 00:18:47,610 --> 00:18:56,070 Which is equal to the total number of particles. 287 00:18:56,070 --> 00:18:57,970 And that's something that's going 288 00:18:57,970 --> 00:19:00,424 to be important for normalizing and understanding. 289 00:19:00,424 --> 00:19:01,840 How much does the whole substance, 290 00:19:01,840 --> 00:19:04,840 how many moles do we have of this stuff. 291 00:19:04,840 --> 00:19:05,960 What is it. 292 00:19:05,960 --> 00:19:08,390 And so this is a number that is going to come up. 293 00:19:08,390 --> 00:19:14,040 Now, I emphasize that this notion of T going to infinity 294 00:19:14,040 --> 00:19:16,890 is just an idealization. 295 00:19:16,890 --> 00:19:20,800 We don't really believe that we're going to wait forever 296 00:19:20,800 --> 00:19:23,580 for this substance to decay. 297 00:19:23,580 --> 00:19:27,980 Nevertheless, as theorists, we write down this quantity. 298 00:19:27,980 --> 00:19:29,110 And we use it. 299 00:19:29,110 --> 00:19:31,970 All the time. 300 00:19:31,970 --> 00:19:34,740 Furthermore, there's other good reasons for using it, 301 00:19:34,740 --> 00:19:36,930 and why physicists accept it immediately. 302 00:19:36,930 --> 00:19:39,850 Even though it's not really completely physically realistic 303 00:19:39,850 --> 00:19:43,890 ever to let time go very, very far into the future. 304 00:19:43,890 --> 00:19:47,710 And the reason is, if you notice this answer here, 305 00:19:47,710 --> 00:19:53,330 look at how much simpler this number is, 1/k, 306 00:19:53,330 --> 00:19:57,940 than the numbers that I got in the intermediate stages here. 307 00:19:57,940 --> 00:20:01,800 These are all ugly, the limits are simple. 308 00:20:01,800 --> 00:20:04,040 And this is a theme that I've been trying 309 00:20:04,040 --> 00:20:05,790 to emphasize all semester. 310 00:20:05,790 --> 00:20:07,755 Namely, that the infinitesimal, the things 311 00:20:07,755 --> 00:20:09,630 that you get when you do differentiation, are 312 00:20:09,630 --> 00:20:10,580 the easier formulas. 313 00:20:10,580 --> 00:20:13,610 The algebraic ones, the things in the process 314 00:20:13,610 --> 00:20:16,500 of getting to the limit, are the ugly ones. 315 00:20:16,500 --> 00:20:18,650 These are the easy ones, these are the hard ones. 316 00:20:18,650 --> 00:20:20,930 So in fact, infinity is basically easier 317 00:20:20,930 --> 00:20:23,040 than any finite number. 318 00:20:23,040 --> 00:20:27,110 And a lot of appealing formulas come from those kinds 319 00:20:27,110 --> 00:20:28,470 of calculations. 320 00:20:28,470 --> 00:20:31,840 Another question. 321 00:20:31,840 --> 00:20:39,310 STUDENT: [INAUDIBLE] 322 00:20:39,310 --> 00:20:43,040 PROFESSOR: The question is, shouldn't the answer be A? 323 00:20:43,040 --> 00:20:47,070 Well, the answer turns out to be A/k. 324 00:20:47,070 --> 00:20:49,660 Which means that when you set up your arithmetic, 325 00:20:49,660 --> 00:20:53,750 and you model this to a collection of particles. 326 00:20:53,750 --> 00:20:56,110 So you said it should be A. But that's 327 00:20:56,110 --> 00:20:58,290 because you made an assumption. 328 00:20:58,290 --> 00:21:01,190 Which was that A was the total number of particles. 329 00:21:01,190 --> 00:21:03,200 But that's just false, right? 330 00:21:03,200 --> 00:21:04,810 This is the total number of particles. 331 00:21:04,810 --> 00:21:06,599 So therefore, if you want to set it up, 332 00:21:06,599 --> 00:21:08,140 you want set up so that this number's 333 00:21:08,140 --> 00:21:11,500 the total number of particles. 334 00:21:11,500 --> 00:21:13,010 And that's how you set up a model, 335 00:21:13,010 --> 00:21:14,593 is you do all the calculations and you 336 00:21:14,593 --> 00:21:16,040 see what it's coming out to be. 337 00:21:16,040 --> 00:21:24,490 And that's why you need to do this kind of calculation. 338 00:21:24,490 --> 00:21:25,085 OK, so. 339 00:21:25,085 --> 00:21:27,710 The main thing is, you shouldn't make assumptions about models. 340 00:21:27,710 --> 00:21:29,950 You have to follow what the calculations tell you. 341 00:21:29,950 --> 00:21:32,110 They're not lying. 342 00:21:32,110 --> 00:21:34,990 OK, so now. 343 00:21:34,990 --> 00:21:36,300 We carried this out. 344 00:21:36,300 --> 00:21:40,640 There's one other example which we talked 345 00:21:40,640 --> 00:21:42,830 about earlier in the class. 346 00:21:42,830 --> 00:21:44,470 And I just wanted to mention it again. 347 00:21:44,470 --> 00:21:48,800 It's probably the most famous after this one. 348 00:21:48,800 --> 00:21:51,390 Namely, the integral from minus infinity to infinity 349 00:21:51,390 --> 00:21:53,740 of e^(-x^2) dx. 350 00:21:53,740 --> 00:21:56,930 Which turns out, amazingly, to be able to be evaluated. 351 00:21:56,930 --> 00:21:59,500 It turns out to be the square root of pi. 352 00:21:59,500 --> 00:22:04,340 So this one is also great. 353 00:22:04,340 --> 00:22:07,350 This is the constant which allows 354 00:22:07,350 --> 00:22:12,230 you to compute all kinds of things in probability. 355 00:22:12,230 --> 00:22:22,760 So this is a key number in probability. 356 00:22:22,760 --> 00:22:25,740 It basically is the key to understanding things 357 00:22:25,740 --> 00:22:28,210 like standard deviation and basically 358 00:22:28,210 --> 00:22:31,010 any other thing in the subject of probability. 359 00:22:31,010 --> 00:22:35,910 It's also what's driving these polls that tell you 360 00:22:35,910 --> 00:22:40,390 within 4% accuracy we know that people are 361 00:22:40,390 --> 00:22:42,430 going to vote this way or that. 362 00:22:42,430 --> 00:22:44,830 So in order to interpret all of those kinds of things, 363 00:22:44,830 --> 00:22:48,360 you need to know this number. 364 00:22:48,360 --> 00:22:51,860 And this number was only calculated 365 00:22:51,860 --> 00:22:58,750 numerically starting in the 1700s or so by people who-- 366 00:22:58,750 --> 00:23:01,050 actually, by one guy whose name was de Moivre, who 367 00:23:01,050 --> 00:23:04,950 was selling his services to various royalty who 368 00:23:04,950 --> 00:23:06,390 were running lotteries. 369 00:23:06,390 --> 00:23:09,190 In those days they ran lotteries, too. 370 00:23:09,190 --> 00:23:13,530 And he was able to tell them what the chances were 371 00:23:13,530 --> 00:23:15,590 of the various games. 372 00:23:15,590 --> 00:23:17,720 And he worked out this number. 373 00:23:17,720 --> 00:23:19,314 He realized that this was the pattern. 374 00:23:19,314 --> 00:23:21,730 Although he didn't know that it was the square root of pi, 375 00:23:21,730 --> 00:23:23,390 he knew it to sufficient accuracy 376 00:23:23,390 --> 00:23:25,580 that he could tell them the correct answer 377 00:23:25,580 --> 00:23:29,770 to how much money their lotteries would make. 378 00:23:29,770 --> 00:23:33,070 And of course we do this nowadays, too. 379 00:23:33,070 --> 00:23:34,530 In all kinds of ways. 380 00:23:34,530 --> 00:23:45,440 Including slightly more legit businesses like insurance. 381 00:23:45,440 --> 00:23:49,920 So now, I'm going to give you some more examples. 382 00:23:49,920 --> 00:23:54,910 And the other examples are much more close 383 00:23:54,910 --> 00:23:59,320 to the edge between infinite and finite. 384 00:23:59,320 --> 00:24:02,500 This distinction between convergence and divergence. 385 00:24:02,500 --> 00:24:07,090 And let me just-- Maybe I'll say one more word about why 386 00:24:07,090 --> 00:24:10,690 we care about this very gross issue of whether something 387 00:24:10,690 --> 00:24:12,630 is finite or infinite. 388 00:24:12,630 --> 00:24:14,670 When you're talking about something 389 00:24:14,670 --> 00:24:18,660 like this normal curve here, there's 390 00:24:18,660 --> 00:24:24,290 an issue of how far out you have to go 391 00:24:24,290 --> 00:24:29,280 before you can ignore the rest. 392 00:24:29,280 --> 00:24:34,399 So we're going to ignore what's called the tail here. 393 00:24:34,399 --> 00:24:36,440 Somehow you want to know that this is negligible. 394 00:24:36,440 --> 00:24:38,790 And you want to know how negligible it is. 395 00:24:38,790 --> 00:24:41,350 And this is the job of a mathematician, 396 00:24:41,350 --> 00:24:44,249 is to know what finite region you have to consider 397 00:24:44,249 --> 00:24:46,290 and which one you're going to carefully calculate 398 00:24:46,290 --> 00:24:46,846 numerically. 399 00:24:46,846 --> 00:24:48,470 And then the rest, you're going to have 400 00:24:48,470 --> 00:24:50,690 to take care of by some theoretical reasoning. 401 00:24:50,690 --> 00:24:52,731 You're going to have to know that these tails are 402 00:24:52,731 --> 00:24:54,360 small enough that they don't matter 403 00:24:54,360 --> 00:24:57,150 in your finite calculation. 404 00:24:57,150 --> 00:24:59,070 And so, we care very much about the tails. 405 00:24:59,070 --> 00:25:02,700 Because they're the only thing that the machine won't tell us. 406 00:25:02,700 --> 00:25:05,090 So that's the part that we have to know. 407 00:25:05,090 --> 00:25:06,830 And these tails are also something 408 00:25:06,830 --> 00:25:09,870 which are discussed all the time in financial mathematics. 409 00:25:09,870 --> 00:25:11,830 They're very worried about fat tails. 410 00:25:11,830 --> 00:25:16,270 That is, unlikely events that nevertheless happen sometimes. 411 00:25:16,270 --> 00:25:18,570 And they get burned fairly regularly with them. 412 00:25:18,570 --> 00:25:25,440 As they have recently, with the mortgage scandal. 413 00:25:25,440 --> 00:25:28,416 So, these things are pretty serious 414 00:25:28,416 --> 00:25:30,540 and they really are spending a lot of time on them. 415 00:25:30,540 --> 00:25:32,922 Of course, there are lots of other practical issues 416 00:25:32,922 --> 00:25:34,130 besides just the mathematics. 417 00:25:34,130 --> 00:25:37,340 But you've got to get the math right, too. 418 00:25:37,340 --> 00:25:40,860 So we're going to now talk about some borderline cases 419 00:25:40,860 --> 00:25:42,950 for these fat tails. 420 00:25:42,950 --> 00:25:45,820 Just how fat do they have to be before they become infinite 421 00:25:45,820 --> 00:25:51,820 and overwhelm the central bump. 422 00:25:51,820 --> 00:25:56,120 So we'll save this for just a second. 423 00:25:56,120 --> 00:25:58,734 And what I'm saving up here is the borderline case, 424 00:25:58,734 --> 00:26:00,150 which I'm going to concentrate on, 425 00:26:00,150 --> 00:26:07,500 which is this moderate rate, which is x to powers. 426 00:26:07,500 --> 00:26:09,570 Here's our next example. 427 00:26:09,570 --> 00:26:13,230 I guess we'll call this Example 3. 428 00:26:13,230 --> 00:26:17,540 It's the integral from 1 to infinity dx / x. 429 00:26:17,540 --> 00:26:20,800 That's the power p = 1. 430 00:26:20,800 --> 00:26:23,980 And this turns out to be a borderline case. 431 00:26:23,980 --> 00:26:26,790 So it's worth carrying out carefully. 432 00:26:26,790 --> 00:26:29,820 Now, again I'm going to do it by the slower method. 433 00:26:29,820 --> 00:26:31,290 Rather than the shorthand method. 434 00:26:31,290 --> 00:26:36,100 But ultimately, you can do it by the short method if you'd like. 435 00:26:36,100 --> 00:26:38,960 I break it up into an integral that 436 00:26:38,960 --> 00:26:44,260 goes up to some large number, N. I see that it's the logarithm 437 00:26:44,260 --> 00:26:47,060 function is the antiderivative. 438 00:26:47,060 --> 00:26:51,740 And so what I get is ln N minus ln 1, which is just 0. 439 00:26:51,740 --> 00:26:53,710 So this is just log N. In any case, 440 00:26:53,710 --> 00:26:57,860 it tends to infinity as N. goes to infinity. 441 00:26:57,860 --> 00:27:01,430 So the conclusion is, since the limit is infinite, 442 00:27:01,430 --> 00:27:12,020 that this thing diverges. 443 00:27:12,020 --> 00:27:16,560 Now, I'm going to do this systematically now 444 00:27:16,560 --> 00:27:20,830 with all powers p, to see what happens. 445 00:27:20,830 --> 00:27:22,050 I'll look at the integral. 446 00:27:22,050 --> 00:27:23,900 Sorry, I'm going to have to start at 1 here. 447 00:27:23,900 --> 00:27:27,820 From 1 to infinity, dx / x^p. 448 00:27:27,820 --> 00:27:29,480 and see what happens with these. 449 00:27:29,480 --> 00:27:32,200 And you'll see that p = 1 is a borderline when 450 00:27:32,200 --> 00:27:35,720 I do this calculation. 451 00:27:35,720 --> 00:27:39,059 This time I'm going to do the calculation the hard way. 452 00:27:39,059 --> 00:27:41,350 But now you're going to have to think and pay attention 453 00:27:41,350 --> 00:27:43,225 to see what it is that I'm doing. 454 00:27:43,225 --> 00:27:45,350 First of all, I'm going to take the antiderivative. 455 00:27:45,350 --> 00:27:53,590 And this is x^(-p), so it's - -p + 1 divided by -p + 1. 456 00:27:53,590 --> 00:28:00,960 That's the antiderivative of the function 1/x^p or x^(-p). 457 00:28:00,960 --> 00:28:07,710 And then I have to evaluate that at 1 and infinity. 458 00:28:07,710 --> 00:28:10,600 So now, I'll write this down. 459 00:28:10,600 --> 00:28:13,990 But I'm going to be particularly careful here. 460 00:28:13,990 --> 00:28:14,880 I'll write it down. 461 00:28:14,880 --> 00:28:27,040 It's infinity to the -p + 1 over -p + 1 minus, 462 00:28:27,040 --> 00:28:29,160 so I plug in 1 here. 463 00:28:29,160 --> 00:28:30,450 So I get 1/(-p+1). 464 00:28:34,870 --> 00:28:36,440 So this is what I'm getting. 465 00:28:36,440 --> 00:28:38,730 Again, what you should be thinking here 466 00:28:38,730 --> 00:28:45,410 is this is a very large number to this power. 467 00:28:45,410 --> 00:28:47,770 Now, there are two cases. 468 00:28:47,770 --> 00:28:48,880 There are two cases. 469 00:28:48,880 --> 00:28:52,530 And they exactly split at p = 1. 470 00:28:52,530 --> 00:28:55,280 When p = 1, this number is 0. 471 00:28:55,280 --> 00:28:57,380 This exponent is 0, and in fact this expression 472 00:28:57,380 --> 00:29:01,310 doesn't make any sense because the denominator is also 0. 473 00:29:01,310 --> 00:29:05,310 But for all of the other values, the denominator makes sense. 474 00:29:05,310 --> 00:29:09,780 But what's going on is that this is infinite 475 00:29:09,780 --> 00:29:13,420 when this exponent is infinity to a positive power. 476 00:29:13,420 --> 00:29:20,070 And it's 0 when it's infinity to a negative power. 477 00:29:20,070 --> 00:29:22,970 So I'm going to say it here, and you must check this at home. 478 00:29:22,970 --> 00:29:24,785 Because this is exactly what I'm going 479 00:29:24,785 --> 00:29:27,060 to ask you about on the exam. 480 00:29:27,060 --> 00:29:28,590 This is it. 481 00:29:28,590 --> 00:29:33,360 This type of thing, maybe with a specific value of p here. 482 00:29:33,360 --> 00:29:45,660 When p < 1, this thing is infinite. 483 00:29:45,660 --> 00:29:53,560 On the other hand, when p > 1, this thing is 0. 484 00:29:53,560 --> 00:29:59,260 So when p > 1, this thing is 0. 485 00:29:59,260 --> 00:30:01,040 It's just equal to 0. 486 00:30:01,040 --> 00:30:09,125 And so the answer is 1/(p-1). 487 00:30:09,125 --> 00:30:10,250 Because that's this number. 488 00:30:10,250 --> 00:30:12,970 Minus the quantity 1/(-p+1). 489 00:30:15,720 --> 00:30:17,480 This is a finite number here. 490 00:30:17,480 --> 00:30:20,390 Notice that the answer would be weird if this thing went away 491 00:30:20,390 --> 00:30:22,840 in the p < 1 case. 492 00:30:22,840 --> 00:30:24,740 Then it would be a negative number. 493 00:30:24,740 --> 00:30:28,420 It would be a very strange answer to this question. 494 00:30:28,420 --> 00:30:29,920 So, in fact that's not what happens. 495 00:30:29,920 --> 00:30:32,477 What happens is that the answer doesn't make sense. 496 00:30:32,477 --> 00:30:33,060 It's infinite. 497 00:30:33,060 --> 00:30:35,620 So let me just write this down again, under here. 498 00:30:35,620 --> 00:30:42,160 This is a test in a particular case. 499 00:30:42,160 --> 00:30:47,190 And here's the conclusion. 500 00:30:47,190 --> 00:30:48,027 Ah. 501 00:30:48,027 --> 00:30:48,610 No, I'm sorry. 502 00:30:48,610 --> 00:31:03,760 I think I was going to write it over on this board here. 503 00:31:03,760 --> 00:31:11,770 So the conclusion is that the integral from 1 to infinity dx 504 00:31:11,770 --> 00:31:21,000 / x^p diverges if p <= 1. 505 00:31:21,000 --> 00:31:33,130 And converges if p > 1. 506 00:31:33,130 --> 00:31:35,550 And in fact, we can actually evaluate it. 507 00:31:35,550 --> 00:31:38,380 It's equal to 1/(p-1). 508 00:31:38,380 --> 00:31:44,050 It's got a nice, clean formula even. 509 00:31:44,050 --> 00:31:45,820 Alright, now let me remind you. 510 00:31:45,820 --> 00:31:47,870 So I didn't spell the word diverges right, did I? 511 00:31:47,870 --> 00:31:49,170 Oh no, that's an r. 512 00:31:49,170 --> 00:31:55,530 I guess that's right. 513 00:31:55,530 --> 00:31:57,040 Diverges if p <= 1. 514 00:31:57,040 --> 00:31:59,910 So really, I needed both of these arguments, 515 00:31:59,910 --> 00:32:02,310 which are sitting above it, in order to do it. 516 00:32:02,310 --> 00:32:06,510 Because the second argument didn't work at all when p = 1 517 00:32:06,510 --> 00:32:09,184 because the formula for the antiderivative is wrong. 518 00:32:09,184 --> 00:32:10,600 The formula for the antiderivative 519 00:32:10,600 --> 00:32:13,570 is given by the log function when p = 1. 520 00:32:13,570 --> 00:32:15,780 So I had to do this calculation too. 521 00:32:15,780 --> 00:32:21,420 This is the borderline case, between p > 1 and p < 1. 522 00:32:21,420 --> 00:32:23,820 When p > 1, we got convergence. 523 00:32:23,820 --> 00:32:27,140 We could calculate the integral. 524 00:32:27,140 --> 00:32:30,290 When p < 1, when we got divergence and we calculated 525 00:32:30,290 --> 00:32:31,510 the integral over there. 526 00:32:31,510 --> 00:32:34,220 And here in the borderline case, we got a logarithm, 527 00:32:34,220 --> 00:32:35,940 and we also got divergence. 528 00:32:35,940 --> 00:32:39,520 So it failed at the edge. 529 00:32:39,520 --> 00:32:46,980 Now, this takes care of all the powers. 530 00:32:46,980 --> 00:32:54,490 Now, there are a number of different things 531 00:32:54,490 --> 00:32:58,320 that one can deduce from this. 532 00:32:58,320 --> 00:33:02,390 And let me carry them out. 533 00:33:02,390 --> 00:33:04,180 So this is more or less the second thing 534 00:33:04,180 --> 00:33:07,030 that you'll want to do. 535 00:33:07,030 --> 00:33:12,440 And I'm going to emphasize maybe one aspect of it. 536 00:33:12,440 --> 00:33:14,510 I guess we'll get rid of this. 537 00:33:14,510 --> 00:33:17,500 But it's still the issue that we're discussing here. 538 00:33:17,500 --> 00:33:20,230 Is whether this area is fat or thin. 539 00:33:20,230 --> 00:33:24,500 I'll remind you of that. 540 00:33:24,500 --> 00:33:29,240 So here's the next idea. 541 00:33:29,240 --> 00:33:34,250 Something called limit comparison. 542 00:33:34,250 --> 00:33:35,850 Limit comparison is what you're going 543 00:33:35,850 --> 00:33:39,100 to use when, instead of being able actually 544 00:33:39,100 --> 00:33:41,670 to calculate the number, you don't yet 545 00:33:41,670 --> 00:33:42,670 know what the number is. 546 00:33:42,670 --> 00:33:44,720 But you can make a comparison to something 547 00:33:44,720 --> 00:33:48,940 whose convergence properties you already understand. 548 00:33:48,940 --> 00:33:50,780 Now, here's the statement. 549 00:33:50,780 --> 00:33:57,090 If a function, f, is similar to a function, asymptotically 550 00:33:57,090 --> 00:34:00,830 the same as a function, g, as x goes to infinity, 551 00:34:00,830 --> 00:34:03,830 I'll remind you what that means in a second. 552 00:34:03,830 --> 00:34:08,890 Then the integral starting at some 553 00:34:08,890 --> 00:34:20,380 point out to infinity of f(x) dx, and the other one, 554 00:34:20,380 --> 00:34:22,230 converge and diverge at the same time. 555 00:34:22,230 --> 00:34:30,320 So both, either, either-- sorry, let's try it the other way. 556 00:34:30,320 --> 00:34:31,940 Either, both. 557 00:34:31,940 --> 00:34:42,700 Either both converge, or both diverge. 558 00:34:42,700 --> 00:34:44,840 They behave exactly the same way. 559 00:34:44,840 --> 00:34:50,800 In terms of whether they're infinite or not. 560 00:34:50,800 --> 00:34:56,650 And, let me remind you what this tilde means. 561 00:34:56,650 --> 00:35:14,750 This thing means that f(x) / g(x) tends to 1. 562 00:35:14,750 --> 00:35:19,820 So if you have a couple of functions like that, 563 00:35:19,820 --> 00:35:21,990 then their behavior is the same. 564 00:35:21,990 --> 00:35:25,050 This is more or less obvious. 565 00:35:25,050 --> 00:35:29,710 It's just because far enough out, 566 00:35:29,710 --> 00:35:34,712 this is for large a, if you like. 567 00:35:34,712 --> 00:35:36,670 We're not paying any attention to what happens. 568 00:35:36,670 --> 00:35:40,400 It just has to do with the tail, and after a while 569 00:35:40,400 --> 00:35:42,290 f(x) and g(x) are comparable to each other. 570 00:35:42,290 --> 00:35:46,980 So their integrals are comparable to each other. 571 00:35:46,980 --> 00:35:51,290 So let's just do a couple of examples here. 572 00:35:51,290 --> 00:35:56,500 If you take the integral from 0 to infinity dx over the square 573 00:35:56,500 --> 00:36:09,400 root of x^2+10, then I claim that the square root of x^2+10 574 00:36:09,400 --> 00:36:16,220 resembles the square root of x^2, which is just x. 575 00:36:16,220 --> 00:36:19,830 So this thing is going to be like-- So now I'm 576 00:36:19,830 --> 00:36:22,570 going to have to do one thing to you here. 577 00:36:22,570 --> 00:36:26,180 Which is, I'm going to change this to 1. 578 00:36:26,180 --> 00:36:28,870 To infinity. 579 00:36:28,870 --> 00:36:30,900 dx/x. 580 00:36:30,900 --> 00:36:35,756 And the reason is that this x = 0 is extraneous. 581 00:36:35,756 --> 00:36:37,380 Doesn't have anything to do with what's 582 00:36:37,380 --> 00:36:39,710 going on with this problem. 583 00:36:39,710 --> 00:36:45,640 This guy here, the piece of it from-- 584 00:36:45,640 --> 00:36:54,660 So we're going to ignore the part integral from 0 to 1 dx / 585 00:36:54,660 --> 00:37:01,480 square root of x^2+10, which is finite anyway. 586 00:37:01,480 --> 00:37:03,340 And unimportant. 587 00:37:03,340 --> 00:37:06,300 Whereas, unfortunately, the integral of dx 588 00:37:06,300 --> 00:37:08,310 will have a singularity at x = 0. 589 00:37:08,310 --> 00:37:12,040 So we can't make the comparison there. 590 00:37:12,040 --> 00:37:14,910 Anyway, this one is infinite. 591 00:37:14,910 --> 00:37:21,490 So this is divergent. 592 00:37:21,490 --> 00:37:27,340 Using what I knew from before. 593 00:37:27,340 --> 00:37:27,840 Yeah. 594 00:37:27,840 --> 00:37:33,680 STUDENT: [INAUDIBLE] 595 00:37:33,680 --> 00:37:39,770 PROFESSOR: The question is, why did we switch from 0 to 1? 596 00:37:39,770 --> 00:37:43,280 So I'm going to say a little bit more about that later. 597 00:37:43,280 --> 00:37:48,750 But let me just make it a warning here. 598 00:37:48,750 --> 00:37:58,010 Which is that this guy here is infinite for other reasons. 599 00:37:58,010 --> 00:38:04,310 Unrelated reasons. 600 00:38:04,310 --> 00:38:07,650 The comparison that we are trying to make is with the tail 601 00:38:07,650 --> 00:38:09,400 as x goes to infinity. 602 00:38:09,400 --> 00:38:11,920 So another way of saying this is that I 603 00:38:11,920 --> 00:38:16,940 should stick an a here and an a here and stay away from 0. 604 00:38:16,940 --> 00:38:18,770 So, say a = 1. 605 00:38:18,770 --> 00:38:21,820 If I make these both 1, that would be OK. 606 00:38:21,820 --> 00:38:24,700 If I make them both 2, that would be OK. 607 00:38:24,700 --> 00:38:27,250 If I make them both 100, that would be OK. 608 00:38:27,250 --> 00:38:29,450 So let's leave it as 100 right now. 609 00:38:29,450 --> 00:38:30,640 And it's acceptable. 610 00:38:30,640 --> 00:38:33,360 I want you to stay away from the origin here. 611 00:38:33,360 --> 00:38:36,820 Because that's another bad point. 612 00:38:36,820 --> 00:38:40,380 And just talk about what's happening with the tail. 613 00:38:40,380 --> 00:38:45,530 So this is a tail, and I also had a different name for it 614 00:38:45,530 --> 00:38:46,580 up top. 615 00:38:46,580 --> 00:38:47,810 Which is emphasizing this. 616 00:38:47,810 --> 00:38:49,770 Which is limit comparison. 617 00:38:49,770 --> 00:38:52,740 It's only what's happening at the very end of the picture 618 00:38:52,740 --> 00:38:53,920 that we're interested in. 619 00:38:53,920 --> 00:38:56,710 So again, this is as x goes to infinity. 620 00:38:56,710 --> 00:38:59,880 That's the limit we're talking about, the limiting behavior. 621 00:38:59,880 --> 00:39:01,770 And we're trying not to pay attention 622 00:39:01,770 --> 00:39:10,890 to what's happening for small values of x. 623 00:39:10,890 --> 00:39:13,450 So to be consistent, if I'm going to do it up 624 00:39:13,450 --> 00:39:17,440 to 100 I'm ignoring what's happening up to the first 100 625 00:39:17,440 --> 00:39:25,240 values. 626 00:39:25,240 --> 00:39:28,950 In any case, this guy diverged. 627 00:39:28,950 --> 00:39:33,120 And let me give you another example. 628 00:39:33,120 --> 00:39:36,050 This one, you could have computed. 629 00:39:36,050 --> 00:39:38,040 This one you could have computed, right? 630 00:39:38,040 --> 00:39:43,930 Because it's a square root of quadratic, 631 00:39:43,930 --> 00:39:48,420 so there's a trig substitution that evaluates this one. 632 00:39:48,420 --> 00:39:51,400 The advantage of this limit comparison method 633 00:39:51,400 --> 00:39:53,432 is, it makes no difference whether you 634 00:39:53,432 --> 00:39:54,640 can compute the thing or not. 635 00:39:54,640 --> 00:39:57,840 You can still decide whether it's finite or infinite, fairly 636 00:39:57,840 --> 00:39:58,820 easily. 637 00:39:58,820 --> 00:40:10,850 So let me give you an example of that. 638 00:40:10,850 --> 00:40:13,030 So here we have another example. 639 00:40:13,030 --> 00:40:21,560 We'll take the integral dx, square root of x^3 + 3. 640 00:40:21,560 --> 00:40:25,080 Let's say, for the sake of argument. 641 00:40:25,080 --> 00:40:28,970 From 0 to infinity. 642 00:40:28,970 --> 00:40:35,200 Let's leave off, let's make it 10 to infinity, whatever. 643 00:40:35,200 --> 00:40:42,360 Now this one is problematic for you. 644 00:40:42,360 --> 00:40:44,910 You're not going to be able to evaluate it, I promise. 645 00:40:44,910 --> 00:40:53,660 So on the other hand 1 over the square root of x^3 + 3 is 646 00:40:53,660 --> 00:40:56,950 similar to 1 over the square root of x^3, 647 00:40:56,950 --> 00:41:00,580 which is 1/x^(3/2). 648 00:41:00,580 --> 00:41:10,680 So this thing is going to resemble this integral here. 649 00:41:10,680 --> 00:41:16,870 Which is convergent. 650 00:41:16,870 --> 00:41:25,660 According to our rule. 651 00:41:25,660 --> 00:41:31,950 So those are the, more or less the main ingredients. 652 00:41:31,950 --> 00:41:34,740 Let me just mention one other integral, which was 653 00:41:34,740 --> 00:41:37,900 the one that we had over here. 654 00:41:37,900 --> 00:41:39,830 This one here. 655 00:41:39,830 --> 00:41:42,300 If you look at this integral, of course 656 00:41:42,300 --> 00:41:45,270 we can compute it so we know the area is finite. 657 00:41:45,270 --> 00:41:51,910 But the way that you would actually carry this out, 658 00:41:51,910 --> 00:41:54,040 if you didn't know the number and you 659 00:41:54,040 --> 00:41:58,860 wanted to check that this integral were finite, 660 00:41:58,860 --> 00:42:00,880 then you would make the following comparison. 661 00:42:00,880 --> 00:42:04,170 This one is not so difficult. First of all, 662 00:42:04,170 --> 00:42:07,300 you would write it as twice the integral from 0 to infinity 663 00:42:07,300 --> 00:42:11,770 of e^(-x^2) dx. 664 00:42:11,770 --> 00:42:15,340 This is a new example here, and we're just 665 00:42:15,340 --> 00:42:18,970 checking for convergence only. 666 00:42:18,970 --> 00:42:25,700 Not evaluation. 667 00:42:25,700 --> 00:42:36,459 And now, I'm going to make a comparison here, 668 00:42:36,459 --> 00:42:38,750 Rather than a limit, comparison I'm actually just going 669 00:42:38,750 --> 00:42:40,041 to make an ordinary comparison. 670 00:42:40,041 --> 00:42:42,170 That's because this thing vanishes so fast. 671 00:42:42,170 --> 00:42:44,387 It's so favorable that we can only put something 672 00:42:44,387 --> 00:42:46,470 on top of it, we can't get something underneath it 673 00:42:46,470 --> 00:42:48,280 that exactly balances with it. 674 00:42:48,280 --> 00:42:50,280 In other words, this wiggle was something 675 00:42:50,280 --> 00:42:53,050 which had the same growth rate as the function involved. 676 00:42:53,050 --> 00:42:54,870 This thing just vanishes incredibly fast. 677 00:42:54,870 --> 00:42:55,830 It's great. 678 00:42:55,830 --> 00:42:58,500 It's too good for us, for this comparison. 679 00:42:58,500 --> 00:43:00,110 So instead what I'm going to make 680 00:43:00,110 --> 00:43:02,810 is the following comparison. 681 00:43:02,810 --> 00:43:07,650 e^(-x^2) <= e^(-x). 682 00:43:07,650 --> 00:43:10,990 At least for x >= 1. 683 00:43:10,990 --> 00:43:20,150 When x >= 1, then x^2 >= x, and so -x^2 < -x. 684 00:43:20,150 --> 00:43:22,830 And so e^(-x^2) is less than this. 685 00:43:22,830 --> 00:43:26,440 So this is the reasoning involved. 686 00:43:26,440 --> 00:43:29,350 And so what we have here is two pieces. 687 00:43:29,350 --> 00:43:33,460 We have 2, the integral from 0 to 1, of e^(-x^2). 688 00:43:33,460 --> 00:43:35,430 That's just a finite part. 689 00:43:35,430 --> 00:43:37,330 And then we have this other part, 690 00:43:37,330 --> 00:43:42,490 which I'm going to replace with the e^(-x) here. 691 00:43:42,490 --> 00:43:50,310 2 times 1 to infinity e^(-x) dx. 692 00:43:50,310 --> 00:43:53,939 So this is, if you like, this is ordinary comparison 693 00:43:53,939 --> 00:43:54,480 of integrals. 694 00:43:54,480 --> 00:43:57,021 It's something that we did way at the beginning of the class. 695 00:43:57,021 --> 00:43:59,750 Or much earlier on, when we were dealing with integrals. 696 00:43:59,750 --> 00:44:04,750 Which is that if you have a larger integrand, then 697 00:44:04,750 --> 00:44:07,519 the integral gets larger. 698 00:44:07,519 --> 00:44:08,810 So we've replaced the integral. 699 00:44:08,810 --> 00:44:11,360 We've got the same integrand on 0 to 1. 700 00:44:11,360 --> 00:44:14,120 And we have a larger integrand on-- 701 00:44:14,120 --> 00:44:20,680 So this one is larger integrand. 702 00:44:20,680 --> 00:44:23,200 And this one we know is finite. 703 00:44:23,200 --> 00:44:24,830 This one is a convergent integral. 704 00:44:24,830 --> 00:44:29,760 So the whole business is convergent. 705 00:44:29,760 --> 00:44:31,970 But of course we replaced it by a much larger thing. 706 00:44:31,970 --> 00:44:34,053 So we're not getting the right number out of this. 707 00:44:34,053 --> 00:44:47,920 We're just showing that it converges. 708 00:44:47,920 --> 00:44:51,060 So these are the main ingredients. 709 00:44:51,060 --> 00:44:54,380 As I say, once the thing gets really, really fast-decaying, 710 00:44:54,380 --> 00:44:57,310 it's relatively straightforward. 711 00:44:57,310 --> 00:45:04,790 There's lots of room to show that it converges. 712 00:45:04,790 --> 00:45:07,020 Now, there's one last item of business 713 00:45:07,020 --> 00:45:10,080 here which I have to promise you. 714 00:45:10,080 --> 00:45:16,630 Which I promised you, which had to do with dealing 715 00:45:16,630 --> 00:45:22,220 with this bottom piece here. 716 00:45:22,220 --> 00:45:24,210 So I have to deal with what happens 717 00:45:24,210 --> 00:45:26,050 when there's a singularity. 718 00:45:26,050 --> 00:45:56,800 This is known as an improper integral of the second type. 719 00:45:56,800 --> 00:46:01,200 And the idea of these examples is the following. 720 00:46:01,200 --> 00:46:06,010 You might have something like this. 721 00:46:06,010 --> 00:46:11,200 Something like this. 722 00:46:11,200 --> 00:46:16,770 Or something like this. 723 00:46:16,770 --> 00:46:20,690 These are typical sorts of examples. 724 00:46:20,690 --> 00:46:27,300 And before actually describing what happens, 725 00:46:27,300 --> 00:46:28,540 I just want to mention. 726 00:46:28,540 --> 00:46:31,020 So first of all, the key point here 727 00:46:31,020 --> 00:46:32,760 is you can just calculate these things. 728 00:46:32,760 --> 00:46:37,940 And plug in 0 and it works and you'll get the right answer. 729 00:46:37,940 --> 00:46:40,254 So you'll determine, you'll figure out, 730 00:46:40,254 --> 00:46:42,170 that it turns out that this one will converge, 731 00:46:42,170 --> 00:46:44,211 this one will diverge, and this one will diverge. 732 00:46:44,211 --> 00:46:46,340 That's what will turn out to happen. 733 00:46:46,340 --> 00:46:50,610 However, I want to warn you that you can fool yourself. 734 00:46:50,610 --> 00:46:53,150 And so let me give you a slightly different example. 735 00:46:53,150 --> 00:46:58,410 Let's consider this integral here. 736 00:46:58,410 --> 00:47:01,360 The integral from -1 to 1 dx / x^2. 737 00:47:05,850 --> 00:47:09,246 If you carry out this integral without thinking, 738 00:47:09,246 --> 00:47:11,370 what will happen is, you'll get the antiderivative, 739 00:47:11,370 --> 00:47:16,250 which is -x^(-1), evaluated at -1 and 1. 740 00:47:16,250 --> 00:47:20,550 And you plug it in. 741 00:47:20,550 --> 00:47:21,520 And what do you get? 742 00:47:21,520 --> 00:47:30,880 You get -1^(-1) minus, uh-oh, minus (-1)^(-1). 743 00:47:30,880 --> 00:47:33,610 There's a lot of -1's in this problem. 744 00:47:33,610 --> 00:47:35,090 OK, so that's -1. 745 00:47:35,090 --> 00:47:37,010 And this one, if you work it all out, 746 00:47:37,010 --> 00:47:39,340 as I sometimes don't get the signs right, but this time 747 00:47:39,340 --> 00:47:41,160 I really paid attention. 748 00:47:41,160 --> 00:47:44,480 It's -1, I'm telling you that's what it is. 749 00:47:44,480 --> 00:47:46,880 So that comes out to be -2. 750 00:47:46,880 --> 00:47:50,450 Now, this is ridiculous. 751 00:47:50,450 --> 00:48:01,400 This function here looks like this. 752 00:48:01,400 --> 00:48:04,140 It's positive, right? 753 00:48:04,140 --> 00:48:06,000 1/x^2 is positive. 754 00:48:06,000 --> 00:48:10,030 How exactly is it that the area between -1 and 1 755 00:48:10,030 --> 00:48:13,990 came out to be a negative number? 756 00:48:13,990 --> 00:48:16,510 That can't be. 757 00:48:16,510 --> 00:48:18,420 There was clearly something wrong with this. 758 00:48:18,420 --> 00:48:21,210 And this is the kind of thing that you'll get regularly 759 00:48:21,210 --> 00:48:25,740 if you don't pay attention to convergence of integrals. 760 00:48:25,740 --> 00:48:29,640 So what's going on here is actually that this area in here 761 00:48:29,640 --> 00:48:33,130 is infinite. 762 00:48:33,130 --> 00:48:38,090 And this calculation that I made is nonsense. 763 00:48:38,090 --> 00:48:41,060 So it doesn't work. 764 00:48:41,060 --> 00:48:42,010 This is wrong. 765 00:48:42,010 --> 00:48:50,237 Because it's divergent. 766 00:48:50,237 --> 00:48:52,070 Actually, when you get to imaginary numbers, 767 00:48:52,070 --> 00:48:56,710 it'll turn out that there's a way of rescuing it. 768 00:48:56,710 --> 00:48:58,980 But, still, it means something totally different when 769 00:48:58,980 --> 00:49:03,160 that integral is thought to -2. 770 00:49:03,160 --> 00:49:04,000 So. 771 00:49:04,000 --> 00:49:07,570 What I want you to do here, so I think 772 00:49:07,570 --> 00:49:11,590 we'll have to finish this up very briefly next time. 773 00:49:11,590 --> 00:49:13,270 We'll do these three calculations 774 00:49:13,270 --> 00:49:20,500 and you'll see that these two guys are divergent and this one 775 00:49:20,500 --> 00:49:21,990 converges. 776 00:49:21,990 --> 00:49:24,110 And we'll do that next time.