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PROFESSOR: Now, today we are
continuing with this last unit.

00:00:30.090 --> 00:00:36.650
Unit 5, continued.

00:00:36.650 --> 00:00:48.490
The informal title of this
unit is Dealing With Infinity.

00:00:48.490 --> 00:00:51.210
That's really the
extra little piece

00:00:51.210 --> 00:00:55.150
that we're putting in to
our discussions of things

00:00:55.150 --> 00:01:01.080
like limits and integrals.

00:01:01.080 --> 00:01:09.680
To start out with today,
I'd like to recall for you,

00:01:09.680 --> 00:01:20.290
L'Hôpital's Rule.

00:01:20.290 --> 00:01:22.260
And in keeping with
the spirit here,

00:01:22.260 --> 00:01:32.670
we're just going to do the
infinity / infinity case.

00:01:32.670 --> 00:01:35.010
I stated this a little
differently last time,

00:01:35.010 --> 00:01:37.750
and I want to state
it again today.

00:01:37.750 --> 00:01:39.870
Just to make clear
what the hypotheses are

00:01:39.870 --> 00:01:43.880
and what the conclusion is.

00:01:43.880 --> 00:01:47.350
We start out with,
really, three hypotheses.

00:01:47.350 --> 00:01:50.160
Two of them are kind of obvious.

00:01:50.160 --> 00:01:55.660
The three hypotheses are
that f(x) tends to infinity,

00:01:55.660 --> 00:01:58.660
g(x) tends to
infinity, that's what

00:01:58.660 --> 00:02:02.380
it means to be in this
infinity / infinity case.

00:02:02.380 --> 00:02:08.590
And then the last assumption
is that f'(x) / g'(x) tends

00:02:08.590 --> 00:02:15.660
to a limit, L. And this is
all as x tends to some a.

00:02:15.660 --> 00:02:18.180
Some limit a.

00:02:18.180 --> 00:02:27.090
And then the conclusion
is that f(x) / g(x) also

00:02:27.090 --> 00:02:37.260
tends to L, as x goes to a.

00:02:37.260 --> 00:02:38.900
Now, so that's the way it is.

00:02:38.900 --> 00:02:40.750
So it's three limits.

00:02:40.750 --> 00:02:43.030
But presumably
these are obvious,

00:02:43.030 --> 00:02:49.180
and this one is exactly what
we were going to check anyway.

00:02:49.180 --> 00:02:52.790
Gives us this one limit.

00:02:52.790 --> 00:02:54.390
So that's the statement.

00:02:54.390 --> 00:02:59.340
And then the other little
interesting point here,

00:02:59.340 --> 00:03:02.900
which is consistent with this
idea of dealing with infinity,

00:03:02.900 --> 00:03:07.200
is that a equals plus
or minus infinity

00:03:07.200 --> 00:03:11.590
and L equals plus or
minus infinity are OK.

00:03:11.590 --> 00:03:15.780
That is, the numbers capital
L, the limit capital L,

00:03:15.780 --> 00:03:21.090
and the number a can
also be infinite.

00:03:21.090 --> 00:03:26.770
Now in recitation yesterday, you
should have discussed something

00:03:26.770 --> 00:03:31.830
about rates of growth,
which follow from what

00:03:31.830 --> 00:03:34.680
I said in lecture last
time and also maybe

00:03:34.680 --> 00:03:37.260
from some more detailed
discussions that you

00:03:37.260 --> 00:03:40.440
had in recitation.

00:03:40.440 --> 00:03:44.130
And I'm going to introduce a
notation to compare functions.

00:03:44.130 --> 00:03:49.580
Namely, we say that f(x)
is a lot less than g(x)

00:03:49.580 --> 00:03:58.060
if-- So this means that the
limit, as it goes to infinity,

00:03:58.060 --> 00:03:59.280
this tends to 0.

00:03:59.280 --> 00:04:04.700
As x goes to infinity,
this would be.

00:04:04.700 --> 00:04:07.490
So this is a notation,
a new notation for us.

00:04:07.490 --> 00:04:10.160
f is a lot less than g.

00:04:10.160 --> 00:04:13.150
And it's meant to be
read only asymptotically.

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It's only in the limit as x goes
to infinity that this happens.

00:04:17.890 --> 00:04:20.120
And implicitly here,
I'm always assuming

00:04:20.120 --> 00:04:23.570
that these are positive
quantities. f and g are

00:04:23.570 --> 00:04:28.130
positive.

00:04:28.130 --> 00:04:31.050
What you saw in
recitation was that you

00:04:31.050 --> 00:04:34.159
can make a systematic comparison
of all the standard functions

00:04:34.159 --> 00:04:34.950
that we know about.

00:04:34.950 --> 00:04:38.050
For example, the log
function goes to infinity.

00:04:38.050 --> 00:04:41.500
But a lot more slowly
than x to a power.

00:04:41.500 --> 00:04:44.920
A lot more slowly then e^x.

00:04:44.920 --> 00:04:48.220
A lot more slowly
than, say, e^(x^2).

00:04:48.220 --> 00:04:50.850
So this one is slow.

00:04:50.850 --> 00:04:54.580
This one is moderate.

00:04:54.580 --> 00:04:56.810
And this one is fast.

00:04:56.810 --> 00:05:00.740
And this one is very fast.

00:05:00.740 --> 00:05:02.900
Going to infinity.

00:05:02.900 --> 00:05:04.650
Tends to infinity,
and this is of course

00:05:04.650 --> 00:05:07.570
as x goes to infinity.

00:05:07.570 --> 00:05:12.270
All of them go to infinity,
but at quite different rates.

00:05:12.270 --> 00:05:14.860
And, analogous to
this, and today

00:05:14.860 --> 00:05:20.430
we're going to be doing this,
needing to do this quite a bit,

00:05:20.430 --> 00:05:24.850
is rates of decay, which are
more or less the opposite

00:05:24.850 --> 00:05:26.460
of rates of growth.

00:05:26.460 --> 00:05:30.270
So rates of decay are rates
at which things tend to 0.

00:05:30.270 --> 00:05:39.820
So the rate of decay, and for
that I'm just going to take

00:05:39.820 --> 00:05:41.890
reciprocals of these numbers.

00:05:41.890 --> 00:05:45.820
So 1 / ln x tends to 0.

00:05:45.820 --> 00:05:47.260
But rather slowly.

00:05:47.260 --> 00:05:51.220
It's much bigger than 1 / x^p.

00:05:51.220 --> 00:05:53.490
Oh, I didn't mention
that this exponent

00:05:53.490 --> 00:05:55.750
p is meant to be positive.

00:05:55.750 --> 00:05:58.360
That's a convention that
I'm using without saying.

00:05:58.360 --> 00:06:00.310
I should've told you that.

00:06:00.310 --> 00:06:06.660
So think x^(1/2), x^1, x^2,
they're all in this sort

00:06:06.660 --> 00:06:09.000
of moderate intermediate range.

00:06:09.000 --> 00:06:15.350
And then that, in turn, goes to
0 but much more slowly then 1 /

00:06:15.350 --> 00:06:19.220
e^x, also known as e^(-x).

00:06:19.220 --> 00:06:27.830
And that, in turn, this guy
here goes to 0 incredibly fast.

00:06:27.830 --> 00:06:28.780
e^(-x^2).

00:06:28.780 --> 00:06:32.950
vanishes really, really fast.

00:06:32.950 --> 00:06:37.130
So this is a review
of L'Hôpital's Rule.

00:06:37.130 --> 00:06:39.860
What we said last time,
and the application of it,

00:06:39.860 --> 00:06:42.920
which is to rates of
growth and tells us

00:06:42.920 --> 00:06:50.860
what these rates of growth are.

00:06:50.860 --> 00:07:01.830
Today, I want to talk
about improper integrals.

00:07:01.830 --> 00:07:05.160
And improper integrals,
we've already

00:07:05.160 --> 00:07:09.050
really seen one or two of
them on your exercises.

00:07:09.050 --> 00:07:11.805
And we mention them a
little bit, briefly.

00:07:11.805 --> 00:07:13.180
I'm just going to
go through them

00:07:13.180 --> 00:07:15.110
more carefully and more
systematically now.

00:07:15.110 --> 00:07:17.530
And we want to get
just exactly what's

00:07:17.530 --> 00:07:19.930
going on with these
rates of decay

00:07:19.930 --> 00:07:21.940
and their relationship
with improper integrals.

00:07:21.940 --> 00:07:26.350
So I need for you to
understand, on the spectrum

00:07:26.350 --> 00:07:28.470
of the range of
functions like this,

00:07:28.470 --> 00:07:32.940
which ones are suitable
for integration

00:07:32.940 --> 00:07:38.320
as x goes to infinity.

00:07:38.320 --> 00:07:43.130
Well, let's start out
with the definition.

00:07:43.130 --> 00:07:47.360
The integral from a
to infinity of f(x) dx

00:07:47.360 --> 00:07:50.170
is, by definition,
the limit as N

00:07:50.170 --> 00:07:56.620
goes to infinity of the
ordinary definite integral

00:07:56.620 --> 00:08:00.430
up to some fixed, finite level.

00:08:00.430 --> 00:08:01.760
That's the definition.

00:08:01.760 --> 00:08:05.990
And there's a word
that we use here,

00:08:05.990 --> 00:08:09.250
which is that we
say the integral,

00:08:09.250 --> 00:08:20.560
so this is terminology for it,
converges if the limit exists.

00:08:20.560 --> 00:08:28.630
And diverges if not.

00:08:28.630 --> 00:08:34.340
Well, these are the
key words for today.

00:08:34.340 --> 00:08:39.870
So here's the issue that
we're going to be addressing.

00:08:39.870 --> 00:08:42.430
Which is whether the
limit exists or not.

00:08:42.430 --> 00:08:50.820
In other words, whether the
integral converges or diverges.

00:08:50.820 --> 00:08:53.800
These notions have
a geometric analog,

00:08:53.800 --> 00:08:56.190
which you should always be
thinking of at the same time

00:08:56.190 --> 00:08:57.730
in the back of your head.

00:08:57.730 --> 00:09:00.390
I'll draw a picture
of the function.

00:09:00.390 --> 00:09:02.510
Here it's starting out at a.

00:09:02.510 --> 00:09:05.790
And maybe it's going
down like this.

00:09:05.790 --> 00:09:09.840
And it's interpreting
it geometrically.

00:09:09.840 --> 00:09:15.520
This would only work
if f is positive.

00:09:15.520 --> 00:09:25.900
Then the convergent case is the
case where the area is finite.

00:09:25.900 --> 00:09:29.660
So the total area is
finite under this curve.

00:09:29.660 --> 00:09:43.750
And the other case is the
total area is infinite.

00:09:43.750 --> 00:09:46.920
I claim that both of
these things are possible.

00:09:46.920 --> 00:09:50.460
Although this thing
goes on forever,

00:09:50.460 --> 00:09:53.070
if you stop it at one
stage, N, then of course

00:09:53.070 --> 00:09:54.310
it's a finite number.

00:09:54.310 --> 00:09:56.270
But as you go further
and further and further,

00:09:56.270 --> 00:09:58.290
there's more and
more and more area.

00:09:58.290 --> 00:10:00.420
And there are two possibilities.

00:10:00.420 --> 00:10:04.700
Either as you go all the
way out here to infinity,

00:10:04.700 --> 00:10:08.900
the total that you get
adds up to a finite total.

00:10:08.900 --> 00:10:11.280
Or else, maybe there's
infinitely much.

00:10:11.280 --> 00:10:13.700
For instance, if it's a
straight line going across,

00:10:13.700 --> 00:10:23.340
there's clearly infinitely
much area underneath.

00:10:23.340 --> 00:10:25.500
So we need to do a
bunch of examples.

00:10:25.500 --> 00:10:28.660
And that's really our
main job for the day,

00:10:28.660 --> 00:10:30.700
and to make sure
that we know exactly

00:10:30.700 --> 00:10:34.810
what to expect in all cases.

00:10:34.810 --> 00:10:42.200
The first example is the
integral from 0 to infinity

00:10:42.200 --> 00:10:45.410
of e^(-kx) dx.

00:10:45.410 --> 00:10:48.550
Where k is going to be
some positive number.

00:10:48.550 --> 00:10:54.080
Some positive constant.

00:10:54.080 --> 00:11:02.919
This is the most fundamental, by
far, of the definite integrals.

00:11:02.919 --> 00:11:03.710
Improper integrals.

00:11:03.710 --> 00:11:07.370
And in order to handle this,
the thing that I need to do is

00:11:07.370 --> 00:11:13.000
to check the integral from
0 up to N, e^(-kx) dx.

00:11:13.000 --> 00:11:15.960
And since this is an easy
integral to evaluate,

00:11:15.960 --> 00:11:17.650
we're going to do it.

00:11:17.650 --> 00:11:22.590
It's -1/k e^(-kx), that's
the antiderivative.

00:11:22.590 --> 00:11:30.370
Evaluated at 0 and N. And that,
if I plug in these values,

00:11:30.370 --> 00:11:39.730
is -1/k e^(-kN), minus--
and if I evaluate it at 0,

00:11:39.730 --> 00:11:42.600
I get a -1/k e^0.

00:11:46.860 --> 00:11:48.690
So there's the answer.

00:11:48.690 --> 00:11:50.180
And now we have to
think about what

00:11:50.180 --> 00:11:54.040
happens as N goes to infinity.

00:11:54.040 --> 00:11:59.050
So as N goes to infinity,
what's happening

00:11:59.050 --> 00:12:03.400
is the second term
here stays unchanged.

00:12:03.400 --> 00:12:06.784
But the first term is e
to some negative power.

00:12:06.784 --> 00:12:08.700
And the exponent is
getting larger and larger.

00:12:08.700 --> 00:12:10.830
That's because k
is positive here.

00:12:10.830 --> 00:12:12.590
You've definitely
got to pay attention.

00:12:12.590 --> 00:12:15.020
Even though I'm doing this
with general variables here,

00:12:15.020 --> 00:12:17.922
you've got to pay attention
to signs of things.

00:12:17.922 --> 00:12:20.130
Because otherwise you'll
always get the wrong answer.

00:12:20.130 --> 00:12:22.710
So you have to pay very
close attention here.

00:12:22.710 --> 00:12:24.990
So this is, if you like,
e to the minus infinity

00:12:24.990 --> 00:12:26.720
in the limit, which is 0.

00:12:26.720 --> 00:12:30.720
And so in the limit,
this thing tends to 0.

00:12:30.720 --> 00:12:33.870
And this thing is
just equal to 1/k.

00:12:33.870 --> 00:12:36.100
And so all told,
the answer is 1/k.

00:12:36.100 --> 00:12:43.610
And that's it.

00:12:43.610 --> 00:12:46.074
Now we're going to
abbreviate this a little bit.

00:12:46.074 --> 00:12:47.490
This thought
process, you're going

00:12:47.490 --> 00:12:50.070
to have to go through every
single time you do this.

00:12:50.070 --> 00:12:52.830
But after a while you
also get good enough at it

00:12:52.830 --> 00:12:55.770
that you can make it a
little bit less cluttered.

00:12:55.770 --> 00:13:09.350
So let me show you a shorthand
for this same calculation.

00:13:09.350 --> 00:13:14.650
Namely, I write 0 to
infinity e^(-kx) dx.

00:13:14.650 --> 00:13:23.890
And that's equal to -1/k
e^(-kx) 0 to infinity.

00:13:23.890 --> 00:13:27.710
That was cute.

00:13:27.710 --> 00:13:35.600
Not small enough, however.

00:13:35.600 --> 00:13:36.374
So, here we are.

00:13:36.374 --> 00:13:38.290
We have the same calculation
as we had before.

00:13:38.290 --> 00:13:40.130
But now we're thinking,
really, in our minds

00:13:40.130 --> 00:13:43.100
that this infinity is some
very, very enormous number.

00:13:43.100 --> 00:13:44.790
And we're going to plug it in.

00:13:44.790 --> 00:13:47.060
And you can either do
this in your head or not.

00:13:47.060 --> 00:13:50.760
You say -1/k e^(-infinity).

00:13:50.760 --> 00:13:53.510
Here's where I've used the
fact that k is positive.

00:13:53.510 --> 00:13:57.300
Because e to the -k times a
large number is minus infinity.

00:13:57.300 --> 00:14:01.570
And then here plus 1/k--
minus -1/k, let write it

00:14:01.570 --> 00:14:05.780
the same way I did before.

00:14:05.780 --> 00:14:11.840
And that's just equal to 0 +
1/k, which is what we want.

00:14:11.840 --> 00:14:17.240
So this is the same calculation,
just slightly abbreviated.

00:14:17.240 --> 00:14:17.740
Yeah.

00:14:17.740 --> 00:14:18.240
Question.

00:14:18.240 --> 00:14:29.050
STUDENT: [INAUDIBLE]

00:14:29.050 --> 00:14:30.385
PROFESSOR: Good question.

00:14:30.385 --> 00:14:31.885
The question is,
what about the case

00:14:31.885 --> 00:14:34.830
when the limit is infinity?

00:14:34.830 --> 00:14:37.710
I'm distinguishing
between something existing

00:14:37.710 --> 00:14:39.690
and its limit being
infinity here.

00:14:39.690 --> 00:14:45.660
Whenever I make a discussion of
limits, I say a finite limit,

00:14:45.660 --> 00:14:49.100
or in this case, it works
for infinite limits.

00:14:49.100 --> 00:14:50.920
So in other words,
when I say exists,

00:14:50.920 --> 00:14:54.970
I mean exists and is finite.

00:14:54.970 --> 00:14:57.535
So here, when I say
that it converges and I

00:14:57.535 --> 00:14:58.910
say the limit
exists, what I mean

00:14:58.910 --> 00:15:00.500
is that it's a finite number.

00:15:00.500 --> 00:15:02.630
And so that's indeed
what I said here.

00:15:02.630 --> 00:15:04.570
The total area is finite.

00:15:04.570 --> 00:15:06.100
And, similarly, over here.

00:15:06.100 --> 00:15:07.720
I might add, however,
that there is

00:15:07.720 --> 00:15:09.770
another part of this subject.

00:15:09.770 --> 00:15:11.410
Which I'm skipping entirely.

00:15:11.410 --> 00:15:13.020
Which is a little bit subtle.

00:15:13.020 --> 00:15:14.330
Which is the following.

00:15:14.330 --> 00:15:17.860
If f changes sign, there
can be some cancellation

00:15:17.860 --> 00:15:19.040
and oscillation.

00:15:19.040 --> 00:15:21.040
And then sometimes
the limit exists,

00:15:21.040 --> 00:15:23.760
but the total area, if you
counted it all positively,

00:15:23.760 --> 00:15:25.680
is actually still infinite.

00:15:25.680 --> 00:15:29.120
And we're going to
avoid that case.

00:15:29.120 --> 00:15:31.820
We're just going to treat
these positive cases.

00:15:31.820 --> 00:15:33.460
So don't worry
about that for now.

00:15:33.460 --> 00:15:35.600
That's the next
layer of complexity

00:15:35.600 --> 00:15:38.432
which we're not
addressing in this class.

00:15:38.432 --> 00:15:39.140
Another question.

00:15:39.140 --> 00:15:45.610
STUDENT: [INAUDIBLE]

00:15:45.610 --> 00:15:48.260
PROFESSOR: The question is,
would this be OK on tests.

00:15:48.260 --> 00:15:49.870
The answer is, absolutely yes.

00:15:49.870 --> 00:15:51.550
I want to encourage
you to do this.

00:15:51.550 --> 00:15:53.660
If you can think
about it correctly.

00:15:53.660 --> 00:15:55.170
The subtle point
is just, you have

00:15:55.170 --> 00:15:57.270
to plug in infinity correctly.

00:15:57.270 --> 00:15:59.830
Namely, you have to
realize that this only

00:15:59.830 --> 00:16:01.115
works if k is positive.

00:16:01.115 --> 00:16:03.240
This is the step where
you're plugging in infinity.

00:16:03.240 --> 00:16:06.630
And I'm letting you put
this infinity up here

00:16:06.630 --> 00:16:08.390
as an endpoint value.

00:16:08.390 --> 00:16:12.470
So in fact that's
exactly the theme.

00:16:12.470 --> 00:16:16.800
The theme is dealing
with infinity here.

00:16:16.800 --> 00:16:18.550
And I want you to be
able to deal with it.

00:16:18.550 --> 00:16:20.440
That's my goal.

00:16:20.440 --> 00:16:32.660
STUDENT: [INAUDIBLE]

00:16:32.660 --> 00:16:35.060
PROFESSOR: OK, so
another question is,

00:16:35.060 --> 00:16:37.940
so let's be sure here
when the limit exists,

00:16:37.940 --> 00:16:40.640
I say it has to be finite.

00:16:40.640 --> 00:16:46.910
That means it's
finite, not infinite.

00:16:46.910 --> 00:16:48.270
The limit can be 0.

00:16:48.270 --> 00:16:50.790
It can also be -1.

00:16:50.790 --> 00:16:51.880
It can be anything.

00:16:51.880 --> 00:16:58.590
Doesn't have to be
a positive number.

00:16:58.590 --> 00:17:04.840
Other questions.

00:17:04.840 --> 00:17:07.180
So we've had our first example.

00:17:07.180 --> 00:17:23.650
And now I just want to add one
physical interpretation here.

00:17:23.650 --> 00:17:29.430
This is Example 1, if you like.

00:17:29.430 --> 00:17:32.930
And this is something that was
on your problem set, remember.

00:17:32.930 --> 00:17:36.680
That we talked about the
probability, or the number,

00:17:36.680 --> 00:17:42.540
if you like, the
number of particles

00:17:42.540 --> 00:18:02.540
on average that decay in
some radioactive substance.

00:18:02.540 --> 00:18:09.870
Say, in time between
0 and some capital

00:18:09.870 --> 00:18:15.990
T. And then that would be
this integral, 0 to capital

00:18:15.990 --> 00:18:22.560
T, some total quantity
times this integral here.

00:18:22.560 --> 00:18:28.000
This is the typical kind
of radioactive decay number

00:18:28.000 --> 00:18:29.800
that one gets.

00:18:29.800 --> 00:18:38.620
Now, in the limit, so this
is some number of particles.

00:18:38.620 --> 00:18:42.720
If the substance is
radioactive, then in the limit,

00:18:42.720 --> 00:18:47.610
we have this.

00:18:47.610 --> 00:18:56.070
Which is equal to the
total number of particles.

00:18:56.070 --> 00:18:57.970
And that's something
that's going

00:18:57.970 --> 00:19:00.424
to be important for
normalizing and understanding.

00:19:00.424 --> 00:19:01.840
How much does the
whole substance,

00:19:01.840 --> 00:19:04.840
how many moles do we
have of this stuff.

00:19:04.840 --> 00:19:05.960
What is it.

00:19:05.960 --> 00:19:08.390
And so this is a number
that is going to come up.

00:19:08.390 --> 00:19:14.040
Now, I emphasize that this
notion of T going to infinity

00:19:14.040 --> 00:19:16.890
is just an idealization.

00:19:16.890 --> 00:19:20.800
We don't really believe that
we're going to wait forever

00:19:20.800 --> 00:19:23.580
for this substance to decay.

00:19:23.580 --> 00:19:27.980
Nevertheless, as theorists,
we write down this quantity.

00:19:27.980 --> 00:19:29.110
And we use it.

00:19:29.110 --> 00:19:31.970
All the time.

00:19:31.970 --> 00:19:34.740
Furthermore, there's other
good reasons for using it,

00:19:34.740 --> 00:19:36.930
and why physicists
accept it immediately.

00:19:36.930 --> 00:19:39.850
Even though it's not really
completely physically realistic

00:19:39.850 --> 00:19:43.890
ever to let time go very,
very far into the future.

00:19:43.890 --> 00:19:47.710
And the reason is, if you
notice this answer here,

00:19:47.710 --> 00:19:53.330
look at how much simpler
this number is, 1/k,

00:19:53.330 --> 00:19:57.940
than the numbers that I got in
the intermediate stages here.

00:19:57.940 --> 00:20:01.800
These are all ugly,
the limits are simple.

00:20:01.800 --> 00:20:04.040
And this is a theme
that I've been trying

00:20:04.040 --> 00:20:05.790
to emphasize all semester.

00:20:05.790 --> 00:20:07.755
Namely, that the
infinitesimal, the things

00:20:07.755 --> 00:20:09.630
that you get when you
do differentiation, are

00:20:09.630 --> 00:20:10.580
the easier formulas.

00:20:10.580 --> 00:20:13.610
The algebraic ones, the
things in the process

00:20:13.610 --> 00:20:16.500
of getting to the limit,
are the ugly ones.

00:20:16.500 --> 00:20:18.650
These are the easy ones,
these are the hard ones.

00:20:18.650 --> 00:20:20.930
So in fact, infinity
is basically easier

00:20:20.930 --> 00:20:23.040
than any finite number.

00:20:23.040 --> 00:20:27.110
And a lot of appealing
formulas come from those kinds

00:20:27.110 --> 00:20:28.470
of calculations.

00:20:28.470 --> 00:20:31.840
Another question.

00:20:31.840 --> 00:20:39.310
STUDENT: [INAUDIBLE]

00:20:39.310 --> 00:20:43.040
PROFESSOR: The question is,
shouldn't the answer be A?

00:20:43.040 --> 00:20:47.070
Well, the answer
turns out to be A/k.

00:20:47.070 --> 00:20:49.660
Which means that when you
set up your arithmetic,

00:20:49.660 --> 00:20:53.750
and you model this to a
collection of particles.

00:20:53.750 --> 00:20:56.110
So you said it should
be A. But that's

00:20:56.110 --> 00:20:58.290
because you made an assumption.

00:20:58.290 --> 00:21:01.190
Which was that A was the
total number of particles.

00:21:01.190 --> 00:21:03.200
But that's just false, right?

00:21:03.200 --> 00:21:04.810
This is the total
number of particles.

00:21:04.810 --> 00:21:06.599
So therefore, if you
want to set it up,

00:21:06.599 --> 00:21:08.140
you want set up so
that this number's

00:21:08.140 --> 00:21:11.500
the total number of particles.

00:21:11.500 --> 00:21:13.010
And that's how you
set up a model,

00:21:13.010 --> 00:21:14.593
is you do all the
calculations and you

00:21:14.593 --> 00:21:16.040
see what it's coming out to be.

00:21:16.040 --> 00:21:24.490
And that's why you need to
do this kind of calculation.

00:21:24.490 --> 00:21:25.085
OK, so.

00:21:25.085 --> 00:21:27.710
The main thing is, you shouldn't
make assumptions about models.

00:21:27.710 --> 00:21:29.950
You have to follow what
the calculations tell you.

00:21:29.950 --> 00:21:32.110
They're not lying.

00:21:32.110 --> 00:21:34.990
OK, so now.

00:21:34.990 --> 00:21:36.300
We carried this out.

00:21:36.300 --> 00:21:40.640
There's one other
example which we talked

00:21:40.640 --> 00:21:42.830
about earlier in the class.

00:21:42.830 --> 00:21:44.470
And I just wanted
to mention it again.

00:21:44.470 --> 00:21:48.800
It's probably the most
famous after this one.

00:21:48.800 --> 00:21:51.390
Namely, the integral from
minus infinity to infinity

00:21:51.390 --> 00:21:53.740
of e^(-x^2) dx.

00:21:53.740 --> 00:21:56.930
Which turns out, amazingly,
to be able to be evaluated.

00:21:56.930 --> 00:21:59.500
It turns out to be
the square root of pi.

00:21:59.500 --> 00:22:04.340
So this one is also great.

00:22:04.340 --> 00:22:07.350
This is the constant
which allows

00:22:07.350 --> 00:22:12.230
you to compute all kinds
of things in probability.

00:22:12.230 --> 00:22:22.760
So this is a key
number in probability.

00:22:22.760 --> 00:22:25.740
It basically is the key
to understanding things

00:22:25.740 --> 00:22:28.210
like standard
deviation and basically

00:22:28.210 --> 00:22:31.010
any other thing in the
subject of probability.

00:22:31.010 --> 00:22:35.910
It's also what's driving
these polls that tell you

00:22:35.910 --> 00:22:40.390
within 4% accuracy we
know that people are

00:22:40.390 --> 00:22:42.430
going to vote this way or that.

00:22:42.430 --> 00:22:44.830
So in order to interpret all
of those kinds of things,

00:22:44.830 --> 00:22:48.360
you need to know this number.

00:22:48.360 --> 00:22:51.860
And this number
was only calculated

00:22:51.860 --> 00:22:58.750
numerically starting in the
1700s or so by people who--

00:22:58.750 --> 00:23:01.050
actually, by one guy whose
name was de Moivre, who

00:23:01.050 --> 00:23:04.950
was selling his services
to various royalty who

00:23:04.950 --> 00:23:06.390
were running lotteries.

00:23:06.390 --> 00:23:09.190
In those days they
ran lotteries, too.

00:23:09.190 --> 00:23:13.530
And he was able to tell
them what the chances were

00:23:13.530 --> 00:23:15.590
of the various games.

00:23:15.590 --> 00:23:17.720
And he worked out this number.

00:23:17.720 --> 00:23:19.314
He realized that
this was the pattern.

00:23:19.314 --> 00:23:21.730
Although he didn't know that
it was the square root of pi,

00:23:21.730 --> 00:23:23.390
he knew it to
sufficient accuracy

00:23:23.390 --> 00:23:25.580
that he could tell
them the correct answer

00:23:25.580 --> 00:23:29.770
to how much money their
lotteries would make.

00:23:29.770 --> 00:23:33.070
And of course we do
this nowadays, too.

00:23:33.070 --> 00:23:34.530
In all kinds of ways.

00:23:34.530 --> 00:23:45.440
Including slightly more legit
businesses like insurance.

00:23:45.440 --> 00:23:49.920
So now, I'm going to give
you some more examples.

00:23:49.920 --> 00:23:54.910
And the other examples
are much more close

00:23:54.910 --> 00:23:59.320
to the edge between
infinite and finite.

00:23:59.320 --> 00:24:02.500
This distinction between
convergence and divergence.

00:24:02.500 --> 00:24:07.090
And let me just-- Maybe I'll
say one more word about why

00:24:07.090 --> 00:24:10.690
we care about this very gross
issue of whether something

00:24:10.690 --> 00:24:12.630
is finite or infinite.

00:24:12.630 --> 00:24:14.670
When you're talking
about something

00:24:14.670 --> 00:24:18.660
like this normal
curve here, there's

00:24:18.660 --> 00:24:24.290
an issue of how far
out you have to go

00:24:24.290 --> 00:24:29.280
before you can ignore the rest.

00:24:29.280 --> 00:24:34.399
So we're going to ignore
what's called the tail here.

00:24:34.399 --> 00:24:36.440
Somehow you want to know
that this is negligible.

00:24:36.440 --> 00:24:38.790
And you want to know
how negligible it is.

00:24:38.790 --> 00:24:41.350
And this is the job
of a mathematician,

00:24:41.350 --> 00:24:44.249
is to know what finite
region you have to consider

00:24:44.249 --> 00:24:46.290
and which one you're going
to carefully calculate

00:24:46.290 --> 00:24:46.846
numerically.

00:24:46.846 --> 00:24:48.470
And then the rest,
you're going to have

00:24:48.470 --> 00:24:50.690
to take care of by some
theoretical reasoning.

00:24:50.690 --> 00:24:52.731
You're going to have to
know that these tails are

00:24:52.731 --> 00:24:54.360
small enough that
they don't matter

00:24:54.360 --> 00:24:57.150
in your finite calculation.

00:24:57.150 --> 00:24:59.070
And so, we care very
much about the tails.

00:24:59.070 --> 00:25:02.700
Because they're the only thing
that the machine won't tell us.

00:25:02.700 --> 00:25:05.090
So that's the part
that we have to know.

00:25:05.090 --> 00:25:06.830
And these tails
are also something

00:25:06.830 --> 00:25:09.870
which are discussed all the
time in financial mathematics.

00:25:09.870 --> 00:25:11.830
They're very worried
about fat tails.

00:25:11.830 --> 00:25:16.270
That is, unlikely events that
nevertheless happen sometimes.

00:25:16.270 --> 00:25:18.570
And they get burned fairly
regularly with them.

00:25:18.570 --> 00:25:25.440
As they have recently,
with the mortgage scandal.

00:25:25.440 --> 00:25:28.416
So, these things
are pretty serious

00:25:28.416 --> 00:25:30.540
and they really are spending
a lot of time on them.

00:25:30.540 --> 00:25:32.922
Of course, there are lots
of other practical issues

00:25:32.922 --> 00:25:34.130
besides just the mathematics.

00:25:34.130 --> 00:25:37.340
But you've got to get
the math right, too.

00:25:37.340 --> 00:25:40.860
So we're going to now talk
about some borderline cases

00:25:40.860 --> 00:25:42.950
for these fat tails.

00:25:42.950 --> 00:25:45.820
Just how fat do they have to
be before they become infinite

00:25:45.820 --> 00:25:51.820
and overwhelm the central bump.

00:25:51.820 --> 00:25:56.120
So we'll save this
for just a second.

00:25:56.120 --> 00:25:58.734
And what I'm saving up here
is the borderline case,

00:25:58.734 --> 00:26:00.150
which I'm going
to concentrate on,

00:26:00.150 --> 00:26:07.500
which is this moderate
rate, which is x to powers.

00:26:07.500 --> 00:26:09.570
Here's our next example.

00:26:09.570 --> 00:26:13.230
I guess we'll call
this Example 3.

00:26:13.230 --> 00:26:17.540
It's the integral from
1 to infinity dx / x.

00:26:17.540 --> 00:26:20.800
That's the power p = 1.

00:26:20.800 --> 00:26:23.980
And this turns out to
be a borderline case.

00:26:23.980 --> 00:26:26.790
So it's worth carrying
out carefully.

00:26:26.790 --> 00:26:29.820
Now, again I'm going to do
it by the slower method.

00:26:29.820 --> 00:26:31.290
Rather than the
shorthand method.

00:26:31.290 --> 00:26:36.100
But ultimately, you can do it by
the short method if you'd like.

00:26:36.100 --> 00:26:38.960
I break it up into
an integral that

00:26:38.960 --> 00:26:44.260
goes up to some large number,
N. I see that it's the logarithm

00:26:44.260 --> 00:26:47.060
function is the antiderivative.

00:26:47.060 --> 00:26:51.740
And so what I get is ln N
minus ln 1, which is just 0.

00:26:51.740 --> 00:26:53.710
So this is just
log N. In any case,

00:26:53.710 --> 00:26:57.860
it tends to infinity
as N. goes to infinity.

00:26:57.860 --> 00:27:01.430
So the conclusion is, since
the limit is infinite,

00:27:01.430 --> 00:27:12.020
that this thing diverges.

00:27:12.020 --> 00:27:16.560
Now, I'm going to do
this systematically now

00:27:16.560 --> 00:27:20.830
with all powers p,
to see what happens.

00:27:20.830 --> 00:27:22.050
I'll look at the integral.

00:27:22.050 --> 00:27:23.900
Sorry, I'm going to
have to start at 1 here.

00:27:23.900 --> 00:27:27.820
From 1 to infinity, dx / x^p.

00:27:27.820 --> 00:27:29.480
and see what happens with these.

00:27:29.480 --> 00:27:32.200
And you'll see that p =
1 is a borderline when

00:27:32.200 --> 00:27:35.720
I do this calculation.

00:27:35.720 --> 00:27:39.059
This time I'm going to do
the calculation the hard way.

00:27:39.059 --> 00:27:41.350
But now you're going to have
to think and pay attention

00:27:41.350 --> 00:27:43.225
to see what it is
that I'm doing.

00:27:43.225 --> 00:27:45.350
First of all, I'm going to
take the antiderivative.

00:27:45.350 --> 00:27:53.590
And this is x^(-p), so it's
- -p + 1 divided by -p + 1.

00:27:53.590 --> 00:28:00.960
That's the antiderivative of
the function 1/x^p or x^(-p).

00:28:00.960 --> 00:28:07.710
And then I have to evaluate
that at 1 and infinity.

00:28:07.710 --> 00:28:10.600
So now, I'll write this down.

00:28:10.600 --> 00:28:13.990
But I'm going to be
particularly careful here.

00:28:13.990 --> 00:28:14.880
I'll write it down.

00:28:14.880 --> 00:28:27.040
It's infinity to the -p
+ 1 over -p + 1 minus,

00:28:27.040 --> 00:28:29.160
so I plug in 1 here.

00:28:29.160 --> 00:28:30.450
So I get 1/(-p+1).

00:28:34.870 --> 00:28:36.440
So this is what I'm getting.

00:28:36.440 --> 00:28:38.730
Again, what you should
be thinking here

00:28:38.730 --> 00:28:45.410
is this is a very large
number to this power.

00:28:45.410 --> 00:28:47.770
Now, there are two cases.

00:28:47.770 --> 00:28:48.880
There are two cases.

00:28:48.880 --> 00:28:52.530
And they exactly split at p = 1.

00:28:52.530 --> 00:28:55.280
When p = 1, this number is 0.

00:28:55.280 --> 00:28:57.380
This exponent is 0, and
in fact this expression

00:28:57.380 --> 00:29:01.310
doesn't make any sense because
the denominator is also 0.

00:29:01.310 --> 00:29:05.310
But for all of the other values,
the denominator makes sense.

00:29:05.310 --> 00:29:09.780
But what's going on is
that this is infinite

00:29:09.780 --> 00:29:13.420
when this exponent is
infinity to a positive power.

00:29:13.420 --> 00:29:20.070
And it's 0 when it's
infinity to a negative power.

00:29:20.070 --> 00:29:22.970
So I'm going to say it here,
and you must check this at home.

00:29:22.970 --> 00:29:24.785
Because this is
exactly what I'm going

00:29:24.785 --> 00:29:27.060
to ask you about on the exam.

00:29:27.060 --> 00:29:28.590
This is it.

00:29:28.590 --> 00:29:33.360
This type of thing, maybe with
a specific value of p here.

00:29:33.360 --> 00:29:45.660
When p < 1, this
thing is infinite.

00:29:45.660 --> 00:29:53.560
On the other hand, when
p > 1, this thing is 0.

00:29:53.560 --> 00:29:59.260
So when p > 1, this thing is 0.

00:29:59.260 --> 00:30:01.040
It's just equal to 0.

00:30:01.040 --> 00:30:09.125
And so the answer is 1/(p-1).

00:30:09.125 --> 00:30:10.250
Because that's this number.

00:30:10.250 --> 00:30:12.970
Minus the quantity 1/(-p+1).

00:30:15.720 --> 00:30:17.480
This is a finite number here.

00:30:17.480 --> 00:30:20.390
Notice that the answer would be
weird if this thing went away

00:30:20.390 --> 00:30:22.840
in the p < 1 case.

00:30:22.840 --> 00:30:24.740
Then it would be
a negative number.

00:30:24.740 --> 00:30:28.420
It would be a very strange
answer to this question.

00:30:28.420 --> 00:30:29.920
So, in fact that's
not what happens.

00:30:29.920 --> 00:30:32.477
What happens is that the
answer doesn't make sense.

00:30:32.477 --> 00:30:33.060
It's infinite.

00:30:33.060 --> 00:30:35.620
So let me just write this
down again, under here.

00:30:35.620 --> 00:30:42.160
This is a test in
a particular case.

00:30:42.160 --> 00:30:47.190
And here's the conclusion.

00:30:47.190 --> 00:30:48.027
Ah.

00:30:48.027 --> 00:30:48.610
No, I'm sorry.

00:30:48.610 --> 00:31:03.760
I think I was going to write
it over on this board here.

00:31:03.760 --> 00:31:11.770
So the conclusion is that the
integral from 1 to infinity dx

00:31:11.770 --> 00:31:21.000
/ x^p diverges if p <= 1.

00:31:21.000 --> 00:31:33.130
And converges if p > 1.

00:31:33.130 --> 00:31:35.550
And in fact, we can
actually evaluate it.

00:31:35.550 --> 00:31:38.380
It's equal to 1/(p-1).

00:31:38.380 --> 00:31:44.050
It's got a nice,
clean formula even.

00:31:44.050 --> 00:31:45.820
Alright, now let me remind you.

00:31:45.820 --> 00:31:47.870
So I didn't spell the word
diverges right, did I?

00:31:47.870 --> 00:31:49.170
Oh no, that's an r.

00:31:49.170 --> 00:31:55.530
I guess that's right.

00:31:55.530 --> 00:31:57.040
Diverges if p <= 1.

00:31:57.040 --> 00:31:59.910
So really, I needed
both of these arguments,

00:31:59.910 --> 00:32:02.310
which are sitting above
it, in order to do it.

00:32:02.310 --> 00:32:06.510
Because the second argument
didn't work at all when p = 1

00:32:06.510 --> 00:32:09.184
because the formula for the
antiderivative is wrong.

00:32:09.184 --> 00:32:10.600
The formula for
the antiderivative

00:32:10.600 --> 00:32:13.570
is given by the log
function when p = 1.

00:32:13.570 --> 00:32:15.780
So I had to do this
calculation too.

00:32:15.780 --> 00:32:21.420
This is the borderline case,
between p > 1 and p < 1.

00:32:21.420 --> 00:32:23.820
When p > 1, we got convergence.

00:32:23.820 --> 00:32:27.140
We could calculate the integral.

00:32:27.140 --> 00:32:30.290
When p < 1, when we got
divergence and we calculated

00:32:30.290 --> 00:32:31.510
the integral over there.

00:32:31.510 --> 00:32:34.220
And here in the borderline
case, we got a logarithm,

00:32:34.220 --> 00:32:35.940
and we also got divergence.

00:32:35.940 --> 00:32:39.520
So it failed at the edge.

00:32:39.520 --> 00:32:46.980
Now, this takes care
of all the powers.

00:32:46.980 --> 00:32:54.490
Now, there are a number
of different things

00:32:54.490 --> 00:32:58.320
that one can deduce from this.

00:32:58.320 --> 00:33:02.390
And let me carry them out.

00:33:02.390 --> 00:33:04.180
So this is more or
less the second thing

00:33:04.180 --> 00:33:07.030
that you'll want to do.

00:33:07.030 --> 00:33:12.440
And I'm going to emphasize
maybe one aspect of it.

00:33:12.440 --> 00:33:14.510
I guess we'll get rid of this.

00:33:14.510 --> 00:33:17.500
But it's still the issue
that we're discussing here.

00:33:17.500 --> 00:33:20.230
Is whether this
area is fat or thin.

00:33:20.230 --> 00:33:24.500
I'll remind you of that.

00:33:24.500 --> 00:33:29.240
So here's the next idea.

00:33:29.240 --> 00:33:34.250
Something called
limit comparison.

00:33:34.250 --> 00:33:35.850
Limit comparison is
what you're going

00:33:35.850 --> 00:33:39.100
to use when, instead
of being able actually

00:33:39.100 --> 00:33:41.670
to calculate the
number, you don't yet

00:33:41.670 --> 00:33:42.670
know what the number is.

00:33:42.670 --> 00:33:44.720
But you can make a
comparison to something

00:33:44.720 --> 00:33:48.940
whose convergence properties
you already understand.

00:33:48.940 --> 00:33:50.780
Now, here's the statement.

00:33:50.780 --> 00:33:57.090
If a function, f, is similar
to a function, asymptotically

00:33:57.090 --> 00:34:00.830
the same as a function,
g, as x goes to infinity,

00:34:00.830 --> 00:34:03.830
I'll remind you what
that means in a second.

00:34:03.830 --> 00:34:08.890
Then the integral
starting at some

00:34:08.890 --> 00:34:20.380
point out to infinity of
f(x) dx, and the other one,

00:34:20.380 --> 00:34:22.230
converge and diverge
at the same time.

00:34:22.230 --> 00:34:30.320
So both, either, either-- sorry,
let's try it the other way.

00:34:30.320 --> 00:34:31.940
Either, both.

00:34:31.940 --> 00:34:42.700
Either both converge,
or both diverge.

00:34:42.700 --> 00:34:44.840
They behave exactly
the same way.

00:34:44.840 --> 00:34:50.800
In terms of whether
they're infinite or not.

00:34:50.800 --> 00:34:56.650
And, let me remind you
what this tilde means.

00:34:56.650 --> 00:35:14.750
This thing means that
f(x) / g(x) tends to 1.

00:35:14.750 --> 00:35:19.820
So if you have a couple
of functions like that,

00:35:19.820 --> 00:35:21.990
then their behavior is the same.

00:35:21.990 --> 00:35:25.050
This is more or less obvious.

00:35:25.050 --> 00:35:29.710
It's just because
far enough out,

00:35:29.710 --> 00:35:34.712
this is for large
a, if you like.

00:35:34.712 --> 00:35:36.670
We're not paying any
attention to what happens.

00:35:36.670 --> 00:35:40.400
It just has to do with the
tail, and after a while

00:35:40.400 --> 00:35:42.290
f(x) and g(x) are
comparable to each other.

00:35:42.290 --> 00:35:46.980
So their integrals are
comparable to each other.

00:35:46.980 --> 00:35:51.290
So let's just do a
couple of examples here.

00:35:51.290 --> 00:35:56.500
If you take the integral from 0
to infinity dx over the square

00:35:56.500 --> 00:36:09.400
root of x^2+10, then I claim
that the square root of x^2+10

00:36:09.400 --> 00:36:16.220
resembles the square root
of x^2, which is just x.

00:36:16.220 --> 00:36:19.830
So this thing is going
to be like-- So now I'm

00:36:19.830 --> 00:36:22.570
going to have to do
one thing to you here.

00:36:22.570 --> 00:36:26.180
Which is, I'm going
to change this to 1.

00:36:26.180 --> 00:36:28.870
To infinity.

00:36:28.870 --> 00:36:30.900
dx/x.

00:36:30.900 --> 00:36:35.756
And the reason is that
this x = 0 is extraneous.

00:36:35.756 --> 00:36:37.380
Doesn't have anything
to do with what's

00:36:37.380 --> 00:36:39.710
going on with this problem.

00:36:39.710 --> 00:36:45.640
This guy here, the
piece of it from--

00:36:45.640 --> 00:36:54.660
So we're going to ignore the
part integral from 0 to 1 dx /

00:36:54.660 --> 00:37:01.480
square root of x^2+10,
which is finite anyway.

00:37:01.480 --> 00:37:03.340
And unimportant.

00:37:03.340 --> 00:37:06.300
Whereas, unfortunately,
the integral of dx

00:37:06.300 --> 00:37:08.310
will have a
singularity at x = 0.

00:37:08.310 --> 00:37:12.040
So we can't make the
comparison there.

00:37:12.040 --> 00:37:14.910
Anyway, this one is infinite.

00:37:14.910 --> 00:37:21.490
So this is divergent.

00:37:21.490 --> 00:37:27.340
Using what I knew from before.

00:37:27.340 --> 00:37:27.840
Yeah.

00:37:27.840 --> 00:37:33.680
STUDENT: [INAUDIBLE]

00:37:33.680 --> 00:37:39.770
PROFESSOR: The question is,
why did we switch from 0 to 1?

00:37:39.770 --> 00:37:43.280
So I'm going to say a little
bit more about that later.

00:37:43.280 --> 00:37:48.750
But let me just make
it a warning here.

00:37:48.750 --> 00:37:58.010
Which is that this guy here
is infinite for other reasons.

00:37:58.010 --> 00:38:04.310
Unrelated reasons.

00:38:04.310 --> 00:38:07.650
The comparison that we are
trying to make is with the tail

00:38:07.650 --> 00:38:09.400
as x goes to infinity.

00:38:09.400 --> 00:38:11.920
So another way of
saying this is that I

00:38:11.920 --> 00:38:16.940
should stick an a here and an
a here and stay away from 0.

00:38:16.940 --> 00:38:18.770
So, say a = 1.

00:38:18.770 --> 00:38:21.820
If I make these both
1, that would be OK.

00:38:21.820 --> 00:38:24.700
If I make them both
2, that would be OK.

00:38:24.700 --> 00:38:27.250
If I make them both
100, that would be OK.

00:38:27.250 --> 00:38:29.450
So let's leave it
as 100 right now.

00:38:29.450 --> 00:38:30.640
And it's acceptable.

00:38:30.640 --> 00:38:33.360
I want you to stay away
from the origin here.

00:38:33.360 --> 00:38:36.820
Because that's
another bad point.

00:38:36.820 --> 00:38:40.380
And just talk about what's
happening with the tail.

00:38:40.380 --> 00:38:45.530
So this is a tail, and I also
had a different name for it

00:38:45.530 --> 00:38:46.580
up top.

00:38:46.580 --> 00:38:47.810
Which is emphasizing this.

00:38:47.810 --> 00:38:49.770
Which is limit comparison.

00:38:49.770 --> 00:38:52.740
It's only what's happening at
the very end of the picture

00:38:52.740 --> 00:38:53.920
that we're interested in.

00:38:53.920 --> 00:38:56.710
So again, this is as
x goes to infinity.

00:38:56.710 --> 00:38:59.880
That's the limit we're talking
about, the limiting behavior.

00:38:59.880 --> 00:39:01.770
And we're trying
not to pay attention

00:39:01.770 --> 00:39:10.890
to what's happening
for small values of x.

00:39:10.890 --> 00:39:13.450
So to be consistent, if
I'm going to do it up

00:39:13.450 --> 00:39:17.440
to 100 I'm ignoring what's
happening up to the first 100

00:39:17.440 --> 00:39:25.240
values.

00:39:25.240 --> 00:39:28.950
In any case, this guy diverged.

00:39:28.950 --> 00:39:33.120
And let me give you
another example.

00:39:33.120 --> 00:39:36.050
This one, you could
have computed.

00:39:36.050 --> 00:39:38.040
This one you could
have computed, right?

00:39:38.040 --> 00:39:43.930
Because it's a square
root of quadratic,

00:39:43.930 --> 00:39:48.420
so there's a trig substitution
that evaluates this one.

00:39:48.420 --> 00:39:51.400
The advantage of this
limit comparison method

00:39:51.400 --> 00:39:53.432
is, it makes no
difference whether you

00:39:53.432 --> 00:39:54.640
can compute the thing or not.

00:39:54.640 --> 00:39:57.840
You can still decide whether
it's finite or infinite, fairly

00:39:57.840 --> 00:39:58.820
easily.

00:39:58.820 --> 00:40:10.850
So let me give you
an example of that.

00:40:10.850 --> 00:40:13.030
So here we have another example.

00:40:13.030 --> 00:40:21.560
We'll take the integral
dx, square root of x^3 + 3.

00:40:21.560 --> 00:40:25.080
Let's say, for the
sake of argument.

00:40:25.080 --> 00:40:28.970
From 0 to infinity.

00:40:28.970 --> 00:40:35.200
Let's leave off, let's make
it 10 to infinity, whatever.

00:40:35.200 --> 00:40:42.360
Now this one is
problematic for you.

00:40:42.360 --> 00:40:44.910
You're not going to be able
to evaluate it, I promise.

00:40:44.910 --> 00:40:53.660
So on the other hand 1 over
the square root of x^3 + 3 is

00:40:53.660 --> 00:40:56.950
similar to 1 over the
square root of x^3,

00:40:56.950 --> 00:41:00.580
which is 1/x^(3/2).

00:41:00.580 --> 00:41:10.680
So this thing is going to
resemble this integral here.

00:41:10.680 --> 00:41:16.870
Which is convergent.

00:41:16.870 --> 00:41:25.660
According to our rule.

00:41:25.660 --> 00:41:31.950
So those are the, more or
less the main ingredients.

00:41:31.950 --> 00:41:34.740
Let me just mention one
other integral, which was

00:41:34.740 --> 00:41:37.900
the one that we had over here.

00:41:37.900 --> 00:41:39.830
This one here.

00:41:39.830 --> 00:41:42.300
If you look at this
integral, of course

00:41:42.300 --> 00:41:45.270
we can compute it so we
know the area is finite.

00:41:45.270 --> 00:41:51.910
But the way that you would
actually carry this out,

00:41:51.910 --> 00:41:54.040
if you didn't know
the number and you

00:41:54.040 --> 00:41:58.860
wanted to check that this
integral were finite,

00:41:58.860 --> 00:42:00.880
then you would make the
following comparison.

00:42:00.880 --> 00:42:04.170
This one is not so
difficult. First of all,

00:42:04.170 --> 00:42:07.300
you would write it as twice
the integral from 0 to infinity

00:42:07.300 --> 00:42:11.770
of e^(-x^2) dx.

00:42:11.770 --> 00:42:15.340
This is a new example
here, and we're just

00:42:15.340 --> 00:42:18.970
checking for convergence only.

00:42:18.970 --> 00:42:25.700
Not evaluation.

00:42:25.700 --> 00:42:36.459
And now, I'm going to
make a comparison here,

00:42:36.459 --> 00:42:38.750
Rather than a limit, comparison
I'm actually just going

00:42:38.750 --> 00:42:40.041
to make an ordinary comparison.

00:42:40.041 --> 00:42:42.170
That's because this
thing vanishes so fast.

00:42:42.170 --> 00:42:44.387
It's so favorable that
we can only put something

00:42:44.387 --> 00:42:46.470
on top of it, we can't get
something underneath it

00:42:46.470 --> 00:42:48.280
that exactly balances with it.

00:42:48.280 --> 00:42:50.280
In other words, this
wiggle was something

00:42:50.280 --> 00:42:53.050
which had the same growth
rate as the function involved.

00:42:53.050 --> 00:42:54.870
This thing just vanishes
incredibly fast.

00:42:54.870 --> 00:42:55.830
It's great.

00:42:55.830 --> 00:42:58.500
It's too good for us,
for this comparison.

00:42:58.500 --> 00:43:00.110
So instead what
I'm going to make

00:43:00.110 --> 00:43:02.810
is the following comparison.

00:43:02.810 --> 00:43:07.650
e^(-x^2) <= e^(-x).

00:43:07.650 --> 00:43:10.990
At least for x >= 1.

00:43:10.990 --> 00:43:20.150
When x >= 1, then x^2
>= x, and so -x^2 < -x.

00:43:20.150 --> 00:43:22.830
And so e^(-x^2)
is less than this.

00:43:22.830 --> 00:43:26.440
So this is the
reasoning involved.

00:43:26.440 --> 00:43:29.350
And so what we have
here is two pieces.

00:43:29.350 --> 00:43:33.460
We have 2, the integral
from 0 to 1, of e^(-x^2).

00:43:33.460 --> 00:43:35.430
That's just a finite part.

00:43:35.430 --> 00:43:37.330
And then we have
this other part,

00:43:37.330 --> 00:43:42.490
which I'm going to replace
with the e^(-x) here.

00:43:42.490 --> 00:43:50.310
2 times 1 to infinity e^(-x) dx.

00:43:50.310 --> 00:43:53.939
So this is, if you like,
this is ordinary comparison

00:43:53.939 --> 00:43:54.480
of integrals.

00:43:54.480 --> 00:43:57.021
It's something that we did way
at the beginning of the class.

00:43:57.021 --> 00:43:59.750
Or much earlier on, when we
were dealing with integrals.

00:43:59.750 --> 00:44:04.750
Which is that if you have
a larger integrand, then

00:44:04.750 --> 00:44:07.519
the integral gets larger.

00:44:07.519 --> 00:44:08.810
So we've replaced the integral.

00:44:08.810 --> 00:44:11.360
We've got the same
integrand on 0 to 1.

00:44:11.360 --> 00:44:14.120
And we have a larger
integrand on--

00:44:14.120 --> 00:44:20.680
So this one is larger integrand.

00:44:20.680 --> 00:44:23.200
And this one we know is finite.

00:44:23.200 --> 00:44:24.830
This one is a
convergent integral.

00:44:24.830 --> 00:44:29.760
So the whole business
is convergent.

00:44:29.760 --> 00:44:31.970
But of course we replaced
it by a much larger thing.

00:44:31.970 --> 00:44:34.053
So we're not getting the
right number out of this.

00:44:34.053 --> 00:44:47.920
We're just showing
that it converges.

00:44:47.920 --> 00:44:51.060
So these are the
main ingredients.

00:44:51.060 --> 00:44:54.380
As I say, once the thing gets
really, really fast-decaying,

00:44:54.380 --> 00:44:57.310
it's relatively straightforward.

00:44:57.310 --> 00:45:04.790
There's lots of room to
show that it converges.

00:45:04.790 --> 00:45:07.020
Now, there's one
last item of business

00:45:07.020 --> 00:45:10.080
here which I have
to promise you.

00:45:10.080 --> 00:45:16.630
Which I promised you, which
had to do with dealing

00:45:16.630 --> 00:45:22.220
with this bottom piece here.

00:45:22.220 --> 00:45:24.210
So I have to deal
with what happens

00:45:24.210 --> 00:45:26.050
when there's a singularity.

00:45:26.050 --> 00:45:56.800
This is known as an improper
integral of the second type.

00:45:56.800 --> 00:46:01.200
And the idea of these
examples is the following.

00:46:01.200 --> 00:46:06.010
You might have
something like this.

00:46:06.010 --> 00:46:11.200
Something like this.

00:46:11.200 --> 00:46:16.770
Or something like this.

00:46:16.770 --> 00:46:20.690
These are typical
sorts of examples.

00:46:20.690 --> 00:46:27.300
And before actually
describing what happens,

00:46:27.300 --> 00:46:28.540
I just want to mention.

00:46:28.540 --> 00:46:31.020
So first of all,
the key point here

00:46:31.020 --> 00:46:32.760
is you can just
calculate these things.

00:46:32.760 --> 00:46:37.940
And plug in 0 and it works and
you'll get the right answer.

00:46:37.940 --> 00:46:40.254
So you'll determine,
you'll figure out,

00:46:40.254 --> 00:46:42.170
that it turns out that
this one will converge,

00:46:42.170 --> 00:46:44.211
this one will diverge,
and this one will diverge.

00:46:44.211 --> 00:46:46.340
That's what will
turn out to happen.

00:46:46.340 --> 00:46:50.610
However, I want to warn you
that you can fool yourself.

00:46:50.610 --> 00:46:53.150
And so let me give you a
slightly different example.

00:46:53.150 --> 00:46:58.410
Let's consider
this integral here.

00:46:58.410 --> 00:47:01.360
The integral from
-1 to 1 dx / x^2.

00:47:05.850 --> 00:47:09.246
If you carry out this
integral without thinking,

00:47:09.246 --> 00:47:11.370
what will happen is, you'll
get the antiderivative,

00:47:11.370 --> 00:47:16.250
which is -x^(-1),
evaluated at -1 and 1.

00:47:16.250 --> 00:47:20.550
And you plug it in.

00:47:20.550 --> 00:47:21.520
And what do you get?

00:47:21.520 --> 00:47:30.880
You get -1^(-1) minus,
uh-oh, minus (-1)^(-1).

00:47:30.880 --> 00:47:33.610
There's a lot of
-1's in this problem.

00:47:33.610 --> 00:47:35.090
OK, so that's -1.

00:47:35.090 --> 00:47:37.010
And this one, if
you work it all out,

00:47:37.010 --> 00:47:39.340
as I sometimes don't get the
signs right, but this time

00:47:39.340 --> 00:47:41.160
I really paid attention.

00:47:41.160 --> 00:47:44.480
It's -1, I'm telling
you that's what it is.

00:47:44.480 --> 00:47:46.880
So that comes out to be -2.

00:47:46.880 --> 00:47:50.450
Now, this is ridiculous.

00:47:50.450 --> 00:48:01.400
This function here
looks like this.

00:48:01.400 --> 00:48:04.140
It's positive, right?

00:48:04.140 --> 00:48:06.000
1/x^2 is positive.

00:48:06.000 --> 00:48:10.030
How exactly is it that
the area between -1 and 1

00:48:10.030 --> 00:48:13.990
came out to be a
negative number?

00:48:13.990 --> 00:48:16.510
That can't be.

00:48:16.510 --> 00:48:18.420
There was clearly
something wrong with this.

00:48:18.420 --> 00:48:21.210
And this is the kind of thing
that you'll get regularly

00:48:21.210 --> 00:48:25.740
if you don't pay attention
to convergence of integrals.

00:48:25.740 --> 00:48:29.640
So what's going on here is
actually that this area in here

00:48:29.640 --> 00:48:33.130
is infinite.

00:48:33.130 --> 00:48:38.090
And this calculation
that I made is nonsense.

00:48:38.090 --> 00:48:41.060
So it doesn't work.

00:48:41.060 --> 00:48:42.010
This is wrong.

00:48:42.010 --> 00:48:50.237
Because it's divergent.

00:48:50.237 --> 00:48:52.070
Actually, when you get
to imaginary numbers,

00:48:52.070 --> 00:48:56.710
it'll turn out that there's
a way of rescuing it.

00:48:56.710 --> 00:48:58.980
But, still, it means something
totally different when

00:48:58.980 --> 00:49:03.160
that integral is thought to -2.

00:49:03.160 --> 00:49:04.000
So.

00:49:04.000 --> 00:49:07.570
What I want you to
do here, so I think

00:49:07.570 --> 00:49:11.590
we'll have to finish this
up very briefly next time.

00:49:11.590 --> 00:49:13.270
We'll do these
three calculations

00:49:13.270 --> 00:49:20.500
and you'll see that these two
guys are divergent and this one

00:49:20.500 --> 00:49:21.990
converges.

00:49:21.990 --> 00:49:24.110
And we'll do that next time.