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