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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.