1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,144 Commons license. 3 00:00:03,144 --> 00:00:05,460 Your support will help MIT OpenCourseWare 4 00:00:05,460 --> 00:00:09,646 continue to offer high quality educational resources for free. 5 00:00:09,646 --> 00:00:12,260 To make a donation or to view additional materials 6 00:00:12,260 --> 00:00:16,150 from hundreds of MIT courses visit MIT OpenCourseWare 7 00:00:16,150 --> 00:00:17,090 at ocw.mit.edu. 8 00:00:21,344 --> 00:00:22,760 PROFESSOR: Today we are continuing 9 00:00:22,760 --> 00:00:24,730 with improper integrals. 10 00:00:24,730 --> 00:00:27,371 I still have a little bit more to tell you about them. 11 00:00:30,080 --> 00:00:33,510 What we were discussing at the very end of last time 12 00:00:33,510 --> 00:00:36,195 was improper integrals. 13 00:00:41,560 --> 00:00:44,570 Now and these are going to be improper integrals 14 00:00:44,570 --> 00:00:48,220 of the second kind. 15 00:00:48,220 --> 00:00:50,990 By second kind I mean that they have a singularity 16 00:00:50,990 --> 00:00:52,095 at a finite place. 17 00:00:54,800 --> 00:00:57,260 That would be something like this. 18 00:00:57,260 --> 00:01:00,140 So here's the definition if you like. 19 00:01:00,140 --> 00:01:03,380 Same sort of thing as we did when the singularity 20 00:01:03,380 --> 00:01:04,120 was at infinity. 21 00:01:04,120 --> 00:01:09,440 So if you have the integral from 0 to 1 of f(x). 22 00:01:09,440 --> 00:01:12,600 This is going to be the same thing as the limit, 23 00:01:12,600 --> 00:01:17,780 as a goes to 0 from above, the integral from a to 1 24 00:01:17,780 --> 00:01:21,150 of f(x) dx. 25 00:01:21,150 --> 00:01:25,920 And the idea here is the same one that we had at infinity. 26 00:01:25,920 --> 00:01:27,320 Let me draw a picture of it. 27 00:01:27,320 --> 00:01:30,030 You have, imagine a function which is coming down like this 28 00:01:30,030 --> 00:01:32,020 and here's the point 1. 29 00:01:32,020 --> 00:01:35,180 And we don't know whether the area enclosed is 30 00:01:35,180 --> 00:01:39,040 going to be infinite or finite and so we cut it off 31 00:01:39,040 --> 00:01:40,680 at some place a. 32 00:01:40,680 --> 00:01:44,220 And we let a go to 0 from above. 33 00:01:44,220 --> 00:01:46,800 So really it's 0+. 34 00:01:46,800 --> 00:01:49,820 So we're coming in from the right here. 35 00:01:49,820 --> 00:01:52,810 And we're counting up the area in this chunk. 36 00:01:52,810 --> 00:01:56,520 And we're seeing as it expands whether it goes to infinity 37 00:01:56,520 --> 00:01:59,290 or whether it tends to some finite limit. 38 00:01:59,290 --> 00:02:02,990 Right, so this is the example and this is the definition. 39 00:02:02,990 --> 00:02:07,000 And just as we did for the other kind of improper integral, 40 00:02:07,000 --> 00:02:13,747 we say that this converges -- so that's the key word here -- 41 00:02:13,747 --> 00:02:24,220 if the limit is finite, exists maybe I should just say 42 00:02:24,220 --> 00:02:30,773 and diverges if not. 43 00:02:35,650 --> 00:02:38,945 Let's just take care of the basic examples. 44 00:02:42,110 --> 00:02:44,740 First of all I wrote this one down last time. 45 00:02:44,740 --> 00:02:47,390 We're going to evaluate this one. 46 00:02:47,390 --> 00:02:51,565 The integral from 0 to 1 of 1 over the square root of x. 47 00:02:55,402 --> 00:02:57,360 And this just, you almost don't notice the fact 48 00:02:57,360 --> 00:02:59,920 that it goes to infinity. 49 00:02:59,920 --> 00:03:02,570 This goes to infinity as x goes to 0. 50 00:03:02,570 --> 00:03:05,860 But if you evaluate it -- first of all we always write this 51 00:03:05,860 --> 00:03:06,860 as a power. 52 00:03:06,860 --> 00:03:08,010 Right? 53 00:03:08,010 --> 00:03:09,870 To get the evaluation. 54 00:03:09,870 --> 00:03:13,060 And then I'm not even going to replace the 0 by an a. 55 00:03:13,060 --> 00:03:14,490 I'm just going to leave it as 0. 56 00:03:14,490 --> 00:03:18,660 The antiderivative here is x^(1/2) times 2. 57 00:03:21,540 --> 00:03:23,750 And then I evaluate that at 0 and 1. 58 00:03:23,750 --> 00:03:25,080 And I get 2. 59 00:03:25,080 --> 00:03:29,090 2 minus 0, which is 2. 60 00:03:29,090 --> 00:03:31,330 All right so this one is convergent. 61 00:03:31,330 --> 00:03:34,180 And not only is it convergent but we can evaluate it. 62 00:03:38,310 --> 00:03:42,140 The second example, being not systematic 63 00:03:42,140 --> 00:03:44,300 but really giving you the principal examples 64 00:03:44,300 --> 00:03:51,650 that we'll be thinking about, is this one here, dx / x. 65 00:03:51,650 --> 00:03:54,200 And this one gives you the antiderivative 66 00:03:54,200 --> 00:03:56,070 as the logarithm. 67 00:03:56,070 --> 00:03:58,180 Evaluated at 0 and 1. 68 00:03:58,180 --> 00:04:00,190 And now again you have to have this thought 69 00:04:00,190 --> 00:04:02,770 process in your mind that you're really taking the limit. 70 00:04:02,770 --> 00:04:06,440 But this is going to be the log of 1 minus the log of 0. 71 00:04:06,440 --> 00:04:07,830 Really the log of 0 from above. 72 00:04:07,830 --> 00:04:10,970 There is no such thing as the log of 0 from below. 73 00:04:10,970 --> 00:04:12,820 And this is minus infinity. 74 00:04:12,820 --> 00:04:19,020 So it's 0 minus minus infinity, which is plus infinity. 75 00:04:19,020 --> 00:04:20,213 And so this one diverges. 76 00:04:29,710 --> 00:04:32,750 All right so what's the general-- 77 00:04:32,750 --> 00:04:39,070 So more or less in general, let's just, for powers anyway, 78 00:04:39,070 --> 00:04:45,220 if you work out this thing for dx / x^p from 0 to 1. 79 00:04:45,220 --> 00:04:51,820 What you're going to find is that it's 1/(1-p) when p is 80 00:04:51,820 --> 00:04:53,020 less than 1. 81 00:04:53,020 --> 00:05:00,240 And it diverges for p >= 1. 82 00:05:02,750 --> 00:05:07,880 Now that's the final result. If you carry out this integration 83 00:05:07,880 --> 00:05:10,920 it's not difficult. 84 00:05:10,920 --> 00:05:14,850 All right so now I just want to try 85 00:05:14,850 --> 00:05:17,270 to help you to remember this. 86 00:05:17,270 --> 00:05:20,500 And to think about how you should think about it. 87 00:05:20,500 --> 00:05:23,890 So I'm going to say it in a few more ways. 88 00:05:23,890 --> 00:05:26,850 All right just repeat what I've said already 89 00:05:26,850 --> 00:05:32,020 but try to get it to percolate and absorb itself. 90 00:05:32,020 --> 00:05:34,270 And in order to do that I have to make 91 00:05:34,270 --> 00:05:37,750 the contrast between the kind of improper integral 92 00:05:37,750 --> 00:05:39,410 that I was dealing with before. 93 00:05:39,410 --> 00:05:43,720 Which was not as x goes to 0 here but as x goes to infinity, 94 00:05:43,720 --> 00:05:45,920 the other side. 95 00:05:45,920 --> 00:05:47,023 Let's make this contrast. 96 00:05:52,470 --> 00:05:55,610 First of all, if I look at the angle 97 00:05:55,610 --> 00:05:57,610 that we have been paying attention to right now. 98 00:05:57,610 --> 00:06:00,580 We've just considered things like this. 99 00:06:00,580 --> 00:06:02,520 1 over x to the 1/2. 100 00:06:02,520 --> 00:06:06,600 Which is a lot smaller than 1/x. 101 00:06:06,600 --> 00:06:10,740 Which is a lot smaller than say 1/x^2. 102 00:06:10,740 --> 00:06:12,050 Which would be another example. 103 00:06:12,050 --> 00:06:14,560 This is as x goes to 0. 104 00:06:19,990 --> 00:06:21,935 So this one's the smallest one. 105 00:06:21,935 --> 00:06:23,310 This one's the next smallest one. 106 00:06:23,310 --> 00:06:26,820 And this one is very large. 107 00:06:26,820 --> 00:06:30,985 On the other hand it goes the other way at infinity. 108 00:06:36,670 --> 00:06:39,936 As x tends to infinity. 109 00:06:39,936 --> 00:06:43,140 All right so try to keep that in mind. 110 00:06:43,140 --> 00:06:48,760 And now I'm going to put a little box around the bad guys 111 00:06:48,760 --> 00:06:50,100 here. 112 00:06:50,100 --> 00:06:54,710 This one is divergent. 113 00:06:54,710 --> 00:06:57,870 And this one is divergent. 114 00:06:57,870 --> 00:06:59,950 And this one is divergent. 115 00:06:59,950 --> 00:07:01,610 And this one is divergent. 116 00:07:01,610 --> 00:07:03,510 The crossover point is 1/x. 117 00:07:03,510 --> 00:07:05,680 When we get smaller than that, we 118 00:07:05,680 --> 00:07:07,390 get to things which are convergent. 119 00:07:07,390 --> 00:07:10,590 When we get smaller than it on this other scale, 120 00:07:10,590 --> 00:07:12,412 it's convergent. 121 00:07:12,412 --> 00:07:13,995 All right so these guys are divergent. 122 00:07:20,300 --> 00:07:23,430 So they're associated with divergent integrals. 123 00:07:23,430 --> 00:07:25,390 The functions themselves are just 124 00:07:25,390 --> 00:07:27,585 tending towards-- well these tend to infinity, 125 00:07:27,585 --> 00:07:29,310 and these tend toward 0. 126 00:07:29,310 --> 00:07:33,830 So I'm not talking about the functions themselves 127 00:07:33,830 --> 00:07:35,190 but the integrals. 128 00:07:35,190 --> 00:07:40,040 Now I want to draw this again here, not small enough. 129 00:07:43,757 --> 00:07:44,840 I want to draw this again. 130 00:07:48,484 --> 00:07:50,150 And, so I'm just going to draw a picture 131 00:07:50,150 --> 00:07:51,760 of what it is that I have here. 132 00:07:51,760 --> 00:07:54,850 But I'm going to combine these two pictures. 133 00:07:54,850 --> 00:08:01,330 So here's the picture for example of y = 1/x. 134 00:08:04,714 --> 00:08:06,700 All right. 135 00:08:06,700 --> 00:08:08,770 That's y y = 1/x. 136 00:08:08,770 --> 00:08:10,470 And that picture is very balanced. 137 00:08:10,470 --> 00:08:12,600 It's symmetric on the two ends. 138 00:08:12,600 --> 00:08:17,660 If I cut it in half then what I get here is two halves. 139 00:08:17,660 --> 00:08:23,700 And this one has infinite area. 140 00:08:23,700 --> 00:08:27,730 That corresponds to the integral from 1 to infinity, 141 00:08:27,730 --> 00:08:30,530 dx / x being infinite. 142 00:08:30,530 --> 00:08:34,810 And the other piece, which -- this one we calculated last 143 00:08:34,810 --> 00:08:37,230 time, this is the one that we just calculated over here 144 00:08:37,230 --> 00:08:42,719 at Example 2 -- has the same property. 145 00:08:42,719 --> 00:08:45,720 It's infinite. 146 00:08:45,720 --> 00:08:48,102 And that's the fact that the integral from 0 to 1 of dx 147 00:08:48,102 --> 00:08:52,100 / x is infinite. 148 00:08:52,100 --> 00:08:56,570 Right, so both, we lose on both ends. 149 00:08:56,570 --> 00:09:02,390 On the other hand if I take something like -- 150 00:09:02,390 --> 00:09:05,480 I'm drawing it the same way but it's really not the same -- 151 00:09:05,480 --> 00:09:08,490 y = 1 over the square root of x. 152 00:09:08,490 --> 00:09:11,010 y = 1 / x^(1/2). 153 00:09:11,010 --> 00:09:17,980 And if I cut that in half here then the x^(1/2) is actually 154 00:09:17,980 --> 00:09:19,890 bigger than this guy. 155 00:09:19,890 --> 00:09:21,740 So this piece is infinite. 156 00:09:26,830 --> 00:09:29,555 And this part over here actually is going 157 00:09:29,555 --> 00:09:31,310 to give us an honest number. 158 00:09:31,310 --> 00:09:34,810 In fact this one is finite. 159 00:09:34,810 --> 00:09:36,550 And we just checked what the number is. 160 00:09:36,550 --> 00:09:38,340 It actually happens to have area 2. 161 00:09:46,250 --> 00:09:49,210 And what's happening here is if you would superimpose 162 00:09:49,210 --> 00:09:51,710 this graph on the other graph what you 163 00:09:51,710 --> 00:09:54,580 would see is that they cross. 164 00:09:54,580 --> 00:09:58,970 And this one sits on top. 165 00:09:58,970 --> 00:10:04,660 So if I drew this one in let's have another color here, 166 00:10:04,660 --> 00:10:05,840 orange let's say. 167 00:10:05,840 --> 00:10:08,630 If this were orange if I set it on top here 168 00:10:08,630 --> 00:10:11,130 it would go this way. 169 00:10:11,130 --> 00:10:14,790 OK and underneath the orange is still infinite. 170 00:10:14,790 --> 00:10:16,070 So both of these are infinite. 171 00:10:16,070 --> 00:10:18,028 On here on the other hand underneath the orange 172 00:10:18,028 --> 00:10:21,700 is infinite but underneath where the green is is finite. 173 00:10:21,700 --> 00:10:23,770 That's a smaller quantity. 174 00:10:23,770 --> 00:10:25,290 Infinity is a lot bigger than 2. 175 00:10:25,290 --> 00:10:27,826 2 is a lot less than infinity. 176 00:10:27,826 --> 00:10:30,835 All right so that's reflected in these comparisons here. 177 00:10:30,835 --> 00:10:33,610 Now if you like if I want to do these in green. 178 00:10:33,610 --> 00:10:39,900 This guy is good and this guy is good. 179 00:10:39,900 --> 00:10:42,960 Well let me just repeat that idea over here in this sort 180 00:10:42,960 --> 00:10:47,930 of reversed picture with y = 1/x^2. 181 00:10:47,930 --> 00:10:53,640 If I chop that in half then the good part is this end here. 182 00:10:53,640 --> 00:10:54,440 This is finite. 183 00:10:56,990 --> 00:10:59,570 And the bad part is this part of here 184 00:10:59,570 --> 00:11:01,310 which is way more singular. 185 00:11:01,310 --> 00:11:02,060 And it's infinite. 186 00:11:07,070 --> 00:11:10,080 All right so again what I've just tried to do 187 00:11:10,080 --> 00:11:16,710 is to give you some geometric sense and also 188 00:11:16,710 --> 00:11:18,580 some visceral sense. 189 00:11:18,580 --> 00:11:23,040 This guy, its tail as it goes out to infinity is much lower. 190 00:11:23,040 --> 00:11:25,470 It's much smaller than 1/x. 191 00:11:25,470 --> 00:11:28,080 And these guys trapped an infinite amount of area. 192 00:11:28,080 --> 00:11:30,110 This one traps only a finite amount of area. 193 00:11:36,676 --> 00:11:38,860 All right so now I'm just going to give one 194 00:11:38,860 --> 00:11:43,090 last example which combines these two types of pictures. 195 00:11:43,090 --> 00:11:45,420 It's really practically the same as what 196 00:11:45,420 --> 00:11:53,320 I've said before but I-- oh have to erase this one too. 197 00:12:01,340 --> 00:12:05,770 So here's another example: if you're in-- So let's 198 00:12:05,770 --> 00:12:08,132 take the following example. 199 00:12:08,132 --> 00:12:09,840 This is somewhat related to the first one 200 00:12:09,840 --> 00:12:11,600 that I gave last time. 201 00:12:11,600 --> 00:12:16,090 If you take a function y = 1/(x-3)^2. 202 00:12:18,770 --> 00:12:21,160 And you think about its integral. 203 00:12:21,160 --> 00:12:24,750 So let's think about the integral from 0 to infinity, 204 00:12:24,750 --> 00:12:27,090 dx / (x-3)^2. 205 00:12:27,090 --> 00:12:30,420 And suppose you were faced with this integral. 206 00:12:30,420 --> 00:12:32,630 In order to understand what it's doing 207 00:12:32,630 --> 00:12:36,310 you have to pay attention to two places where it can go wrong. 208 00:12:36,310 --> 00:12:39,530 We're going to split into two pieces. 209 00:12:39,530 --> 00:12:44,349 I'm going say break it up into this one here up to 5, 210 00:12:44,349 --> 00:12:45,390 for the sake of argument. 211 00:12:48,090 --> 00:12:49,715 And say from 5 to infinity. 212 00:12:53,672 --> 00:12:54,960 All right. 213 00:12:54,960 --> 00:12:56,260 So these are the two chunks. 214 00:12:56,260 --> 00:12:58,810 Now why did I break it up into those two pieces? 215 00:12:58,810 --> 00:13:00,670 Because what's happening with this function 216 00:13:00,670 --> 00:13:05,250 is that it's going up like this at 3. 217 00:13:05,250 --> 00:13:08,210 And so if I look at the two halves here. 218 00:13:08,210 --> 00:13:09,710 I'm going to draw them again and I'm 219 00:13:09,710 --> 00:13:12,560 going to illustrate them with the colors we've chosen, 220 00:13:12,560 --> 00:13:15,350 which are I guess red and green. 221 00:13:15,350 --> 00:13:20,660 What you'll discover is that this one 222 00:13:20,660 --> 00:13:30,540 here, which corresponds to this piece here, is infinite. 223 00:13:30,540 --> 00:13:32,320 And it's infinite because there's 224 00:13:32,320 --> 00:13:34,270 a square in the denominator. 225 00:13:34,270 --> 00:13:39,190 And as x goes to 3 this is very much like if we 226 00:13:39,190 --> 00:13:40,670 shifted the 3 to 0. 227 00:13:40,670 --> 00:13:42,560 Very much like this 1/x^2 here. 228 00:13:42,560 --> 00:13:44,247 But not in this context. 229 00:13:44,247 --> 00:13:46,330 In the other context where it's going to infinity. 230 00:13:49,460 --> 00:13:52,360 This is the same as at the picture 231 00:13:52,360 --> 00:13:57,398 directly above with the infinite part in red. 232 00:13:57,398 --> 00:13:59,650 All right. 233 00:13:59,650 --> 00:14:04,915 And this part here, this part is finite. 234 00:14:04,915 --> 00:14:06,240 All right. 235 00:14:06,240 --> 00:14:08,960 So since we have an infinite part plus a finite part 236 00:14:08,960 --> 00:14:15,070 the conclusion is that this thing, well this guy converges. 237 00:14:15,070 --> 00:14:18,850 And this one diverges. 238 00:14:21,470 --> 00:14:23,930 But the total unfortunately diverges. 239 00:14:23,930 --> 00:14:25,760 Right, because it's got one infinity in it. 240 00:14:25,760 --> 00:14:28,370 So this thing diverges. 241 00:14:31,964 --> 00:14:33,380 And that's what happened last time 242 00:14:33,380 --> 00:14:34,870 when we got a crazy number. 243 00:14:34,870 --> 00:14:37,480 If you integrated this you would get some negative number. 244 00:14:37,480 --> 00:14:39,850 If you wrote down the formulas carelessly. 245 00:14:39,850 --> 00:14:42,250 And the reason is that the calculation actually 246 00:14:42,250 --> 00:14:44,710 is nonsense. 247 00:14:44,710 --> 00:14:47,890 So you've gotta be aware, if you encounter 248 00:14:47,890 --> 00:14:51,600 a singularity in the middle, not to ignore it. 249 00:14:51,600 --> 00:14:52,100 Yeah. 250 00:14:52,100 --> 00:14:52,620 Question. 251 00:14:52,620 --> 00:14:53,786 AUDIENCE: [INAUDIBLE PHRASE] 252 00:14:56,430 --> 00:14:59,710 PROFESSOR: Why do we say that the whole thing diverges? 253 00:14:59,710 --> 00:15:02,490 The reason why we say that is the area under the whole curve 254 00:15:02,490 --> 00:15:03,690 is infinite. 255 00:15:03,690 --> 00:15:06,130 It's the sum of this piece plus this piece. 256 00:15:06,130 --> 00:15:08,218 And so the total is infinite. 257 00:15:08,218 --> 00:15:09,384 AUDIENCE: [INAUDIBLE PHRASE] 258 00:15:17,032 --> 00:15:17,990 PROFESSOR: We're stuck. 259 00:15:17,990 --> 00:15:19,323 This is an ill-defined integral. 260 00:15:19,323 --> 00:15:22,092 It's one where your red flashing warning sign should be on. 261 00:15:22,092 --> 00:15:23,800 Because you're not going to get the right 262 00:15:23,800 --> 00:15:24,758 answer by computing it. 263 00:15:24,758 --> 00:15:26,860 You'll never get an answer. 264 00:15:26,860 --> 00:15:29,230 Similarly you'll never get an answer with this. 265 00:15:29,230 --> 00:15:32,700 And you will get an answer with that. 266 00:15:32,700 --> 00:15:33,200 OK? 267 00:15:37,220 --> 00:15:38,944 Yeah another question. 268 00:15:38,944 --> 00:15:40,110 AUDIENCE: [INAUDIBLE PHRASE] 269 00:15:45,760 --> 00:15:47,690 PROFESSOR: So the question is, if you 270 00:15:47,690 --> 00:15:50,010 have a little glance at an integral, 271 00:15:50,010 --> 00:15:54,750 how are you going to decide where you should be heading? 272 00:15:54,750 --> 00:15:58,280 So I'm going to answer that orally. 273 00:15:58,280 --> 00:16:04,010 Although you know, but I'll say one little hint here. 274 00:16:04,010 --> 00:16:08,070 So you always have to check x going to infinity and x 275 00:16:08,070 --> 00:16:10,740 going to minus infinity, if they're in there. 276 00:16:10,740 --> 00:16:15,460 And you also have to check any singularity, like x going to 3 277 00:16:15,460 --> 00:16:16,419 for sure in this case. 278 00:16:16,419 --> 00:16:18,210 You have to pay attention to all the places 279 00:16:18,210 --> 00:16:19,376 where the thing is infinite. 280 00:16:19,376 --> 00:16:22,330 And then you want to focus in on each one separately. 281 00:16:22,330 --> 00:16:26,920 And decide what's going on it at that particular place. 282 00:16:26,920 --> 00:16:34,530 When it's a negative power-- So remember dx / x as x goes to 0 283 00:16:34,530 --> 00:16:36,980 is bad. 284 00:16:36,980 --> 00:16:39,510 And dx / x^2 is bad. 285 00:16:39,510 --> 00:16:40,490 dx / x^3 is bad. 286 00:16:40,490 --> 00:16:42,600 All of them are even worse. 287 00:16:42,600 --> 00:16:49,650 So anything of this form is bad: n = 1, 2, 3. 288 00:16:49,650 --> 00:16:51,980 These are the red box kinds. 289 00:16:51,980 --> 00:16:55,030 All right. 290 00:16:55,030 --> 00:16:56,940 That means that any of the integrals 291 00:16:56,940 --> 00:16:59,990 that we did in partial fractions which 292 00:16:59,990 --> 00:17:02,080 had a root, which had a factor of something 293 00:17:02,080 --> 00:17:03,160 in the denominator. 294 00:17:03,160 --> 00:17:04,790 Those are all divergent integrals 295 00:17:04,790 --> 00:17:06,670 if you cross the singularly. 296 00:17:06,670 --> 00:17:09,357 Not a single one of them makes sense across the singularity. 297 00:17:09,357 --> 00:17:09,856 Right? 298 00:17:12,692 --> 00:17:14,150 If you have square roots and things 299 00:17:14,150 --> 00:17:16,108 like that then you can repair things like that. 300 00:17:16,108 --> 00:17:18,030 And there's some interesting examples of that. 301 00:17:18,030 --> 00:17:21,080 Such as with the arcsine function and so forth. 302 00:17:21,080 --> 00:17:25,940 Where you have an improper integral which is really OK. 303 00:17:25,940 --> 00:17:26,730 All right. 304 00:17:26,730 --> 00:17:29,880 So that's the best I can do. 305 00:17:29,880 --> 00:17:32,260 It's obviously something you get experience with. 306 00:17:32,260 --> 00:17:34,110 All right. 307 00:17:34,110 --> 00:17:39,350 Now I'm going to move on and this is more or less 308 00:17:39,350 --> 00:17:42,150 our last topic. 309 00:17:42,150 --> 00:17:43,900 Yay, but not quite. 310 00:17:43,900 --> 00:17:46,730 Well, so I should say it's our penultimate topic. 311 00:17:46,730 --> 00:17:49,430 Right because we have one more lecture. 312 00:17:49,430 --> 00:17:52,010 All right. 313 00:17:52,010 --> 00:17:54,750 So that our next topic is series. 314 00:17:54,750 --> 00:17:58,810 Now we'll do it in a sort of a concrete way today. 315 00:17:58,810 --> 00:18:02,775 And then we'll do what are known as power series tomorrow. 316 00:18:05,400 --> 00:18:06,760 So let me tell you about series. 317 00:18:20,490 --> 00:18:22,604 Remember we're talking about infinity 318 00:18:22,604 --> 00:18:23,687 and dealing with infinity. 319 00:18:26,772 --> 00:18:28,730 So we're not just talking about any old series. 320 00:18:28,730 --> 00:18:30,230 We're talking about infinite series. 321 00:18:32,650 --> 00:18:37,400 There is one infinite series which is probably, 322 00:18:37,400 --> 00:18:39,000 which is without question the most 323 00:18:39,000 --> 00:18:41,980 important and useful series. 324 00:18:41,980 --> 00:18:44,690 And that's the geometric series but I'm 325 00:18:44,690 --> 00:18:48,405 going to introduce it concretely first in a particular case. 326 00:18:51,990 --> 00:18:54,260 If I draw a picture of this sum. 327 00:18:54,260 --> 00:18:56,480 Which in principle goes on forever. 328 00:18:56,480 --> 00:18:59,670 You can see that it goes someplace fairly easily 329 00:18:59,670 --> 00:19:02,730 by marking out what's happening on the number line. 330 00:19:02,730 --> 00:19:06,630 The first step takes us to 1 from 0. 331 00:19:06,630 --> 00:19:11,790 And then if I add this half, I get to 3/2. 332 00:19:11,790 --> 00:19:16,240 Right, so the first step was 1 and the second step was 1/2. 333 00:19:16,240 --> 00:19:20,940 Now if I add this quarter in, which is the next piece then 334 00:19:20,940 --> 00:19:22,400 I get some place here. 335 00:19:22,400 --> 00:19:28,440 But what I want to observe is that I got, 336 00:19:28,440 --> 00:19:31,130 I can look at it from the other point of view. 337 00:19:31,130 --> 00:19:36,240 I got, when I move this quarter I got half way to 2 here. 338 00:19:36,240 --> 00:19:39,420 I'm putting 2 in green because I want you to think 339 00:19:39,420 --> 00:19:42,950 of it as being the good kind. 340 00:19:42,950 --> 00:19:43,970 Right. 341 00:19:43,970 --> 00:19:45,420 The kind that has a number. 342 00:19:45,420 --> 00:19:47,280 And not one of the red kinds. 343 00:19:47,280 --> 00:19:49,920 We're getting there and we're almost there. 344 00:19:49,920 --> 00:19:52,620 So the next stage we get half way again. 345 00:19:52,620 --> 00:19:54,605 That's the eighth and so forth. 346 00:19:54,605 --> 00:19:56,490 And eventually we get to 2. 347 00:19:56,490 --> 00:20:00,476 So this sum we write equals two. 348 00:20:00,476 --> 00:20:02,590 All right that's kind of a paradox 349 00:20:02,590 --> 00:20:04,130 because we never get to 2. 350 00:20:04,130 --> 00:20:08,080 This is the paradox that Zeno fussed with. 351 00:20:08,080 --> 00:20:12,807 And his conclusion, you know, with the rabbit and the hare. 352 00:20:12,807 --> 00:20:14,140 No, the rabbit and the tortoise. 353 00:20:14,140 --> 00:20:18,890 Sorry hare chasing-- anyway, the rabbit chasing the tortoise. 354 00:20:18,890 --> 00:20:21,050 His conclusion-- you know, I don't 355 00:20:21,050 --> 00:20:23,770 know if you're aware of this, but he understood this paradox. 356 00:20:23,770 --> 00:20:25,811 And he said you know it doesn't look like it ever 357 00:20:25,811 --> 00:20:29,180 gets there because they're infinitely many times 358 00:20:29,180 --> 00:20:32,330 between the time-- you know that the tortoise is always behind, 359 00:20:32,330 --> 00:20:34,580 always behind, always behind, always behind. 360 00:20:34,580 --> 00:20:36,970 So therefore it's impossible that the tortoise 361 00:20:36,970 --> 00:20:38,710 catches up right. 362 00:20:38,710 --> 00:20:41,600 So do you know what his conclusion was? 363 00:20:41,600 --> 00:20:45,280 Time does not exist. 364 00:20:45,280 --> 00:20:48,007 That was actually literally his conclusion. 365 00:20:48,007 --> 00:20:49,840 Because he didn't understand the possibility 366 00:20:49,840 --> 00:20:50,799 of a continuum of time. 367 00:20:50,799 --> 00:20:53,090 Because there were infinitely many things that happened 368 00:20:53,090 --> 00:20:56,610 before the tortoise caught up. 369 00:20:56,610 --> 00:20:57,790 So that was the reasoning. 370 00:20:57,790 --> 00:21:00,430 I mean it's a long time ago but you know people didn't-- he 371 00:21:00,430 --> 00:21:02,430 didn't believe in continuum. 372 00:21:02,430 --> 00:21:03,480 All right. 373 00:21:03,480 --> 00:21:06,940 So anyway that's a small point. 374 00:21:06,940 --> 00:21:19,410 Now the general case here of a geometric series 375 00:21:19,410 --> 00:21:22,820 is where I put in a number a instead of 1/2 here. 376 00:21:22,820 --> 00:21:23,870 So what we had before. 377 00:21:23,870 --> 00:21:26,890 So that's 1 + a + a^2... 378 00:21:26,890 --> 00:21:31,900 Isn't quite the most general but anyway I'll write this down. 379 00:21:31,900 --> 00:21:34,630 And you're certainly going to want to remember that 380 00:21:34,630 --> 00:21:39,510 the formula for this in the limit is 1/(1-a). 381 00:21:39,510 --> 00:21:44,000 And I remind you that this only works when the absolute value 382 00:21:44,000 --> 00:21:45,810 is strictly less than 1. 383 00:21:45,810 --> 00:21:47,970 In other words when -1 is strictly less than a 384 00:21:47,970 --> 00:21:51,130 is less than 1. 385 00:21:51,130 --> 00:21:53,090 And that's really the issue that we're going 386 00:21:53,090 --> 00:21:54,310 to want to worry about now. 387 00:21:54,310 --> 00:21:58,270 What we're worrying about is this notion of convergence. 388 00:21:58,270 --> 00:22:05,300 And what goes wrong when there isn't convergence, 389 00:22:05,300 --> 00:22:07,360 when there's a divergence. 390 00:22:07,360 --> 00:22:13,220 So let me illustrate the divergences before going on. 391 00:22:13,220 --> 00:22:15,900 And this is what we have to avoid if we're 392 00:22:15,900 --> 00:22:18,990 going to understand series. 393 00:22:18,990 --> 00:22:21,890 So here's an example when a = 1. 394 00:22:21,890 --> 00:22:26,620 You get 1 + 1 + 1 plus et cetera. 395 00:22:26,620 --> 00:22:29,990 And that's equal to 1/(1-1). 396 00:22:29,990 --> 00:22:32,370 Which is 1 over 0. 397 00:22:32,370 --> 00:22:33,760 So this is not bad. 398 00:22:33,760 --> 00:22:34,861 It's almost right. 399 00:22:34,861 --> 00:22:35,360 Right? 400 00:22:35,360 --> 00:22:37,750 It's sort of infinity equals infinity. 401 00:22:37,750 --> 00:22:39,850 At the edge here we managed to get something 402 00:22:39,850 --> 00:22:42,340 which is sort of almost right. 403 00:22:42,340 --> 00:22:46,100 But you know, it's, we don't consider this to be logically 404 00:22:46,100 --> 00:22:47,540 to make complete sense. 405 00:22:47,540 --> 00:22:51,200 So it's a little dangerous. 406 00:22:51,200 --> 00:22:52,960 And so we just say that it diverges. 407 00:22:52,960 --> 00:22:54,100 And we get rid of this. 408 00:22:54,100 --> 00:22:55,990 So we're still putting it in red. 409 00:22:55,990 --> 00:22:58,540 All right. 410 00:22:58,540 --> 00:22:59,975 The bad guy here. 411 00:22:59,975 --> 00:23:00,850 So this one diverges. 412 00:23:04,670 --> 00:23:12,650 Similarly if I take a equals -1, I get 1 - 1 + 1 - 1 + 1... 413 00:23:12,650 --> 00:23:15,450 Because the odd and the even powers in that formula 414 00:23:15,450 --> 00:23:17,270 alternate sign. 415 00:23:17,270 --> 00:23:19,750 And this bounces back and forth. 416 00:23:19,750 --> 00:23:21,640 It never settles down. 417 00:23:21,640 --> 00:23:23,346 It starts at 1. 418 00:23:23,346 --> 00:23:25,720 And then it gets down to 0 and then it goes back up to 1, 419 00:23:25,720 --> 00:23:28,400 down to 0, back up to 1. 420 00:23:28,400 --> 00:23:29,517 It doesn't settle down. 421 00:23:29,517 --> 00:23:30,600 It bounces back and forth. 422 00:23:30,600 --> 00:23:31,580 It oscillates. 423 00:23:31,580 --> 00:23:34,740 On the other hand if you compare the right hand side. 424 00:23:34,740 --> 00:23:35,980 What's the right hand side? 425 00:23:35,980 --> 00:23:36,970 It's 1 / (1-(-1)). 426 00:23:39,730 --> 00:23:41,515 Which is 1/2. 427 00:23:41,515 --> 00:23:42,460 All right. 428 00:23:42,460 --> 00:23:45,169 So if you just paid attention to the formula, which 429 00:23:45,169 --> 00:23:47,710 is what we were doing when we integrated without thinking too 430 00:23:47,710 --> 00:23:50,070 hard about this, you get a number 431 00:23:50,070 --> 00:23:51,320 here but in fact that's wrong. 432 00:23:51,320 --> 00:23:53,153 Actually it's kind of an interesting number. 433 00:23:53,153 --> 00:23:56,380 It's halfway between the two, between 0 and 1. 434 00:23:56,380 --> 00:23:58,505 So again there's some sort of vague sense 435 00:23:58,505 --> 00:24:01,966 in which this is trying to be this answer. 436 00:24:01,966 --> 00:24:04,580 All right. 437 00:24:04,580 --> 00:24:08,744 It's not so bad but we're still going to put this in a red box. 438 00:24:08,744 --> 00:24:10,030 All right. 439 00:24:10,030 --> 00:24:12,710 because this is what we called divergence. 440 00:24:12,710 --> 00:24:16,130 So both of these cases are divergent. 441 00:24:16,130 --> 00:24:20,490 It only really works when alpha-- when a is less than 1. 442 00:24:20,490 --> 00:24:23,480 I'm going to add one more case just 443 00:24:23,480 --> 00:24:30,040 to see that mathematicians are slightly curious about what 444 00:24:30,040 --> 00:24:32,020 goes on in other cases. 445 00:24:32,020 --> 00:24:37,570 So this is 1 + 2 + 2^2 + 2^3 plus etc.. 446 00:24:37,570 --> 00:24:41,980 And that should be equal to -- according to this formula -- 447 00:24:41,980 --> 00:24:44,750 1/(1-2). 448 00:24:44,750 --> 00:24:48,242 Which is -1. 449 00:24:48,242 --> 00:24:49,860 All right. 450 00:24:49,860 --> 00:24:53,460 Now this one is clearly wrong, right? 451 00:24:53,460 --> 00:24:55,530 This one is totally wrong. 452 00:24:58,170 --> 00:24:59,465 It certainly diverges. 453 00:24:59,465 --> 00:25:02,370 The left hand side is obviously infinite. 454 00:25:02,370 --> 00:25:04,070 The right hand side is way off. 455 00:25:04,070 --> 00:25:05,960 It's -1. 456 00:25:05,960 --> 00:25:10,350 On the other hand it turns out actually 457 00:25:10,350 --> 00:25:13,360 that mathematicians have ways of making sense out of these. 458 00:25:13,360 --> 00:25:15,590 In number theory there's a strange system 459 00:25:15,590 --> 00:25:18,160 where this is actually true. 460 00:25:18,160 --> 00:25:21,530 And what happens in that system is 461 00:25:21,530 --> 00:25:24,030 that what you have to throw out is the idea 462 00:25:24,030 --> 00:25:27,050 that 0 is less than 1. 463 00:25:27,050 --> 00:25:29,980 There is no such thing as negative numbers. 464 00:25:29,980 --> 00:25:32,090 So this number exists. 465 00:25:32,090 --> 00:25:35,700 And it's the additive inverse of 1. 466 00:25:35,700 --> 00:25:39,780 It has this arithmetic property but the statement 467 00:25:39,780 --> 00:25:43,330 that this is, that 1 is bigger than 0 does not make sense. 468 00:25:43,330 --> 00:25:45,440 So you have your choice, either this diverges 469 00:25:45,440 --> 00:25:48,630 or you have to throw out something like this. 470 00:25:48,630 --> 00:25:51,510 So that's a very curious thing in higher mathematics. 471 00:25:51,510 --> 00:25:56,410 Which if you get to number theory there's fun stuff there. 472 00:25:56,410 --> 00:25:58,920 All right. 473 00:25:58,920 --> 00:26:02,740 OK but for our purposes these things are all out. 474 00:26:02,740 --> 00:26:03,300 All right. 475 00:26:03,300 --> 00:26:04,010 They're gone. 476 00:26:04,010 --> 00:26:05,160 We're not considering them. 477 00:26:05,160 --> 00:26:09,550 Only a between -1 and 1. 478 00:26:09,550 --> 00:26:10,070 All right. 479 00:26:13,910 --> 00:26:18,190 Now I want to do something systematic. 480 00:26:18,190 --> 00:26:21,530 And it's more or less on the lines of the powers 481 00:26:21,530 --> 00:26:23,090 that I'm erasing right now. 482 00:26:26,630 --> 00:26:28,590 I want to tell you about series which are 483 00:26:28,590 --> 00:26:30,420 kind of borderline convergent. 484 00:26:30,420 --> 00:26:33,720 And then next time when we talk about powers series we'll 485 00:26:33,720 --> 00:26:35,810 come back to this very important series which 486 00:26:35,810 --> 00:26:37,080 is the most important one. 487 00:26:40,680 --> 00:26:47,400 So now let's talk about some series-- er, general notations. 488 00:26:47,400 --> 00:26:49,970 And this will help you with the last bit. 489 00:26:53,810 --> 00:26:56,760 This is going to be pretty much the same as what 490 00:26:56,760 --> 00:27:00,890 we did for improper integrals. 491 00:27:00,890 --> 00:27:04,320 Namely, first of all I'm going to have S_N which 492 00:27:04,320 --> 00:27:09,760 is the sum of a_n, n equals 0 to capital N. 493 00:27:09,760 --> 00:27:12,220 And this is what we're calling a partial sum. 494 00:27:18,050 --> 00:27:23,980 And then the full limit, which is capital S, if you like. 495 00:27:23,980 --> 00:27:29,980 a_n, n equals 0 to infinity, is just the limit as N goes 496 00:27:29,980 --> 00:27:32,010 to infinity of the S_N's. 497 00:27:36,240 --> 00:27:39,770 And then we have the same kind of notation that we had before. 498 00:27:39,770 --> 00:27:42,860 Which is there are these two choices which 499 00:27:42,860 --> 00:27:45,890 is that if the limit exists. 500 00:27:50,450 --> 00:27:51,830 That's the green choice. 501 00:27:51,830 --> 00:27:54,460 And we say it converges. 502 00:27:54,460 --> 00:28:00,830 So we say the series converges. 503 00:28:00,830 --> 00:28:06,616 And then the other case which is the limit does not exist. 504 00:28:10,782 --> 00:28:12,240 And we can say the series diverges. 505 00:28:20,560 --> 00:28:21,343 Question. 506 00:28:21,343 --> 00:28:22,509 AUDIENCE: [INAUDIBLE PHRASE] 507 00:28:26,480 --> 00:28:29,290 PROFESSOR: The question was how did I get to this? 508 00:28:29,290 --> 00:28:31,840 And I will do that next time but in fact of course 509 00:28:31,840 --> 00:28:33,240 you've seen it in high school. 510 00:28:33,240 --> 00:28:35,930 Right this is-- Yeah. 511 00:28:35,930 --> 00:28:36,860 Yeah. 512 00:28:36,860 --> 00:28:40,000 We'll do that next time. 513 00:28:40,000 --> 00:28:42,222 The question was how did we arrive-- sorry I 514 00:28:42,222 --> 00:28:43,430 didn't tell you the question. 515 00:28:43,430 --> 00:28:44,360 The question was how do we arrive 516 00:28:44,360 --> 00:28:46,590 at this formula on the right hand side here. 517 00:28:46,590 --> 00:28:48,240 But we'll talk about that next time. 518 00:28:53,060 --> 00:28:54,020 All right. 519 00:28:54,020 --> 00:28:59,930 So here's the basic definition and what 520 00:28:59,930 --> 00:29:02,430 we're going to recognize about series. 521 00:29:02,430 --> 00:29:06,835 And I'm going to give you a few examples 522 00:29:06,835 --> 00:29:08,460 and then we'll do something systematic. 523 00:29:12,070 --> 00:29:14,495 So the first example-- well the first example 524 00:29:14,495 --> 00:29:16,290 is the geometric series. 525 00:29:16,290 --> 00:29:19,430 But the first example that I'm going to discuss now 526 00:29:19,430 --> 00:29:23,352 and in a little bit of detail is this sum 1/n^2, 527 00:29:23,352 --> 00:29:24,310 n equals 1 to infinity. 528 00:29:28,580 --> 00:29:34,050 It turns out that this series is very analogous -- 529 00:29:34,050 --> 00:29:36,690 and we'll develop this analogy carefully -- 530 00:29:36,690 --> 00:29:41,065 the integral from 1 to x, dx / x^2. 531 00:29:41,065 --> 00:29:46,220 And we're going to develop this analogy in detail later 532 00:29:46,220 --> 00:29:47,920 in this lecture. 533 00:29:47,920 --> 00:29:49,770 And this one is one of the ones-- 534 00:29:49,770 --> 00:29:51,890 so now you have to go back and actually remember, 535 00:29:51,890 --> 00:29:54,335 this is one of the ones you really want to memorize. 536 00:29:54,335 --> 00:29:56,460 And you should especially pay attention to the ones 537 00:29:56,460 --> 00:29:58,507 with an infinity in them. 538 00:29:58,507 --> 00:29:59,465 This one is convergent. 539 00:30:03,270 --> 00:30:04,520 And this series is convergent. 540 00:30:04,520 --> 00:30:11,070 Now it turns out that evaluating this is very easy. 541 00:30:11,070 --> 00:30:12,820 This is 1. 542 00:30:12,820 --> 00:30:15,310 It's easy to calculate. 543 00:30:15,310 --> 00:30:19,500 Evaluating this is very tricky. 544 00:30:19,500 --> 00:30:21,610 And Euler did it. 545 00:30:21,610 --> 00:30:26,440 And the answer is pi^2 / 6. 546 00:30:26,440 --> 00:30:29,050 That's an amazing calculation. 547 00:30:29,050 --> 00:30:33,570 And it was done very early in the history of mathematics. 548 00:30:33,570 --> 00:30:38,110 If you look at another example-- so maybe example two here, 549 00:30:38,110 --> 00:30:45,790 if you look at 1/n^3, n equals-- well you can't start here at 0 550 00:30:45,790 --> 00:30:46,580 by the way. 551 00:30:46,580 --> 00:30:48,930 I get to start wherever I want in these series. 552 00:30:48,930 --> 00:30:49,920 Here I start with 0. 553 00:30:49,920 --> 00:30:51,220 Here I started with 1. 554 00:30:51,220 --> 00:30:52,880 And notice the reason why I started-- 555 00:30:52,880 --> 00:30:56,650 it was a bad idea to start with 0 was that 1 over 0 556 00:30:56,650 --> 00:30:57,860 is undefined. 557 00:30:57,860 --> 00:30:58,360 Right? 558 00:30:58,360 --> 00:31:00,443 So I'm just starting where it's convenient for me. 559 00:31:00,443 --> 00:31:03,560 And since I'm interested mostly in the tail behavior 560 00:31:03,560 --> 00:31:06,150 it doesn't matter to me so much where I start. 561 00:31:06,150 --> 00:31:07,800 Although if I want an exact answer 562 00:31:07,800 --> 00:31:10,165 I need to start exactly at n = 1. 563 00:31:10,165 --> 00:31:10,890 All right. 564 00:31:10,890 --> 00:31:17,932 This one is similar to this integral here. 565 00:31:17,932 --> 00:31:18,700 All right. 566 00:31:18,700 --> 00:31:20,440 Which is convergent again. 567 00:31:20,440 --> 00:31:22,390 So there's a number that you get. 568 00:31:22,390 --> 00:31:27,180 And let's see what is it something like 2/3 or something 569 00:31:27,180 --> 00:31:30,190 like that, all right, for this for this number. 570 00:31:30,190 --> 00:31:32,440 Or 1/3. 571 00:31:32,440 --> 00:31:33,110 What is it? 572 00:31:33,110 --> 00:31:33,690 No 1/2. 573 00:31:33,690 --> 00:31:35,000 I guess it's 1/2. 574 00:31:35,000 --> 00:31:37,110 This one is 1/2. 575 00:31:37,110 --> 00:31:38,780 You check that, I'm not positive, 576 00:31:38,780 --> 00:31:40,600 but anyway just doing it in my head quickly 577 00:31:40,600 --> 00:31:42,150 it seems to be 1/2. 578 00:31:42,150 --> 00:31:44,100 Anyway it's an easy number to calculate. 579 00:31:44,100 --> 00:31:49,710 This one over here stumped mathematicians basically 580 00:31:49,710 --> 00:31:51,870 for all time. 581 00:31:51,870 --> 00:31:56,250 It doesn't have any kind of elementary form like this. 582 00:31:56,250 --> 00:31:59,842 And it was only very recently proved to be rational. 583 00:31:59,842 --> 00:32:01,550 People couldn't even couldn't even decide 584 00:32:01,550 --> 00:32:05,260 whether this was a rational number or not. 585 00:32:05,260 --> 00:32:07,830 But anyway that's been resolved; it is an irrational number 586 00:32:07,830 --> 00:32:09,540 which is what people suspected. 587 00:32:09,540 --> 00:32:10,650 Yeah question. 588 00:32:10,650 --> 00:32:11,816 AUDIENCE: [INAUDIBLE PHRASE] 589 00:32:14,360 --> 00:32:16,220 PROFESSOR: Yeah sorry. 590 00:32:16,220 --> 00:32:16,720 OK. 591 00:32:19,820 --> 00:32:25,070 I violated a rule of mathematics-- 592 00:32:25,070 --> 00:32:26,720 you said why is this similar? 593 00:32:26,720 --> 00:32:29,140 I thought that similar was something else. 594 00:32:29,140 --> 00:32:30,380 And you're absolutely right. 595 00:32:30,380 --> 00:32:33,520 And I violated a rule of mathematics. 596 00:32:33,520 --> 00:32:37,825 Which is that I used this symbol for two different things. 597 00:32:41,196 --> 00:32:42,820 I should have written this symbol here. 598 00:32:42,820 --> 00:32:43,530 All right. 599 00:32:43,530 --> 00:32:45,260 I'll create a new symbol here. 600 00:32:45,260 --> 00:32:48,920 The question of whether this converges or this converges. 601 00:32:48,920 --> 00:32:51,920 These are the the same type of question. 602 00:32:51,920 --> 00:32:54,090 And we'll see why they're the same question it 603 00:32:54,090 --> 00:32:55,000 in a few minutes. 604 00:32:55,000 --> 00:32:58,190 But in fact the wiggle I used, "similar", 605 00:32:58,190 --> 00:33:02,360 I used for the connection between functions. 606 00:33:02,360 --> 00:33:09,070 The things that are really similar are that 1/n resembles 607 00:33:09,070 --> 00:33:10,230 1/x^2. 608 00:33:10,230 --> 00:33:12,066 So I apologize I didn't-- 609 00:33:12,066 --> 00:33:13,232 AUDIENCE: [INAUDIBLE PHRASE] 610 00:33:15,890 --> 00:33:17,890 PROFESSOR: Oh you thought that this 611 00:33:17,890 --> 00:33:19,210 was the definition of that. 612 00:33:19,210 --> 00:33:21,460 That's actually the reason why these things correspond 613 00:33:21,460 --> 00:33:21,990 so closely. 614 00:33:21,990 --> 00:33:25,560 That is that the Riemann sum is close to this. 615 00:33:25,560 --> 00:33:27,520 But that doesn't mean they're equal. 616 00:33:27,520 --> 00:33:31,245 The Riemann sum only works when the delta x goes to 0. 617 00:33:31,245 --> 00:33:33,870 The way that we're going to get a connection between these two, 618 00:33:33,870 --> 00:33:38,710 as we will just a second, is with a Riemann sum with-- What 619 00:33:38,710 --> 00:33:47,180 we're going to use is a Riemann sum with delta x = 1. 620 00:33:47,180 --> 00:33:49,617 All right and then that will be the connection between. 621 00:33:49,617 --> 00:33:53,240 All right that's absolutely right. 622 00:33:53,240 --> 00:33:53,740 All right. 623 00:33:57,020 --> 00:34:00,280 So in order to illustrate exactly this idea 624 00:34:00,280 --> 00:34:02,110 that you've just come up with, and in fact 625 00:34:02,110 --> 00:34:04,490 that we're going to use, we'll do the same thing 626 00:34:04,490 --> 00:34:07,740 but we're going to do it on the example sum 1/n. 627 00:34:13,380 --> 00:34:20,170 So here's Example 3 and it's going to be sum 1/n, 628 00:34:20,170 --> 00:34:21,369 n equals 1 to infinity. 629 00:34:24,090 --> 00:34:27,040 And what we're now going to see is 630 00:34:27,040 --> 00:34:29,340 that it corresponds to this integral here. 631 00:34:32,620 --> 00:34:34,850 And we're going to show therefore 632 00:34:34,850 --> 00:34:37,870 that this thing diverges. 633 00:34:37,870 --> 00:34:40,300 But we're going to do this more carefully. 634 00:34:40,300 --> 00:34:42,750 We're going to do this in some detail 635 00:34:42,750 --> 00:34:45,880 so that you see what it is, that the correspondence is 636 00:34:45,880 --> 00:34:47,390 between these quantities. 637 00:34:47,390 --> 00:34:49,890 And the same sort of reasoning applies 638 00:34:49,890 --> 00:34:51,275 to these other examples. 639 00:34:55,820 --> 00:34:59,180 So here we go. 640 00:34:59,180 --> 00:35:04,290 I'm going to take the integral and draw 641 00:35:04,290 --> 00:35:07,220 the picture of the Riemann sum. 642 00:35:07,220 --> 00:35:13,840 So here's the level 1 and here's the function y = 1/x. 643 00:35:13,840 --> 00:35:15,730 And I'm going to take the Riemann sum. 644 00:35:21,980 --> 00:35:25,200 With delta x = 1. 645 00:35:25,200 --> 00:35:28,820 And that's going to be closely connected to the series 646 00:35:28,820 --> 00:35:29,570 that I have. 647 00:35:32,160 --> 00:35:35,670 But now I have to decide whether I want a lower Riemann 648 00:35:35,670 --> 00:35:37,797 sum or an upper Riemann sum. 649 00:35:37,797 --> 00:35:39,630 And actually I'm going to check both of them 650 00:35:39,630 --> 00:35:41,213 because both of them are illuminating. 651 00:35:44,620 --> 00:35:47,000 First we'll do the upper Riemann's sum. 652 00:35:47,000 --> 00:35:48,700 Now that's this staircase here. 653 00:35:51,780 --> 00:35:54,600 So we'll call this the upper Riemann's sum. 654 00:35:58,130 --> 00:35:59,710 And let's check what its levels are. 655 00:35:59,710 --> 00:36:01,560 This is not to scale. 656 00:36:01,560 --> 00:36:03,250 This level should be 1/2. 657 00:36:03,250 --> 00:36:05,245 So if this is 1 and this is 2 and that level 658 00:36:05,245 --> 00:36:10,340 was supposed to be 1/2 and this next level should be 1/3. 659 00:36:10,340 --> 00:36:12,830 That's how the Riemann sums are working out. 660 00:36:17,040 --> 00:36:21,770 And now I have the following phenomenon. 661 00:36:21,770 --> 00:36:24,640 Let's cut it off at the nth stage. 662 00:36:24,640 --> 00:36:27,095 So that means that I'm going, the integral 663 00:36:27,095 --> 00:36:30,760 is from 1 to n, dx / x. 664 00:36:30,760 --> 00:36:33,690 And the Riemann sum is something that's bigger than it. 665 00:36:33,690 --> 00:36:40,300 Because the areas are enclosing the area of the curved region. 666 00:36:40,300 --> 00:36:43,510 And that's going to be the area of the first box which 667 00:36:43,510 --> 00:36:50,330 is 1, plus the area of the second box which is 1/2, 668 00:36:50,330 --> 00:36:54,150 plus the area of the third box which is 1/3. 669 00:36:54,150 --> 00:37:00,840 All the way up the last one, but the last one starts at N-1. 670 00:37:00,840 --> 00:37:03,450 So it has 1/(N-1). 671 00:37:03,450 --> 00:37:05,780 There are not N boxes here. 672 00:37:05,780 --> 00:37:07,960 There are only N-1 boxes. 673 00:37:07,960 --> 00:37:11,830 Because the distance between 1 and N is N-1. 674 00:37:11,830 --> 00:37:13,555 Right so this is N-1 terms. 675 00:37:17,330 --> 00:37:25,140 However, if I use the notation for partial sum. 676 00:37:25,140 --> 00:37:33,367 Which is 1 + 1/2 plus all the way up to 1/(n-1) 1 + 1/n. 677 00:37:33,367 --> 00:37:35,200 In other words I go out to the Nth one which 678 00:37:35,200 --> 00:37:37,370 is what I would ordinarily do. 679 00:37:37,370 --> 00:37:42,900 Then this sum that I have here certainly is less than S_N. 680 00:37:42,900 --> 00:37:47,850 Because there's one more term there. 681 00:37:47,850 --> 00:37:50,430 And so here I have an integral which 682 00:37:50,430 --> 00:37:53,570 is underneath this sum S_N. 683 00:38:01,700 --> 00:38:19,031 Now this is going to allow us to prove conclusively 684 00:38:19,031 --> 00:38:21,156 that the-- So I'm just going to rewrite this, prove 685 00:38:21,156 --> 00:38:22,614 conclusively that the sum diverges. 686 00:38:22,614 --> 00:38:23,340 Why is that? 687 00:38:23,340 --> 00:38:26,000 Because this term here we can calculate. 688 00:38:26,000 --> 00:38:29,470 This is log x evaluated at 1 and n. 689 00:38:29,470 --> 00:38:34,850 Which is the same thing as log N minus 0. 690 00:38:34,850 --> 00:38:39,270 All right, the quantity log N - log 1 which is 0. 691 00:38:39,270 --> 00:38:46,892 And so what we have here is that log N is less than S_N. 692 00:38:46,892 --> 00:38:51,455 All right and clearly this goes to infinity right. 693 00:38:51,455 --> 00:38:57,830 As N goes to infinity this thing goes to infinity. 694 00:38:57,830 --> 00:38:58,435 So we're done. 695 00:38:58,435 --> 00:39:00,330 All right we've shown divergence. 696 00:39:08,730 --> 00:39:15,040 Now the way I'm going to use the lower Riemann's sum 697 00:39:15,040 --> 00:39:20,010 is to recognize that we've captured 698 00:39:20,010 --> 00:39:21,960 the rate appropriately. 699 00:39:21,960 --> 00:39:24,400 That is not only do I have a lower bound like this 700 00:39:24,400 --> 00:39:27,860 but I have an upper bound which is very similar. 701 00:39:27,860 --> 00:39:30,440 So if I use the upper Riemann-- oh sorry, 702 00:39:30,440 --> 00:39:39,540 lower Riemann sum again with delta x = 1. 703 00:39:43,760 --> 00:39:53,946 Then I have that the integral from 1 to n of dx 704 00:39:53,946 --> 00:39:57,970 / x is bigger than-- Well what are the terms going 705 00:39:57,970 --> 00:40:00,640 to be if fit them underneath? 706 00:40:00,640 --> 00:40:03,210 If I fit them underneath I'm missing the first term. 707 00:40:03,210 --> 00:40:05,600 That is the box is going to be half height. 708 00:40:05,600 --> 00:40:08,300 It's going to be this lower piece. 709 00:40:08,300 --> 00:40:10,480 So I'm missing this first term. 710 00:40:10,480 --> 00:40:16,765 So it'll be a 1/2 + 1/3 plus... 711 00:40:16,765 --> 00:40:18,150 All right, it will keep on going. 712 00:40:18,150 --> 00:40:22,150 But now the last one instead of being 1/(N-1), 713 00:40:22,150 --> 00:40:25,850 it's going to be 1 over N. This is again a total of the N-1 714 00:40:25,850 --> 00:40:27,050 terms. 715 00:40:27,050 --> 00:40:28,430 This is the lower Riemann sum. 716 00:40:31,190 --> 00:40:38,808 And now we can recognize that this is exactly equal to-- well 717 00:40:38,808 --> 00:40:40,558 so I'll put it over here-- this is exactly 718 00:40:40,558 --> 00:40:43,520 equal to S_N minus 1, minus the first term. 719 00:40:43,520 --> 00:40:47,110 So we missed the first term but we got all the rest of them. 720 00:40:47,110 --> 00:40:49,360 So if I put this to the other side 721 00:40:49,360 --> 00:40:53,746 remember this is log N. All right. 722 00:40:53,746 --> 00:40:55,120 If I put this to the other side I 723 00:40:55,120 --> 00:40:57,130 have the other side of this bound. 724 00:40:57,130 --> 00:41:07,010 I have that S S_N is less than, if I reverse it, log N + 1. 725 00:41:07,010 --> 00:41:09,040 And so I've trapped it on the other side. 726 00:41:09,040 --> 00:41:10,950 And here I have the lower bound. 727 00:41:10,950 --> 00:41:13,170 So I'm going to combine those together. 728 00:41:13,170 --> 00:41:18,030 So all told I have this correspondence here. 729 00:41:18,030 --> 00:41:23,150 It is the size of log N is trapped between the-- sorry, 730 00:41:23,150 --> 00:41:26,150 the size of S_N, which is relatively hard to calculate 731 00:41:26,150 --> 00:41:30,780 and understand exactly, is trapped between log N 732 00:41:30,780 --> 00:41:34,260 and log N + 1. 733 00:41:34,260 --> 00:41:35,160 Yeah question. 734 00:41:35,160 --> 00:41:36,326 AUDIENCE: [INAUDIBLE PHRASE] 735 00:41:46,039 --> 00:41:47,580 PROFESSOR: This step here is the step 736 00:41:47,580 --> 00:41:49,550 that you're concerned about. 737 00:41:49,550 --> 00:41:54,160 So this step is a geometric argument which 738 00:41:54,160 --> 00:41:57,070 is analogous to this step. 739 00:41:57,070 --> 00:42:01,320 All right it's the same type of argument. 740 00:42:01,320 --> 00:42:04,720 And in this case it's that the rectangles are on top 741 00:42:04,720 --> 00:42:07,270 and so the area represented on the right hand side 742 00:42:07,270 --> 00:42:09,840 is less than the area represented on this side. 743 00:42:09,840 --> 00:42:11,560 And this is the same type of thing 744 00:42:11,560 --> 00:42:14,400 except that the rectangles are underneath. 745 00:42:14,400 --> 00:42:17,310 So the sum of the areas of the rectangles 746 00:42:17,310 --> 00:42:19,190 is less than the area under the curve. 747 00:42:23,741 --> 00:42:24,240 All right. 748 00:42:24,240 --> 00:42:27,380 So I've now trapped this quantity. 749 00:42:27,380 --> 00:42:34,440 And I'm now going to state the sort of general results. 750 00:42:38,420 --> 00:42:42,230 So here's what's known as integral comparison. 751 00:42:42,230 --> 00:42:44,900 It's this double arrow correspondence 752 00:42:44,900 --> 00:42:54,580 in the general case, for a very general case. 753 00:42:54,580 --> 00:42:57,130 There are actually even more cases where it works. 754 00:42:57,130 --> 00:43:01,220 But this is a good case and convenient. 755 00:43:01,220 --> 00:43:02,845 Now this is called integral comparison. 756 00:43:07,060 --> 00:43:12,030 And it comes with hypotheses but it follows the same argument 757 00:43:12,030 --> 00:43:13,495 that I just gave. 758 00:43:13,495 --> 00:43:27,540 If f(x) is decreasing and it's positive, 759 00:43:27,540 --> 00:43:37,520 then the sum f(n), n equals 1 to infinity, 760 00:43:37,520 --> 00:43:44,570 minus the integral from 1 to infinity of f(x) dx 761 00:43:44,570 --> 00:43:45,420 is less than f(1). 762 00:43:50,060 --> 00:43:51,590 That's basically what we showed. 763 00:43:51,590 --> 00:43:54,570 We showed that the difference between S_N and log N 764 00:43:54,570 --> 00:43:55,930 was at most 1. 765 00:43:59,321 --> 00:43:59,820 All right. 766 00:43:59,820 --> 00:44:11,590 Now if both of them are-- And the sum 767 00:44:11,590 --> 00:44:25,380 and the integral converge or diverge together. 768 00:44:25,380 --> 00:44:27,750 That is they either both converge or both diverge. 769 00:44:27,750 --> 00:44:30,780 This is the type of test that we like because then we can just 770 00:44:30,780 --> 00:44:33,000 convert the question of convergence over here 771 00:44:33,000 --> 00:44:37,770 to this question of convergence over on the other side. 772 00:44:37,770 --> 00:44:41,590 Now I remind you that it's incredibly hard 773 00:44:41,590 --> 00:44:44,870 to calculate these numbers. 774 00:44:44,870 --> 00:44:47,220 Whereas these numbers are easier to calculate. 775 00:44:47,220 --> 00:44:50,390 Our goal is to reduce things to simpler things. 776 00:44:50,390 --> 00:44:52,910 And in this case sums, infinite sums 777 00:44:52,910 --> 00:44:54,720 are much harder than infinite integrals. 778 00:45:00,080 --> 00:45:03,150 All right so that's the integral comparison. 779 00:45:03,150 --> 00:45:12,140 And now I have one last bit on comparisons 780 00:45:12,140 --> 00:45:13,840 that I need to tell you about. 781 00:45:13,840 --> 00:45:16,310 And this is very much like what we did with integrals. 782 00:45:16,310 --> 00:45:18,240 Which is a so called limit comparison. 783 00:45:29,200 --> 00:45:31,190 The limit comparison says the following: 784 00:45:31,190 --> 00:45:45,210 if f(n) is similar to g(n) -- recall that means f(n) / g(n) 785 00:45:45,210 --> 00:45:51,760 tends to 1 as n goes to infinity -- 786 00:45:51,760 --> 00:45:55,000 and we're in the positive case. 787 00:45:55,000 --> 00:45:57,535 So let's just say g(n) is positive. 788 00:46:03,920 --> 00:46:05,490 Then-- that doesn't even, well-- then 789 00:46:05,490 --> 00:46:15,055 sum f(n), sum g(n) either both-- same thing 790 00:46:15,055 --> 00:46:21,884 as above, either both converge or both diverge. 791 00:46:27,488 --> 00:46:28,770 All right. 792 00:46:28,770 --> 00:46:30,730 This is just saying that if they behave 793 00:46:30,730 --> 00:46:33,920 the same way in the tail, which is all we really care about, 794 00:46:33,920 --> 00:46:40,820 then they have similar behavior, similar convergence properties. 795 00:46:44,107 --> 00:46:45,690 And let me give you a couple examples. 796 00:46:50,370 --> 00:46:56,580 So here's one example: if you take the sum 1 over n^2 + 1, 797 00:46:56,580 --> 00:46:57,080 square root. 798 00:47:01,890 --> 00:47:05,350 This is going to be replaced by something simpler. 799 00:47:05,350 --> 00:47:07,240 Which is the main term here. 800 00:47:07,240 --> 00:47:10,460 Which is 1 over square root of n^2, 801 00:47:10,460 --> 00:47:15,100 which we recognize as sum 1/n, which diverges. 802 00:47:17,920 --> 00:47:20,440 So this guy is one of the red guys. 803 00:47:24,300 --> 00:47:26,410 On the red team. 804 00:47:26,410 --> 00:47:30,330 Now we have another example. 805 00:47:33,370 --> 00:47:38,960 Which is let's say the square root of n, I don't know, 806 00:47:38,960 --> 00:47:43,080 to the fifth minus n^2. 807 00:47:43,080 --> 00:47:44,665 Now if you have something where it's 808 00:47:44,665 --> 00:47:46,290 negative in the denominator you kind of 809 00:47:46,290 --> 00:47:49,540 do have to watch out that denominator makes sense. 810 00:47:49,540 --> 00:47:50,570 It isn't 0. 811 00:47:50,570 --> 00:47:53,042 So we're going to be careful and start this at n = 2. 812 00:47:53,042 --> 00:48:00,370 In which case, the first term, I don't like 1/0 813 00:48:00,370 --> 00:48:01,720 as a term in my series. 814 00:48:01,720 --> 00:48:04,130 So I'm just going to be a little careful about how-- 815 00:48:04,130 --> 00:48:05,710 as I said I was kind of lazy here. 816 00:48:05,710 --> 00:48:09,930 I could have started this one at 0 for instance. 817 00:48:09,930 --> 00:48:10,810 All right. 818 00:48:10,810 --> 00:48:14,110 So here's the picture. 819 00:48:14,110 --> 00:48:19,070 Now this I just replace by its main term which is 1 over n^5, 820 00:48:19,070 --> 00:48:20,340 square root. 821 00:48:20,340 --> 00:48:25,975 Which is sum 1/n^(5/2), which converges. 822 00:48:28,705 --> 00:48:29,520 All right. 823 00:48:29,520 --> 00:48:30,800 The power is bigger than 1. 824 00:48:30,800 --> 00:48:33,960 1 is the divider for these things and it just misses. 825 00:48:33,960 --> 00:48:38,752 This one converges. 826 00:48:38,752 --> 00:48:41,800 All right so these are the typical ways 827 00:48:41,800 --> 00:48:47,355 in which these convergence processes are used. 828 00:48:47,355 --> 00:48:47,870 All right. 829 00:48:47,870 --> 00:48:49,930 So I have one more thing for you. 830 00:48:49,930 --> 00:48:52,600 Which is an advertisement for next time. 831 00:48:52,600 --> 00:48:56,360 And I have this demo here which I will grab. 832 00:48:56,360 --> 00:48:58,300 But you will see this next time. 833 00:48:58,300 --> 00:49:00,880 So here's a question for you to think about overnight 834 00:49:00,880 --> 00:49:04,240 but don't ask friends, you have to think about it yourself. 835 00:49:04,240 --> 00:49:05,492 So here's the problem. 836 00:49:05,492 --> 00:49:08,000 Here are some blocks which I acquired 837 00:49:08,000 --> 00:49:09,390 when my kids left home. 838 00:49:12,190 --> 00:49:20,550 Anyway yeah that'll happen to you too in about four years. 839 00:49:20,550 --> 00:49:25,929 So now here you are, these are blocks. 840 00:49:25,929 --> 00:49:27,470 So now here's the question that we're 841 00:49:27,470 --> 00:49:29,546 going to deal with next time. 842 00:49:29,546 --> 00:49:30,920 I'm going to build it, maybe I'll 843 00:49:30,920 --> 00:49:32,961 put it over here because I want to have some room 844 00:49:32,961 --> 00:49:34,890 to head this way. 845 00:49:34,890 --> 00:49:41,320 I want to stack them up so that-- oh didn't work. 846 00:49:41,320 --> 00:49:44,430 Going to stack them up in the following way. 847 00:49:44,430 --> 00:49:48,280 I want to do it so that the top one is completely 848 00:49:48,280 --> 00:49:51,540 to the right of the bottom one. 849 00:49:51,540 --> 00:49:53,170 That's the question can I do that? 850 00:49:53,170 --> 00:49:57,870 Can I get-- Can I build this up? 851 00:49:57,870 --> 00:50:01,980 So let's see here. 852 00:50:01,980 --> 00:50:04,590 I just seem to be missing-- but anyway what I'm going to do 853 00:50:04,590 --> 00:50:05,965 is I'm going to try to build this 854 00:50:05,965 --> 00:50:10,880 and we're going to see how far we can get with this next time.