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PROFESSOR: Today
we are continuing
00:00:22.760 --> 00:00:24.730
with improper integrals.
00:00:24.730 --> 00:00:27.371
I still have a little bit
more to tell you about them.
00:00:30.080 --> 00:00:33.510
What we were discussing at
the very end of last time
00:00:33.510 --> 00:00:36.195
was improper integrals.
00:00:41.560 --> 00:00:44.570
Now and these are going
to be improper integrals
00:00:44.570 --> 00:00:48.220
of the second kind.
00:00:48.220 --> 00:00:50.990
By second kind I mean that
they have a singularity
00:00:50.990 --> 00:00:52.095
at a finite place.
00:00:54.800 --> 00:00:57.260
That would be
something like this.
00:00:57.260 --> 00:01:00.140
So here's the
definition if you like.
00:01:00.140 --> 00:01:03.380
Same sort of thing as we
did when the singularity
00:01:03.380 --> 00:01:04.120
was at infinity.
00:01:04.120 --> 00:01:09.440
So if you have the integral
from 0 to 1 of f(x).
00:01:09.440 --> 00:01:12.600
This is going to be the
same thing as the limit,
00:01:12.600 --> 00:01:17.780
as a goes to 0 from above,
the integral from a to 1
00:01:17.780 --> 00:01:21.150
of f(x) dx.
00:01:21.150 --> 00:01:25.920
And the idea here is the same
one that we had at infinity.
00:01:25.920 --> 00:01:27.320
Let me draw a picture of it.
00:01:27.320 --> 00:01:30.030
You have, imagine a function
which is coming down like this
00:01:30.030 --> 00:01:32.020
and here's the point 1.
00:01:32.020 --> 00:01:35.180
And we don't know whether
the area enclosed is
00:01:35.180 --> 00:01:39.040
going to be infinite or
finite and so we cut it off
00:01:39.040 --> 00:01:40.680
at some place a.
00:01:40.680 --> 00:01:44.220
And we let a go to 0 from above.
00:01:44.220 --> 00:01:46.800
So really it's 0+.
00:01:46.800 --> 00:01:49.820
So we're coming in
from the right here.
00:01:49.820 --> 00:01:52.810
And we're counting up
the area in this chunk.
00:01:52.810 --> 00:01:56.520
And we're seeing as it expands
whether it goes to infinity
00:01:56.520 --> 00:01:59.290
or whether it tends
to some finite limit.
00:01:59.290 --> 00:02:02.990
Right, so this is the example
and this is the definition.
00:02:02.990 --> 00:02:07.000
And just as we did for the
other kind of improper integral,
00:02:07.000 --> 00:02:13.747
we say that this converges --
so that's the key word here --
00:02:13.747 --> 00:02:24.220
if the limit is finite,
exists maybe I should just say
00:02:24.220 --> 00:02:30.773
and diverges if not.
00:02:35.650 --> 00:02:38.945
Let's just take care
of the basic examples.
00:02:42.110 --> 00:02:44.740
First of all I wrote
this one down last time.
00:02:44.740 --> 00:02:47.390
We're going to
evaluate this one.
00:02:47.390 --> 00:02:51.565
The integral from 0 to 1 of
1 over the square root of x.
00:02:55.402 --> 00:02:57.360
And this just, you almost
don't notice the fact
00:02:57.360 --> 00:02:59.920
that it goes to infinity.
00:02:59.920 --> 00:03:02.570
This goes to infinity
as x goes to 0.
00:03:02.570 --> 00:03:05.860
But if you evaluate it -- first
of all we always write this
00:03:05.860 --> 00:03:06.860
as a power.
00:03:06.860 --> 00:03:08.010
Right?
00:03:08.010 --> 00:03:09.870
To get the evaluation.
00:03:09.870 --> 00:03:13.060
And then I'm not even going
to replace the 0 by an a.
00:03:13.060 --> 00:03:14.490
I'm just going to leave it as 0.
00:03:14.490 --> 00:03:18.660
The antiderivative here
is x^(1/2) times 2.
00:03:21.540 --> 00:03:23.750
And then I evaluate
that at 0 and 1.
00:03:23.750 --> 00:03:25.080
And I get 2.
00:03:25.080 --> 00:03:29.090
2 minus 0, which is 2.
00:03:29.090 --> 00:03:31.330
All right so this
one is convergent.
00:03:31.330 --> 00:03:34.180
And not only is it convergent
but we can evaluate it.
00:03:38.310 --> 00:03:42.140
The second example,
being not systematic
00:03:42.140 --> 00:03:44.300
but really giving you
the principal examples
00:03:44.300 --> 00:03:51.650
that we'll be thinking about,
is this one here, dx / x.
00:03:51.650 --> 00:03:54.200
And this one gives
you the antiderivative
00:03:54.200 --> 00:03:56.070
as the logarithm.
00:03:56.070 --> 00:03:58.180
Evaluated at 0 and 1.
00:03:58.180 --> 00:04:00.190
And now again you have
to have this thought
00:04:00.190 --> 00:04:02.770
process in your mind that
you're really taking the limit.
00:04:02.770 --> 00:04:06.440
But this is going to be the
log of 1 minus the log of 0.
00:04:06.440 --> 00:04:07.830
Really the log of 0 from above.
00:04:07.830 --> 00:04:10.970
There is no such thing as
the log of 0 from below.
00:04:10.970 --> 00:04:12.820
And this is minus infinity.
00:04:12.820 --> 00:04:19.020
So it's 0 minus minus infinity,
which is plus infinity.
00:04:19.020 --> 00:04:20.213
And so this one diverges.
00:04:29.710 --> 00:04:32.750
All right so what's
the general--
00:04:32.750 --> 00:04:39.070
So more or less in general,
let's just, for powers anyway,
00:04:39.070 --> 00:04:45.220
if you work out this thing
for dx / x^p from 0 to 1.
00:04:45.220 --> 00:04:51.820
What you're going to find is
that it's 1/(1-p) when p is
00:04:51.820 --> 00:04:53.020
less than 1.
00:04:53.020 --> 00:05:00.240
And it diverges for p >= 1.
00:05:02.750 --> 00:05:07.880
Now that's the final result. If
you carry out this integration
00:05:07.880 --> 00:05:10.920
it's not difficult.
00:05:10.920 --> 00:05:14.850
All right so now
I just want to try
00:05:14.850 --> 00:05:17.270
to help you to remember this.
00:05:17.270 --> 00:05:20.500
And to think about how
you should think about it.
00:05:20.500 --> 00:05:23.890
So I'm going to say
it in a few more ways.
00:05:23.890 --> 00:05:26.850
All right just repeat
what I've said already
00:05:26.850 --> 00:05:32.020
but try to get it to
percolate and absorb itself.
00:05:32.020 --> 00:05:34.270
And in order to do
that I have to make
00:05:34.270 --> 00:05:37.750
the contrast between the
kind of improper integral
00:05:37.750 --> 00:05:39.410
that I was dealing with before.
00:05:39.410 --> 00:05:43.720
Which was not as x goes to 0
here but as x goes to infinity,
00:05:43.720 --> 00:05:45.920
the other side.
00:05:45.920 --> 00:05:47.023
Let's make this contrast.
00:05:52.470 --> 00:05:55.610
First of all, if I
look at the angle
00:05:55.610 --> 00:05:57.610
that we have been paying
attention to right now.
00:05:57.610 --> 00:06:00.580
We've just considered
things like this.
00:06:00.580 --> 00:06:02.520
1 over x to the 1/2.
00:06:02.520 --> 00:06:06.600
Which is a lot smaller than 1/x.
00:06:06.600 --> 00:06:10.740
Which is a lot smaller
than say 1/x^2.
00:06:10.740 --> 00:06:12.050
Which would be another example.
00:06:12.050 --> 00:06:14.560
This is as x goes to 0.
00:06:19.990 --> 00:06:21.935
So this one's the smallest one.
00:06:21.935 --> 00:06:23.310
This one's the
next smallest one.
00:06:23.310 --> 00:06:26.820
And this one is very large.
00:06:26.820 --> 00:06:30.985
On the other hand it goes
the other way at infinity.
00:06:36.670 --> 00:06:39.936
As x tends to infinity.
00:06:39.936 --> 00:06:43.140
All right so try to
keep that in mind.
00:06:43.140 --> 00:06:48.760
And now I'm going to put a
little box around the bad guys
00:06:48.760 --> 00:06:50.100
here.
00:06:50.100 --> 00:06:54.710
This one is divergent.
00:06:54.710 --> 00:06:57.870
And this one is divergent.
00:06:57.870 --> 00:06:59.950
And this one is divergent.
00:06:59.950 --> 00:07:01.610
And this one is divergent.
00:07:01.610 --> 00:07:03.510
The crossover point is 1/x.
00:07:03.510 --> 00:07:05.680
When we get smaller
than that, we
00:07:05.680 --> 00:07:07.390
get to things which
are convergent.
00:07:07.390 --> 00:07:10.590
When we get smaller than
it on this other scale,
00:07:10.590 --> 00:07:12.412
it's convergent.
00:07:12.412 --> 00:07:13.995
All right so these
guys are divergent.
00:07:20.300 --> 00:07:23.430
So they're associated
with divergent integrals.
00:07:23.430 --> 00:07:25.390
The functions
themselves are just
00:07:25.390 --> 00:07:27.585
tending towards-- well
these tend to infinity,
00:07:27.585 --> 00:07:29.310
and these tend toward 0.
00:07:29.310 --> 00:07:33.830
So I'm not talking about
the functions themselves
00:07:33.830 --> 00:07:35.190
but the integrals.
00:07:35.190 --> 00:07:40.040
Now I want to draw this
again here, not small enough.
00:07:43.757 --> 00:07:44.840
I want to draw this again.
00:07:48.484 --> 00:07:50.150
And, so I'm just going
to draw a picture
00:07:50.150 --> 00:07:51.760
of what it is that I have here.
00:07:51.760 --> 00:07:54.850
But I'm going to combine
these two pictures.
00:07:54.850 --> 00:08:01.330
So here's the picture
for example of y = 1/x.
00:08:04.714 --> 00:08:06.700
All right.
00:08:06.700 --> 00:08:08.770
That's y y = 1/x.
00:08:08.770 --> 00:08:10.470
And that picture
is very balanced.
00:08:10.470 --> 00:08:12.600
It's symmetric on the two ends.
00:08:12.600 --> 00:08:17.660
If I cut it in half then what
I get here is two halves.
00:08:17.660 --> 00:08:23.700
And this one has infinite area.
00:08:23.700 --> 00:08:27.730
That corresponds to the
integral from 1 to infinity,
00:08:27.730 --> 00:08:30.530
dx / x being infinite.
00:08:30.530 --> 00:08:34.810
And the other piece, which --
this one we calculated last
00:08:34.810 --> 00:08:37.230
time, this is the one that
we just calculated over here
00:08:37.230 --> 00:08:42.719
at Example 2 -- has
the same property.
00:08:42.719 --> 00:08:45.720
It's infinite.
00:08:45.720 --> 00:08:48.102
And that's the fact that the
integral from 0 to 1 of dx
00:08:48.102 --> 00:08:52.100
/ x is infinite.
00:08:52.100 --> 00:08:56.570
Right, so both, we
lose on both ends.
00:08:56.570 --> 00:09:02.390
On the other hand if I
take something like --
00:09:02.390 --> 00:09:05.480
I'm drawing it the same way
but it's really not the same --
00:09:05.480 --> 00:09:08.490
y = 1 over the square root of x.
00:09:08.490 --> 00:09:11.010
y = 1 / x^(1/2).
00:09:11.010 --> 00:09:17.980
And if I cut that in half here
then the x^(1/2) is actually
00:09:17.980 --> 00:09:19.890
bigger than this guy.
00:09:19.890 --> 00:09:21.740
So this piece is infinite.
00:09:26.830 --> 00:09:29.555
And this part over
here actually is going
00:09:29.555 --> 00:09:31.310
to give us an honest number.
00:09:31.310 --> 00:09:34.810
In fact this one is finite.
00:09:34.810 --> 00:09:36.550
And we just checked
what the number is.
00:09:36.550 --> 00:09:38.340
It actually happens
to have area 2.
00:09:46.250 --> 00:09:49.210
And what's happening here
is if you would superimpose
00:09:49.210 --> 00:09:51.710
this graph on the
other graph what you
00:09:51.710 --> 00:09:54.580
would see is that they cross.
00:09:54.580 --> 00:09:58.970
And this one sits on top.
00:09:58.970 --> 00:10:04.660
So if I drew this one in
let's have another color here,
00:10:04.660 --> 00:10:05.840
orange let's say.
00:10:05.840 --> 00:10:08.630
If this were orange if
I set it on top here
00:10:08.630 --> 00:10:11.130
it would go this way.
00:10:11.130 --> 00:10:14.790
OK and underneath the
orange is still infinite.
00:10:14.790 --> 00:10:16.070
So both of these are infinite.
00:10:16.070 --> 00:10:18.028
On here on the other hand
underneath the orange
00:10:18.028 --> 00:10:21.700
is infinite but underneath
where the green is is finite.
00:10:21.700 --> 00:10:23.770
That's a smaller quantity.
00:10:23.770 --> 00:10:25.290
Infinity is a lot bigger than 2.
00:10:25.290 --> 00:10:27.826
2 is a lot less than infinity.
00:10:27.826 --> 00:10:30.835
All right so that's reflected
in these comparisons here.
00:10:30.835 --> 00:10:33.610
Now if you like if I want
to do these in green.
00:10:33.610 --> 00:10:39.900
This guy is good and
this guy is good.
00:10:39.900 --> 00:10:42.960
Well let me just repeat that
idea over here in this sort
00:10:42.960 --> 00:10:47.930
of reversed picture
with y = 1/x^2.
00:10:47.930 --> 00:10:53.640
If I chop that in half then
the good part is this end here.
00:10:53.640 --> 00:10:54.440
This is finite.
00:10:56.990 --> 00:10:59.570
And the bad part is
this part of here
00:10:59.570 --> 00:11:01.310
which is way more singular.
00:11:01.310 --> 00:11:02.060
And it's infinite.
00:11:07.070 --> 00:11:10.080
All right so again what
I've just tried to do
00:11:10.080 --> 00:11:16.710
is to give you some
geometric sense and also
00:11:16.710 --> 00:11:18.580
some visceral sense.
00:11:18.580 --> 00:11:23.040
This guy, its tail as it goes
out to infinity is much lower.
00:11:23.040 --> 00:11:25.470
It's much smaller than 1/x.
00:11:25.470 --> 00:11:28.080
And these guys trapped an
infinite amount of area.
00:11:28.080 --> 00:11:30.110
This one traps only a
finite amount of area.
00:11:36.676 --> 00:11:38.860
All right so now I'm
just going to give one
00:11:38.860 --> 00:11:43.090
last example which combines
these two types of pictures.
00:11:43.090 --> 00:11:45.420
It's really practically
the same as what
00:11:45.420 --> 00:11:53.320
I've said before but I-- oh
have to erase this one too.
00:12:01.340 --> 00:12:05.770
So here's another example:
if you're in-- So let's
00:12:05.770 --> 00:12:08.132
take the following example.
00:12:08.132 --> 00:12:09.840
This is somewhat
related to the first one
00:12:09.840 --> 00:12:11.600
that I gave last time.
00:12:11.600 --> 00:12:16.090
If you take a function
y = 1/(x-3)^2.
00:12:18.770 --> 00:12:21.160
And you think
about its integral.
00:12:21.160 --> 00:12:24.750
So let's think about the
integral from 0 to infinity,
00:12:24.750 --> 00:12:27.090
dx / (x-3)^2.
00:12:27.090 --> 00:12:30.420
And suppose you were
faced with this integral.
00:12:30.420 --> 00:12:32.630
In order to understand
what it's doing
00:12:32.630 --> 00:12:36.310
you have to pay attention to two
places where it can go wrong.
00:12:36.310 --> 00:12:39.530
We're going to split
into two pieces.
00:12:39.530 --> 00:12:44.349
I'm going say break it up
into this one here up to 5,
00:12:44.349 --> 00:12:45.390
for the sake of argument.
00:12:48.090 --> 00:12:49.715
And say from 5 to infinity.
00:12:53.672 --> 00:12:54.960
All right.
00:12:54.960 --> 00:12:56.260
So these are the two chunks.
00:12:56.260 --> 00:12:58.810
Now why did I break it
up into those two pieces?
00:12:58.810 --> 00:13:00.670
Because what's happening
with this function
00:13:00.670 --> 00:13:05.250
is that it's going
up like this at 3.
00:13:05.250 --> 00:13:08.210
And so if I look at
the two halves here.
00:13:08.210 --> 00:13:09.710
I'm going to draw
them again and I'm
00:13:09.710 --> 00:13:12.560
going to illustrate them
with the colors we've chosen,
00:13:12.560 --> 00:13:15.350
which are I guess red and green.
00:13:15.350 --> 00:13:20.660
What you'll discover
is that this one
00:13:20.660 --> 00:13:30.540
here, which corresponds to
this piece here, is infinite.
00:13:30.540 --> 00:13:32.320
And it's infinite
because there's
00:13:32.320 --> 00:13:34.270
a square in the denominator.
00:13:34.270 --> 00:13:39.190
And as x goes to 3 this
is very much like if we
00:13:39.190 --> 00:13:40.670
shifted the 3 to 0.
00:13:40.670 --> 00:13:42.560
Very much like this 1/x^2 here.
00:13:42.560 --> 00:13:44.247
But not in this context.
00:13:44.247 --> 00:13:46.330
In the other context where
it's going to infinity.
00:13:49.460 --> 00:13:52.360
This is the same
as at the picture
00:13:52.360 --> 00:13:57.398
directly above with the
infinite part in red.
00:13:57.398 --> 00:13:59.650
All right.
00:13:59.650 --> 00:14:04.915
And this part here,
this part is finite.
00:14:04.915 --> 00:14:06.240
All right.
00:14:06.240 --> 00:14:08.960
So since we have an infinite
part plus a finite part
00:14:08.960 --> 00:14:15.070
the conclusion is that this
thing, well this guy converges.
00:14:15.070 --> 00:14:18.850
And this one diverges.
00:14:21.470 --> 00:14:23.930
But the total
unfortunately diverges.
00:14:23.930 --> 00:14:25.760
Right, because it's
got one infinity in it.
00:14:25.760 --> 00:14:28.370
So this thing diverges.
00:14:31.964 --> 00:14:33.380
And that's what
happened last time
00:14:33.380 --> 00:14:34.870
when we got a crazy number.
00:14:34.870 --> 00:14:37.480
If you integrated this you
would get some negative number.
00:14:37.480 --> 00:14:39.850
If you wrote down the
formulas carelessly.
00:14:39.850 --> 00:14:42.250
And the reason is that
the calculation actually
00:14:42.250 --> 00:14:44.710
is nonsense.
00:14:44.710 --> 00:14:47.890
So you've gotta be
aware, if you encounter
00:14:47.890 --> 00:14:51.600
a singularity in the
middle, not to ignore it.
00:14:51.600 --> 00:14:52.100
Yeah.
00:14:52.100 --> 00:14:52.620
Question.
00:14:52.620 --> 00:14:53.786
AUDIENCE: [INAUDIBLE PHRASE]
00:14:56.430 --> 00:14:59.710
PROFESSOR: Why do we say that
the whole thing diverges?
00:14:59.710 --> 00:15:02.490
The reason why we say that is
the area under the whole curve
00:15:02.490 --> 00:15:03.690
is infinite.
00:15:03.690 --> 00:15:06.130
It's the sum of this
piece plus this piece.
00:15:06.130 --> 00:15:08.218
And so the total is infinite.
00:15:08.218 --> 00:15:09.384
AUDIENCE: [INAUDIBLE PHRASE]
00:15:17.032 --> 00:15:17.990
PROFESSOR: We're stuck.
00:15:17.990 --> 00:15:19.323
This is an ill-defined integral.
00:15:19.323 --> 00:15:22.092
It's one where your red flashing
warning sign should be on.
00:15:22.092 --> 00:15:23.800
Because you're not
going to get the right
00:15:23.800 --> 00:15:24.758
answer by computing it.
00:15:24.758 --> 00:15:26.860
You'll never get an answer.
00:15:26.860 --> 00:15:29.230
Similarly you'll never
get an answer with this.
00:15:29.230 --> 00:15:32.700
And you will get an
answer with that.
00:15:32.700 --> 00:15:33.200
OK?
00:15:37.220 --> 00:15:38.944
Yeah another question.
00:15:38.944 --> 00:15:40.110
AUDIENCE: [INAUDIBLE PHRASE]
00:15:45.760 --> 00:15:47.690
PROFESSOR: So the
question is, if you
00:15:47.690 --> 00:15:50.010
have a little glance
at an integral,
00:15:50.010 --> 00:15:54.750
how are you going to decide
where you should be heading?
00:15:54.750 --> 00:15:58.280
So I'm going to
answer that orally.
00:15:58.280 --> 00:16:04.010
Although you know, but I'll
say one little hint here.
00:16:04.010 --> 00:16:08.070
So you always have to check
x going to infinity and x
00:16:08.070 --> 00:16:10.740
going to minus infinity,
if they're in there.
00:16:10.740 --> 00:16:15.460
And you also have to check any
singularity, like x going to 3
00:16:15.460 --> 00:16:16.419
for sure in this case.
00:16:16.419 --> 00:16:18.210
You have to pay attention
to all the places
00:16:18.210 --> 00:16:19.376
where the thing is infinite.
00:16:19.376 --> 00:16:22.330
And then you want to focus
in on each one separately.
00:16:22.330 --> 00:16:26.920
And decide what's going on
it at that particular place.
00:16:26.920 --> 00:16:34.530
When it's a negative power-- So
remember dx / x as x goes to 0
00:16:34.530 --> 00:16:36.980
is bad.
00:16:36.980 --> 00:16:39.510
And dx / x^2 is bad.
00:16:39.510 --> 00:16:40.490
dx / x^3 is bad.
00:16:40.490 --> 00:16:42.600
All of them are even worse.
00:16:42.600 --> 00:16:49.650
So anything of this form
is bad: n = 1, 2, 3.
00:16:49.650 --> 00:16:51.980
These are the red box kinds.
00:16:51.980 --> 00:16:55.030
All right.
00:16:55.030 --> 00:16:56.940
That means that any
of the integrals
00:16:56.940 --> 00:16:59.990
that we did in partial
fractions which
00:16:59.990 --> 00:17:02.080
had a root, which had
a factor of something
00:17:02.080 --> 00:17:03.160
in the denominator.
00:17:03.160 --> 00:17:04.790
Those are all
divergent integrals
00:17:04.790 --> 00:17:06.670
if you cross the singularly.
00:17:06.670 --> 00:17:09.357
Not a single one of them makes
sense across the singularity.
00:17:09.357 --> 00:17:09.856
Right?
00:17:12.692 --> 00:17:14.150
If you have square
roots and things
00:17:14.150 --> 00:17:16.108
like that then you can
repair things like that.
00:17:16.108 --> 00:17:18.030
And there's some interesting
examples of that.
00:17:18.030 --> 00:17:21.080
Such as with the arcsine
function and so forth.
00:17:21.080 --> 00:17:25.940
Where you have an improper
integral which is really OK.
00:17:25.940 --> 00:17:26.730
All right.
00:17:26.730 --> 00:17:29.880
So that's the best I can do.
00:17:29.880 --> 00:17:32.260
It's obviously something
you get experience with.
00:17:32.260 --> 00:17:34.110
All right.
00:17:34.110 --> 00:17:39.350
Now I'm going to move on
and this is more or less
00:17:39.350 --> 00:17:42.150
our last topic.
00:17:42.150 --> 00:17:43.900
Yay, but not quite.
00:17:43.900 --> 00:17:46.730
Well, so I should say it's
our penultimate topic.
00:17:46.730 --> 00:17:49.430
Right because we have
one more lecture.
00:17:49.430 --> 00:17:52.010
All right.
00:17:52.010 --> 00:17:54.750
So that our next
topic is series.
00:17:54.750 --> 00:17:58.810
Now we'll do it in a sort
of a concrete way today.
00:17:58.810 --> 00:18:02.775
And then we'll do what are
known as power series tomorrow.
00:18:05.400 --> 00:18:06.760
So let me tell you about series.
00:18:20.490 --> 00:18:22.604
Remember we're
talking about infinity
00:18:22.604 --> 00:18:23.687
and dealing with infinity.
00:18:26.772 --> 00:18:28.730
So we're not just talking
about any old series.
00:18:28.730 --> 00:18:30.230
We're talking about
infinite series.
00:18:32.650 --> 00:18:37.400
There is one infinite
series which is probably,
00:18:37.400 --> 00:18:39.000
which is without
question the most
00:18:39.000 --> 00:18:41.980
important and useful series.
00:18:41.980 --> 00:18:44.690
And that's the
geometric series but I'm
00:18:44.690 --> 00:18:48.405
going to introduce it concretely
first in a particular case.
00:18:51.990 --> 00:18:54.260
If I draw a picture of this sum.
00:18:54.260 --> 00:18:56.480
Which in principle
goes on forever.
00:18:56.480 --> 00:18:59.670
You can see that it goes
someplace fairly easily
00:18:59.670 --> 00:19:02.730
by marking out what's
happening on the number line.
00:19:02.730 --> 00:19:06.630
The first step takes
us to 1 from 0.
00:19:06.630 --> 00:19:11.790
And then if I add this
half, I get to 3/2.
00:19:11.790 --> 00:19:16.240
Right, so the first step was
1 and the second step was 1/2.
00:19:16.240 --> 00:19:20.940
Now if I add this quarter in,
which is the next piece then
00:19:20.940 --> 00:19:22.400
I get some place here.
00:19:22.400 --> 00:19:28.440
But what I want to
observe is that I got,
00:19:28.440 --> 00:19:31.130
I can look at it from
the other point of view.
00:19:31.130 --> 00:19:36.240
I got, when I move this quarter
I got half way to 2 here.
00:19:36.240 --> 00:19:39.420
I'm putting 2 in green
because I want you to think
00:19:39.420 --> 00:19:42.950
of it as being the good kind.
00:19:42.950 --> 00:19:43.970
Right.
00:19:43.970 --> 00:19:45.420
The kind that has a number.
00:19:45.420 --> 00:19:47.280
And not one of the red kinds.
00:19:47.280 --> 00:19:49.920
We're getting there
and we're almost there.
00:19:49.920 --> 00:19:52.620
So the next stage we
get half way again.
00:19:52.620 --> 00:19:54.605
That's the eighth and so forth.
00:19:54.605 --> 00:19:56.490
And eventually we get to 2.
00:19:56.490 --> 00:20:00.476
So this sum we write equals two.
00:20:00.476 --> 00:20:02.590
All right that's
kind of a paradox
00:20:02.590 --> 00:20:04.130
because we never get to 2.
00:20:04.130 --> 00:20:08.080
This is the paradox
that Zeno fussed with.
00:20:08.080 --> 00:20:12.807
And his conclusion, you know,
with the rabbit and the hare.
00:20:12.807 --> 00:20:14.140
No, the rabbit and the tortoise.
00:20:14.140 --> 00:20:18.890
Sorry hare chasing-- anyway,
the rabbit chasing the tortoise.
00:20:18.890 --> 00:20:21.050
His conclusion--
you know, I don't
00:20:21.050 --> 00:20:23.770
know if you're aware of this,
but he understood this paradox.
00:20:23.770 --> 00:20:25.811
And he said you know it
doesn't look like it ever
00:20:25.811 --> 00:20:29.180
gets there because they're
infinitely many times
00:20:29.180 --> 00:20:32.330
between the time-- you know that
the tortoise is always behind,
00:20:32.330 --> 00:20:34.580
always behind, always
behind, always behind.
00:20:34.580 --> 00:20:36.970
So therefore it's
impossible that the tortoise
00:20:36.970 --> 00:20:38.710
catches up right.
00:20:38.710 --> 00:20:41.600
So do you know what
his conclusion was?
00:20:41.600 --> 00:20:45.280
Time does not exist.
00:20:45.280 --> 00:20:48.007
That was actually
literally his conclusion.
00:20:48.007 --> 00:20:49.840
Because he didn't
understand the possibility
00:20:49.840 --> 00:20:50.799
of a continuum of time.
00:20:50.799 --> 00:20:53.090
Because there were infinitely
many things that happened
00:20:53.090 --> 00:20:56.610
before the tortoise caught up.
00:20:56.610 --> 00:20:57.790
So that was the reasoning.
00:20:57.790 --> 00:21:00.430
I mean it's a long time ago
but you know people didn't-- he
00:21:00.430 --> 00:21:02.430
didn't believe in continuum.
00:21:02.430 --> 00:21:03.480
All right.
00:21:03.480 --> 00:21:06.940
So anyway that's a small point.
00:21:06.940 --> 00:21:19.410
Now the general case here
of a geometric series
00:21:19.410 --> 00:21:22.820
is where I put in a number
a instead of 1/2 here.
00:21:22.820 --> 00:21:23.870
So what we had before.
00:21:23.870 --> 00:21:26.890
So that's 1 + a + a^2...
00:21:26.890 --> 00:21:31.900
Isn't quite the most general
but anyway I'll write this down.
00:21:31.900 --> 00:21:34.630
And you're certainly going
to want to remember that
00:21:34.630 --> 00:21:39.510
the formula for this in
the limit is 1/(1-a).
00:21:39.510 --> 00:21:44.000
And I remind you that this only
works when the absolute value
00:21:44.000 --> 00:21:45.810
is strictly less than 1.
00:21:45.810 --> 00:21:47.970
In other words when -1
is strictly less than a
00:21:47.970 --> 00:21:51.130
is less than 1.
00:21:51.130 --> 00:21:53.090
And that's really the
issue that we're going
00:21:53.090 --> 00:21:54.310
to want to worry about now.
00:21:54.310 --> 00:21:58.270
What we're worrying about is
this notion of convergence.
00:21:58.270 --> 00:22:05.300
And what goes wrong when
there isn't convergence,
00:22:05.300 --> 00:22:07.360
when there's a divergence.
00:22:07.360 --> 00:22:13.220
So let me illustrate the
divergences before going on.
00:22:13.220 --> 00:22:15.900
And this is what we
have to avoid if we're
00:22:15.900 --> 00:22:18.990
going to understand series.
00:22:18.990 --> 00:22:21.890
So here's an example when a = 1.
00:22:21.890 --> 00:22:26.620
You get 1 + 1 +
1 plus et cetera.
00:22:26.620 --> 00:22:29.990
And that's equal to 1/(1-1).
00:22:29.990 --> 00:22:32.370
Which is 1 over 0.
00:22:32.370 --> 00:22:33.760
So this is not bad.
00:22:33.760 --> 00:22:34.861
It's almost right.
00:22:34.861 --> 00:22:35.360
Right?
00:22:35.360 --> 00:22:37.750
It's sort of infinity
equals infinity.
00:22:37.750 --> 00:22:39.850
At the edge here we
managed to get something
00:22:39.850 --> 00:22:42.340
which is sort of almost right.
00:22:42.340 --> 00:22:46.100
But you know, it's, we don't
consider this to be logically
00:22:46.100 --> 00:22:47.540
to make complete sense.
00:22:47.540 --> 00:22:51.200
So it's a little dangerous.
00:22:51.200 --> 00:22:52.960
And so we just say
that it diverges.
00:22:52.960 --> 00:22:54.100
And we get rid of this.
00:22:54.100 --> 00:22:55.990
So we're still
putting it in red.
00:22:55.990 --> 00:22:58.540
All right.
00:22:58.540 --> 00:22:59.975
The bad guy here.
00:22:59.975 --> 00:23:00.850
So this one diverges.
00:23:04.670 --> 00:23:12.650
Similarly if I take a equals
-1, I get 1 - 1 + 1 - 1 + 1...
00:23:12.650 --> 00:23:15.450
Because the odd and the
even powers in that formula
00:23:15.450 --> 00:23:17.270
alternate sign.
00:23:17.270 --> 00:23:19.750
And this bounces back and forth.
00:23:19.750 --> 00:23:21.640
It never settles down.
00:23:21.640 --> 00:23:23.346
It starts at 1.
00:23:23.346 --> 00:23:25.720
And then it gets down to 0
and then it goes back up to 1,
00:23:25.720 --> 00:23:28.400
down to 0, back up to 1.
00:23:28.400 --> 00:23:29.517
It doesn't settle down.
00:23:29.517 --> 00:23:30.600
It bounces back and forth.
00:23:30.600 --> 00:23:31.580
It oscillates.
00:23:31.580 --> 00:23:34.740
On the other hand if you
compare the right hand side.
00:23:34.740 --> 00:23:35.980
What's the right hand side?
00:23:35.980 --> 00:23:36.970
It's 1 / (1-(-1)).
00:23:39.730 --> 00:23:41.515
Which is 1/2.
00:23:41.515 --> 00:23:42.460
All right.
00:23:42.460 --> 00:23:45.169
So if you just paid attention
to the formula, which
00:23:45.169 --> 00:23:47.710
is what we were doing when we
integrated without thinking too
00:23:47.710 --> 00:23:50.070
hard about this,
you get a number
00:23:50.070 --> 00:23:51.320
here but in fact that's wrong.
00:23:51.320 --> 00:23:53.153
Actually it's kind of
an interesting number.
00:23:53.153 --> 00:23:56.380
It's halfway between the
two, between 0 and 1.
00:23:56.380 --> 00:23:58.505
So again there's some
sort of vague sense
00:23:58.505 --> 00:24:01.966
in which this is trying
to be this answer.
00:24:01.966 --> 00:24:04.580
All right.
00:24:04.580 --> 00:24:08.744
It's not so bad but we're still
going to put this in a red box.
00:24:08.744 --> 00:24:10.030
All right.
00:24:10.030 --> 00:24:12.710
because this is what
we called divergence.
00:24:12.710 --> 00:24:16.130
So both of these
cases are divergent.
00:24:16.130 --> 00:24:20.490
It only really works when
alpha-- when a is less than 1.
00:24:20.490 --> 00:24:23.480
I'm going to add
one more case just
00:24:23.480 --> 00:24:30.040
to see that mathematicians are
slightly curious about what
00:24:30.040 --> 00:24:32.020
goes on in other cases.
00:24:32.020 --> 00:24:37.570
So this is 1 + 2 +
2^2 + 2^3 plus etc..
00:24:37.570 --> 00:24:41.980
And that should be equal to --
according to this formula --
00:24:41.980 --> 00:24:44.750
1/(1-2).
00:24:44.750 --> 00:24:48.242
Which is -1.
00:24:48.242 --> 00:24:49.860
All right.
00:24:49.860 --> 00:24:53.460
Now this one is
clearly wrong, right?
00:24:53.460 --> 00:24:55.530
This one is totally wrong.
00:24:58.170 --> 00:24:59.465
It certainly diverges.
00:24:59.465 --> 00:25:02.370
The left hand side is
obviously infinite.
00:25:02.370 --> 00:25:04.070
The right hand side is way off.
00:25:04.070 --> 00:25:05.960
It's -1.
00:25:05.960 --> 00:25:10.350
On the other hand it
turns out actually
00:25:10.350 --> 00:25:13.360
that mathematicians have ways
of making sense out of these.
00:25:13.360 --> 00:25:15.590
In number theory
there's a strange system
00:25:15.590 --> 00:25:18.160
where this is actually true.
00:25:18.160 --> 00:25:21.530
And what happens
in that system is
00:25:21.530 --> 00:25:24.030
that what you have to
throw out is the idea
00:25:24.030 --> 00:25:27.050
that 0 is less than 1.
00:25:27.050 --> 00:25:29.980
There is no such thing
as negative numbers.
00:25:29.980 --> 00:25:32.090
So this number exists.
00:25:32.090 --> 00:25:35.700
And it's the additive
inverse of 1.
00:25:35.700 --> 00:25:39.780
It has this arithmetic
property but the statement
00:25:39.780 --> 00:25:43.330
that this is, that 1 is bigger
than 0 does not make sense.
00:25:43.330 --> 00:25:45.440
So you have your choice,
either this diverges
00:25:45.440 --> 00:25:48.630
or you have to throw
out something like this.
00:25:48.630 --> 00:25:51.510
So that's a very curious
thing in higher mathematics.
00:25:51.510 --> 00:25:56.410
Which if you get to number
theory there's fun stuff there.
00:25:56.410 --> 00:25:58.920
All right.
00:25:58.920 --> 00:26:02.740
OK but for our purposes
these things are all out.
00:26:02.740 --> 00:26:03.300
All right.
00:26:03.300 --> 00:26:04.010
They're gone.
00:26:04.010 --> 00:26:05.160
We're not considering them.
00:26:05.160 --> 00:26:09.550
Only a between -1 and 1.
00:26:09.550 --> 00:26:10.070
All right.
00:26:13.910 --> 00:26:18.190
Now I want to do
something systematic.
00:26:18.190 --> 00:26:21.530
And it's more or less on
the lines of the powers
00:26:21.530 --> 00:26:23.090
that I'm erasing right now.
00:26:26.630 --> 00:26:28.590
I want to tell you
about series which are
00:26:28.590 --> 00:26:30.420
kind of borderline convergent.
00:26:30.420 --> 00:26:33.720
And then next time when we
talk about powers series we'll
00:26:33.720 --> 00:26:35.810
come back to this very
important series which
00:26:35.810 --> 00:26:37.080
is the most important one.
00:26:40.680 --> 00:26:47.400
So now let's talk about some
series-- er, general notations.
00:26:47.400 --> 00:26:49.970
And this will help
you with the last bit.
00:26:53.810 --> 00:26:56.760
This is going to be pretty
much the same as what
00:26:56.760 --> 00:27:00.890
we did for improper integrals.
00:27:00.890 --> 00:27:04.320
Namely, first of all I'm
going to have S_N which
00:27:04.320 --> 00:27:09.760
is the sum of a_n, n
equals 0 to capital N.
00:27:09.760 --> 00:27:12.220
And this is what we're
calling a partial sum.
00:27:18.050 --> 00:27:23.980
And then the full limit, which
is capital S, if you like.
00:27:23.980 --> 00:27:29.980
a_n, n equals 0 to infinity,
is just the limit as N goes
00:27:29.980 --> 00:27:32.010
to infinity of the S_N's.
00:27:36.240 --> 00:27:39.770
And then we have the same kind
of notation that we had before.
00:27:39.770 --> 00:27:42.860
Which is there are
these two choices which
00:27:42.860 --> 00:27:45.890
is that if the limit exists.
00:27:50.450 --> 00:27:51.830
That's the green choice.
00:27:51.830 --> 00:27:54.460
And we say it converges.
00:27:54.460 --> 00:28:00.830
So we say the series converges.
00:28:00.830 --> 00:28:06.616
And then the other case which
is the limit does not exist.
00:28:10.782 --> 00:28:12.240
And we can say the
series diverges.
00:28:20.560 --> 00:28:21.343
Question.
00:28:21.343 --> 00:28:22.509
AUDIENCE: [INAUDIBLE PHRASE]
00:28:26.480 --> 00:28:29.290
PROFESSOR: The question
was how did I get to this?
00:28:29.290 --> 00:28:31.840
And I will do that next
time but in fact of course
00:28:31.840 --> 00:28:33.240
you've seen it in high school.
00:28:33.240 --> 00:28:35.930
Right this is-- Yeah.
00:28:35.930 --> 00:28:36.860
Yeah.
00:28:36.860 --> 00:28:40.000
We'll do that next time.
00:28:40.000 --> 00:28:42.222
The question was how
did we arrive-- sorry I
00:28:42.222 --> 00:28:43.430
didn't tell you the question.
00:28:43.430 --> 00:28:44.360
The question was
how do we arrive
00:28:44.360 --> 00:28:46.590
at this formula on the
right hand side here.
00:28:46.590 --> 00:28:48.240
But we'll talk about
that next time.
00:28:53.060 --> 00:28:54.020
All right.
00:28:54.020 --> 00:28:59.930
So here's the basic
definition and what
00:28:59.930 --> 00:29:02.430
we're going to
recognize about series.
00:29:02.430 --> 00:29:06.835
And I'm going to give
you a few examples
00:29:06.835 --> 00:29:08.460
and then we'll do
something systematic.
00:29:12.070 --> 00:29:14.495
So the first example--
well the first example
00:29:14.495 --> 00:29:16.290
is the geometric series.
00:29:16.290 --> 00:29:19.430
But the first example that
I'm going to discuss now
00:29:19.430 --> 00:29:23.352
and in a little bit of
detail is this sum 1/n^2,
00:29:23.352 --> 00:29:24.310
n equals 1 to infinity.
00:29:28.580 --> 00:29:34.050
It turns out that this
series is very analogous --
00:29:34.050 --> 00:29:36.690
and we'll develop this
analogy carefully --
00:29:36.690 --> 00:29:41.065
the integral from
1 to x, dx / x^2.
00:29:41.065 --> 00:29:46.220
And we're going to develop
this analogy in detail later
00:29:46.220 --> 00:29:47.920
in this lecture.
00:29:47.920 --> 00:29:49.770
And this one is
one of the ones--
00:29:49.770 --> 00:29:51.890
so now you have to go back
and actually remember,
00:29:51.890 --> 00:29:54.335
this is one of the ones you
really want to memorize.
00:29:54.335 --> 00:29:56.460
And you should especially
pay attention to the ones
00:29:56.460 --> 00:29:58.507
with an infinity in them.
00:29:58.507 --> 00:29:59.465
This one is convergent.
00:30:03.270 --> 00:30:04.520
And this series is convergent.
00:30:04.520 --> 00:30:11.070
Now it turns out that
evaluating this is very easy.
00:30:11.070 --> 00:30:12.820
This is 1.
00:30:12.820 --> 00:30:15.310
It's easy to calculate.
00:30:15.310 --> 00:30:19.500
Evaluating this is very tricky.
00:30:19.500 --> 00:30:21.610
And Euler did it.
00:30:21.610 --> 00:30:26.440
And the answer is pi^2 / 6.
00:30:26.440 --> 00:30:29.050
That's an amazing calculation.
00:30:29.050 --> 00:30:33.570
And it was done very early in
the history of mathematics.
00:30:33.570 --> 00:30:38.110
If you look at another example--
so maybe example two here,
00:30:38.110 --> 00:30:45.790
if you look at 1/n^3, n equals--
well you can't start here at 0
00:30:45.790 --> 00:30:46.580
by the way.
00:30:46.580 --> 00:30:48.930
I get to start wherever
I want in these series.
00:30:48.930 --> 00:30:49.920
Here I start with 0.
00:30:49.920 --> 00:30:51.220
Here I started with 1.
00:30:51.220 --> 00:30:52.880
And notice the reason
why I started--
00:30:52.880 --> 00:30:56.650
it was a bad idea to start
with 0 was that 1 over 0
00:30:56.650 --> 00:30:57.860
is undefined.
00:30:57.860 --> 00:30:58.360
Right?
00:30:58.360 --> 00:31:00.443
So I'm just starting where
it's convenient for me.
00:31:00.443 --> 00:31:03.560
And since I'm interested
mostly in the tail behavior
00:31:03.560 --> 00:31:06.150
it doesn't matter to me
so much where I start.
00:31:06.150 --> 00:31:07.800
Although if I want
an exact answer
00:31:07.800 --> 00:31:10.165
I need to start
exactly at n = 1.
00:31:10.165 --> 00:31:10.890
All right.
00:31:10.890 --> 00:31:17.932
This one is similar
to this integral here.
00:31:17.932 --> 00:31:18.700
All right.
00:31:18.700 --> 00:31:20.440
Which is convergent again.
00:31:20.440 --> 00:31:22.390
So there's a number
that you get.
00:31:22.390 --> 00:31:27.180
And let's see what is it
something like 2/3 or something
00:31:27.180 --> 00:31:30.190
like that, all right,
for this for this number.
00:31:30.190 --> 00:31:32.440
Or 1/3.
00:31:32.440 --> 00:31:33.110
What is it?
00:31:33.110 --> 00:31:33.690
No 1/2.
00:31:33.690 --> 00:31:35.000
I guess it's 1/2.
00:31:35.000 --> 00:31:37.110
This one is 1/2.
00:31:37.110 --> 00:31:38.780
You check that,
I'm not positive,
00:31:38.780 --> 00:31:40.600
but anyway just doing
it in my head quickly
00:31:40.600 --> 00:31:42.150
it seems to be 1/2.
00:31:42.150 --> 00:31:44.100
Anyway it's an easy
number to calculate.
00:31:44.100 --> 00:31:49.710
This one over here stumped
mathematicians basically
00:31:49.710 --> 00:31:51.870
for all time.
00:31:51.870 --> 00:31:56.250
It doesn't have any kind of
elementary form like this.
00:31:56.250 --> 00:31:59.842
And it was only very recently
proved to be rational.
00:31:59.842 --> 00:32:01.550
People couldn't even
couldn't even decide
00:32:01.550 --> 00:32:05.260
whether this was a
rational number or not.
00:32:05.260 --> 00:32:07.830
But anyway that's been resolved;
it is an irrational number
00:32:07.830 --> 00:32:09.540
which is what people suspected.
00:32:09.540 --> 00:32:10.650
Yeah question.
00:32:10.650 --> 00:32:11.816
AUDIENCE: [INAUDIBLE PHRASE]
00:32:14.360 --> 00:32:16.220
PROFESSOR: Yeah sorry.
00:32:16.220 --> 00:32:16.720
OK.
00:32:19.820 --> 00:32:25.070
I violated a rule
of mathematics--
00:32:25.070 --> 00:32:26.720
you said why is this similar?
00:32:26.720 --> 00:32:29.140
I thought that similar
was something else.
00:32:29.140 --> 00:32:30.380
And you're absolutely right.
00:32:30.380 --> 00:32:33.520
And I violated a
rule of mathematics.
00:32:33.520 --> 00:32:37.825
Which is that I used this
symbol for two different things.
00:32:41.196 --> 00:32:42.820
I should have written
this symbol here.
00:32:42.820 --> 00:32:43.530
All right.
00:32:43.530 --> 00:32:45.260
I'll create a new symbol here.
00:32:45.260 --> 00:32:48.920
The question of whether this
converges or this converges.
00:32:48.920 --> 00:32:51.920
These are the the
same type of question.
00:32:51.920 --> 00:32:54.090
And we'll see why they're
the same question it
00:32:54.090 --> 00:32:55.000
in a few minutes.
00:32:55.000 --> 00:32:58.190
But in fact the wiggle
I used, "similar",
00:32:58.190 --> 00:33:02.360
I used for the connection
between functions.
00:33:02.360 --> 00:33:09.070
The things that are really
similar are that 1/n resembles
00:33:09.070 --> 00:33:10.230
1/x^2.
00:33:10.230 --> 00:33:12.066
So I apologize I didn't--
00:33:12.066 --> 00:33:13.232
AUDIENCE: [INAUDIBLE PHRASE]
00:33:15.890 --> 00:33:17.890
PROFESSOR: Oh you
thought that this
00:33:17.890 --> 00:33:19.210
was the definition of that.
00:33:19.210 --> 00:33:21.460
That's actually the reason
why these things correspond
00:33:21.460 --> 00:33:21.990
so closely.
00:33:21.990 --> 00:33:25.560
That is that the Riemann
sum is close to this.
00:33:25.560 --> 00:33:27.520
But that doesn't
mean they're equal.
00:33:27.520 --> 00:33:31.245
The Riemann sum only works
when the delta x goes to 0.
00:33:31.245 --> 00:33:33.870
The way that we're going to get
a connection between these two,
00:33:33.870 --> 00:33:38.710
as we will just a second, is
with a Riemann sum with-- What
00:33:38.710 --> 00:33:47.180
we're going to use is a
Riemann sum with delta x = 1.
00:33:47.180 --> 00:33:49.617
All right and then that will
be the connection between.
00:33:49.617 --> 00:33:53.240
All right that's
absolutely right.
00:33:53.240 --> 00:33:53.740
All right.
00:33:57.020 --> 00:34:00.280
So in order to illustrate
exactly this idea
00:34:00.280 --> 00:34:02.110
that you've just come
up with, and in fact
00:34:02.110 --> 00:34:04.490
that we're going to use,
we'll do the same thing
00:34:04.490 --> 00:34:07.740
but we're going to do it
on the example sum 1/n.
00:34:13.380 --> 00:34:20.170
So here's Example 3 and
it's going to be sum 1/n,
00:34:20.170 --> 00:34:21.369
n equals 1 to infinity.
00:34:24.090 --> 00:34:27.040
And what we're now
going to see is
00:34:27.040 --> 00:34:29.340
that it corresponds
to this integral here.
00:34:32.620 --> 00:34:34.850
And we're going
to show therefore
00:34:34.850 --> 00:34:37.870
that this thing diverges.
00:34:37.870 --> 00:34:40.300
But we're going to do
this more carefully.
00:34:40.300 --> 00:34:42.750
We're going to do
this in some detail
00:34:42.750 --> 00:34:45.880
so that you see what it is,
that the correspondence is
00:34:45.880 --> 00:34:47.390
between these quantities.
00:34:47.390 --> 00:34:49.890
And the same sort
of reasoning applies
00:34:49.890 --> 00:34:51.275
to these other examples.
00:34:55.820 --> 00:34:59.180
So here we go.
00:34:59.180 --> 00:35:04.290
I'm going to take
the integral and draw
00:35:04.290 --> 00:35:07.220
the picture of the Riemann sum.
00:35:07.220 --> 00:35:13.840
So here's the level 1 and
here's the function y = 1/x.
00:35:13.840 --> 00:35:15.730
And I'm going to
take the Riemann sum.
00:35:21.980 --> 00:35:25.200
With delta x = 1.
00:35:25.200 --> 00:35:28.820
And that's going to be closely
connected to the series
00:35:28.820 --> 00:35:29.570
that I have.
00:35:32.160 --> 00:35:35.670
But now I have to decide
whether I want a lower Riemann
00:35:35.670 --> 00:35:37.797
sum or an upper Riemann sum.
00:35:37.797 --> 00:35:39.630
And actually I'm going
to check both of them
00:35:39.630 --> 00:35:41.213
because both of them
are illuminating.
00:35:44.620 --> 00:35:47.000
First we'll do the
upper Riemann's sum.
00:35:47.000 --> 00:35:48.700
Now that's this staircase here.
00:35:51.780 --> 00:35:54.600
So we'll call this the
upper Riemann's sum.
00:35:58.130 --> 00:35:59.710
And let's check
what its levels are.
00:35:59.710 --> 00:36:01.560
This is not to scale.
00:36:01.560 --> 00:36:03.250
This level should be 1/2.
00:36:03.250 --> 00:36:05.245
So if this is 1 and
this is 2 and that level
00:36:05.245 --> 00:36:10.340
was supposed to be 1/2 and
this next level should be 1/3.
00:36:10.340 --> 00:36:12.830
That's how the Riemann
sums are working out.
00:36:17.040 --> 00:36:21.770
And now I have the
following phenomenon.
00:36:21.770 --> 00:36:24.640
Let's cut it off
at the nth stage.
00:36:24.640 --> 00:36:27.095
So that means that I'm
going, the integral
00:36:27.095 --> 00:36:30.760
is from 1 to n, dx / x.
00:36:30.760 --> 00:36:33.690
And the Riemann sum is
something that's bigger than it.
00:36:33.690 --> 00:36:40.300
Because the areas are enclosing
the area of the curved region.
00:36:40.300 --> 00:36:43.510
And that's going to be the
area of the first box which
00:36:43.510 --> 00:36:50.330
is 1, plus the area of the
second box which is 1/2,
00:36:50.330 --> 00:36:54.150
plus the area of the
third box which is 1/3.
00:36:54.150 --> 00:37:00.840
All the way up the last one,
but the last one starts at N-1.
00:37:00.840 --> 00:37:03.450
So it has 1/(N-1).
00:37:03.450 --> 00:37:05.780
There are not N boxes here.
00:37:05.780 --> 00:37:07.960
There are only N-1 boxes.
00:37:07.960 --> 00:37:11.830
Because the distance
between 1 and N is N-1.
00:37:11.830 --> 00:37:13.555
Right so this is N-1 terms.
00:37:17.330 --> 00:37:25.140
However, if I use the
notation for partial sum.
00:37:25.140 --> 00:37:33.367
Which is 1 + 1/2 plus all the
way up to 1/(n-1) 1 + 1/n.
00:37:33.367 --> 00:37:35.200
In other words I go out
to the Nth one which
00:37:35.200 --> 00:37:37.370
is what I would ordinarily do.
00:37:37.370 --> 00:37:42.900
Then this sum that I have here
certainly is less than S_N.
00:37:42.900 --> 00:37:47.850
Because there's one
more term there.
00:37:47.850 --> 00:37:50.430
And so here I have
an integral which
00:37:50.430 --> 00:37:53.570
is underneath this sum S_N.
00:38:01.700 --> 00:38:19.031
Now this is going to allow
us to prove conclusively
00:38:19.031 --> 00:38:21.156
that the-- So I'm just
going to rewrite this, prove
00:38:21.156 --> 00:38:22.614
conclusively that
the sum diverges.
00:38:22.614 --> 00:38:23.340
Why is that?
00:38:23.340 --> 00:38:26.000
Because this term
here we can calculate.
00:38:26.000 --> 00:38:29.470
This is log x
evaluated at 1 and n.
00:38:29.470 --> 00:38:34.850
Which is the same
thing as log N minus 0.
00:38:34.850 --> 00:38:39.270
All right, the quantity
log N - log 1 which is 0.
00:38:39.270 --> 00:38:46.892
And so what we have here is
that log N is less than S_N.
00:38:46.892 --> 00:38:51.455
All right and clearly this
goes to infinity right.
00:38:51.455 --> 00:38:57.830
As N goes to infinity this
thing goes to infinity.
00:38:57.830 --> 00:38:58.435
So we're done.
00:38:58.435 --> 00:39:00.330
All right we've
shown divergence.
00:39:08.730 --> 00:39:15.040
Now the way I'm going to
use the lower Riemann's sum
00:39:15.040 --> 00:39:20.010
is to recognize
that we've captured
00:39:20.010 --> 00:39:21.960
the rate appropriately.
00:39:21.960 --> 00:39:24.400
That is not only do I have
a lower bound like this
00:39:24.400 --> 00:39:27.860
but I have an upper bound
which is very similar.
00:39:27.860 --> 00:39:30.440
So if I use the upper
Riemann-- oh sorry,
00:39:30.440 --> 00:39:39.540
lower Riemann sum
again with delta x = 1.
00:39:43.760 --> 00:39:53.946
Then I have that the
integral from 1 to n of dx
00:39:53.946 --> 00:39:57.970
/ x is bigger than-- Well
what are the terms going
00:39:57.970 --> 00:40:00.640
to be if fit them underneath?
00:40:00.640 --> 00:40:03.210
If I fit them underneath
I'm missing the first term.
00:40:03.210 --> 00:40:05.600
That is the box is
going to be half height.
00:40:05.600 --> 00:40:08.300
It's going to be
this lower piece.
00:40:08.300 --> 00:40:10.480
So I'm missing this first term.
00:40:10.480 --> 00:40:16.765
So it'll be a 1/2 + 1/3 plus...
00:40:16.765 --> 00:40:18.150
All right, it will
keep on going.
00:40:18.150 --> 00:40:22.150
But now the last one
instead of being 1/(N-1),
00:40:22.150 --> 00:40:25.850
it's going to be 1 over N. This
is again a total of the N-1
00:40:25.850 --> 00:40:27.050
terms.
00:40:27.050 --> 00:40:28.430
This is the lower Riemann sum.
00:40:31.190 --> 00:40:38.808
And now we can recognize that
this is exactly equal to-- well
00:40:38.808 --> 00:40:40.558
so I'll put it over
here-- this is exactly
00:40:40.558 --> 00:40:43.520
equal to S_N minus 1,
minus the first term.
00:40:43.520 --> 00:40:47.110
So we missed the first term but
we got all the rest of them.
00:40:47.110 --> 00:40:49.360
So if I put this
to the other side
00:40:49.360 --> 00:40:53.746
remember this is
log N. All right.
00:40:53.746 --> 00:40:55.120
If I put this to
the other side I
00:40:55.120 --> 00:40:57.130
have the other
side of this bound.
00:40:57.130 --> 00:41:07.010
I have that S S_N is less than,
if I reverse it, log N + 1.
00:41:07.010 --> 00:41:09.040
And so I've trapped
it on the other side.
00:41:09.040 --> 00:41:10.950
And here I have the lower bound.
00:41:10.950 --> 00:41:13.170
So I'm going to
combine those together.
00:41:13.170 --> 00:41:18.030
So all told I have this
correspondence here.
00:41:18.030 --> 00:41:23.150
It is the size of log N is
trapped between the-- sorry,
00:41:23.150 --> 00:41:26.150
the size of S_N, which is
relatively hard to calculate
00:41:26.150 --> 00:41:30.780
and understand exactly,
is trapped between log N
00:41:30.780 --> 00:41:34.260
and log N + 1.
00:41:34.260 --> 00:41:35.160
Yeah question.
00:41:35.160 --> 00:41:36.326
AUDIENCE: [INAUDIBLE PHRASE]
00:41:46.039 --> 00:41:47.580
PROFESSOR: This step
here is the step
00:41:47.580 --> 00:41:49.550
that you're concerned about.
00:41:49.550 --> 00:41:54.160
So this step is a
geometric argument which
00:41:54.160 --> 00:41:57.070
is analogous to this step.
00:41:57.070 --> 00:42:01.320
All right it's the
same type of argument.
00:42:01.320 --> 00:42:04.720
And in this case it's that
the rectangles are on top
00:42:04.720 --> 00:42:07.270
and so the area represented
on the right hand side
00:42:07.270 --> 00:42:09.840
is less than the area
represented on this side.
00:42:09.840 --> 00:42:11.560
And this is the
same type of thing
00:42:11.560 --> 00:42:14.400
except that the
rectangles are underneath.
00:42:14.400 --> 00:42:17.310
So the sum of the
areas of the rectangles
00:42:17.310 --> 00:42:19.190
is less than the
area under the curve.
00:42:23.741 --> 00:42:24.240
All right.
00:42:24.240 --> 00:42:27.380
So I've now trapped
this quantity.
00:42:27.380 --> 00:42:34.440
And I'm now going to state
the sort of general results.
00:42:38.420 --> 00:42:42.230
So here's what's known
as integral comparison.
00:42:42.230 --> 00:42:44.900
It's this double
arrow correspondence
00:42:44.900 --> 00:42:54.580
in the general case,
for a very general case.
00:42:54.580 --> 00:42:57.130
There are actually even
more cases where it works.
00:42:57.130 --> 00:43:01.220
But this is a good
case and convenient.
00:43:01.220 --> 00:43:02.845
Now this is called
integral comparison.
00:43:07.060 --> 00:43:12.030
And it comes with hypotheses
but it follows the same argument
00:43:12.030 --> 00:43:13.495
that I just gave.
00:43:13.495 --> 00:43:27.540
If f(x) is decreasing
and it's positive,
00:43:27.540 --> 00:43:37.520
then the sum f(n), n
equals 1 to infinity,
00:43:37.520 --> 00:43:44.570
minus the integral from
1 to infinity of f(x) dx
00:43:44.570 --> 00:43:45.420
is less than f(1).
00:43:50.060 --> 00:43:51.590
That's basically what we showed.
00:43:51.590 --> 00:43:54.570
We showed that the difference
between S_N and log N
00:43:54.570 --> 00:43:55.930
was at most 1.
00:43:59.321 --> 00:43:59.820
All right.
00:43:59.820 --> 00:44:11.590
Now if both of them
are-- And the sum
00:44:11.590 --> 00:44:25.380
and the integral converge
or diverge together.
00:44:25.380 --> 00:44:27.750
That is they either both
converge or both diverge.
00:44:27.750 --> 00:44:30.780
This is the type of test that
we like because then we can just
00:44:30.780 --> 00:44:33.000
convert the question of
convergence over here
00:44:33.000 --> 00:44:37.770
to this question of convergence
over on the other side.
00:44:37.770 --> 00:44:41.590
Now I remind you that
it's incredibly hard
00:44:41.590 --> 00:44:44.870
to calculate these numbers.
00:44:44.870 --> 00:44:47.220
Whereas these numbers
are easier to calculate.
00:44:47.220 --> 00:44:50.390
Our goal is to reduce
things to simpler things.
00:44:50.390 --> 00:44:52.910
And in this case
sums, infinite sums
00:44:52.910 --> 00:44:54.720
are much harder than
infinite integrals.
00:45:00.080 --> 00:45:03.150
All right so that's the
integral comparison.
00:45:03.150 --> 00:45:12.140
And now I have one
last bit on comparisons
00:45:12.140 --> 00:45:13.840
that I need to tell you about.
00:45:13.840 --> 00:45:16.310
And this is very much like
what we did with integrals.
00:45:16.310 --> 00:45:18.240
Which is a so called
limit comparison.
00:45:29.200 --> 00:45:31.190
The limit comparison
says the following:
00:45:31.190 --> 00:45:45.210
if f(n) is similar to g(n) --
recall that means f(n) / g(n)
00:45:45.210 --> 00:45:51.760
tends to 1 as n
goes to infinity --
00:45:51.760 --> 00:45:55.000
and we're in the positive case.
00:45:55.000 --> 00:45:57.535
So let's just say
g(n) is positive.
00:46:03.920 --> 00:46:05.490
Then-- that doesn't
even, well-- then
00:46:05.490 --> 00:46:15.055
sum f(n), sum g(n)
either both-- same thing
00:46:15.055 --> 00:46:21.884
as above, either both
converge or both diverge.
00:46:27.488 --> 00:46:28.770
All right.
00:46:28.770 --> 00:46:30.730
This is just saying
that if they behave
00:46:30.730 --> 00:46:33.920
the same way in the tail, which
is all we really care about,
00:46:33.920 --> 00:46:40.820
then they have similar behavior,
similar convergence properties.
00:46:44.107 --> 00:46:45.690
And let me give you
a couple examples.
00:46:50.370 --> 00:46:56.580
So here's one example: if you
take the sum 1 over n^2 + 1,
00:46:56.580 --> 00:46:57.080
square root.
00:47:01.890 --> 00:47:05.350
This is going to be replaced
by something simpler.
00:47:05.350 --> 00:47:07.240
Which is the main term here.
00:47:07.240 --> 00:47:10.460
Which is 1 over
square root of n^2,
00:47:10.460 --> 00:47:15.100
which we recognize as
sum 1/n, which diverges.
00:47:17.920 --> 00:47:20.440
So this guy is one
of the red guys.
00:47:24.300 --> 00:47:26.410
On the red team.
00:47:26.410 --> 00:47:30.330
Now we have another example.
00:47:33.370 --> 00:47:38.960
Which is let's say the square
root of n, I don't know,
00:47:38.960 --> 00:47:43.080
to the fifth minus n^2.
00:47:43.080 --> 00:47:44.665
Now if you have
something where it's
00:47:44.665 --> 00:47:46.290
negative in the
denominator you kind of
00:47:46.290 --> 00:47:49.540
do have to watch out that
denominator makes sense.
00:47:49.540 --> 00:47:50.570
It isn't 0.
00:47:50.570 --> 00:47:53.042
So we're going to be careful
and start this at n = 2.
00:47:53.042 --> 00:48:00.370
In which case, the first
term, I don't like 1/0
00:48:00.370 --> 00:48:01.720
as a term in my series.
00:48:01.720 --> 00:48:04.130
So I'm just going to be a
little careful about how--
00:48:04.130 --> 00:48:05.710
as I said I was
kind of lazy here.
00:48:05.710 --> 00:48:09.930
I could have started this
one at 0 for instance.
00:48:09.930 --> 00:48:10.810
All right.
00:48:10.810 --> 00:48:14.110
So here's the picture.
00:48:14.110 --> 00:48:19.070
Now this I just replace by its
main term which is 1 over n^5,
00:48:19.070 --> 00:48:20.340
square root.
00:48:20.340 --> 00:48:25.975
Which is sum 1/n^(5/2),
which converges.
00:48:28.705 --> 00:48:29.520
All right.
00:48:29.520 --> 00:48:30.800
The power is bigger than 1.
00:48:30.800 --> 00:48:33.960
1 is the divider for these
things and it just misses.
00:48:33.960 --> 00:48:38.752
This one converges.
00:48:38.752 --> 00:48:41.800
All right so these
are the typical ways
00:48:41.800 --> 00:48:47.355
in which these convergence
processes are used.
00:48:47.355 --> 00:48:47.870
All right.
00:48:47.870 --> 00:48:49.930
So I have one more
thing for you.
00:48:49.930 --> 00:48:52.600
Which is an advertisement
for next time.
00:48:52.600 --> 00:48:56.360
And I have this demo
here which I will grab.
00:48:56.360 --> 00:48:58.300
But you will see this next time.
00:48:58.300 --> 00:49:00.880
So here's a question for
you to think about overnight
00:49:00.880 --> 00:49:04.240
but don't ask friends, you have
to think about it yourself.
00:49:04.240 --> 00:49:05.492
So here's the problem.
00:49:05.492 --> 00:49:08.000
Here are some blocks
which I acquired
00:49:08.000 --> 00:49:09.390
when my kids left home.
00:49:12.190 --> 00:49:20.550
Anyway yeah that'll happen to
you too in about four years.
00:49:20.550 --> 00:49:25.929
So now here you are,
these are blocks.
00:49:25.929 --> 00:49:27.470
So now here's the
question that we're
00:49:27.470 --> 00:49:29.546
going to deal with next time.
00:49:29.546 --> 00:49:30.920
I'm going to build
it, maybe I'll
00:49:30.920 --> 00:49:32.961
put it over here because
I want to have some room
00:49:32.961 --> 00:49:34.890
to head this way.
00:49:34.890 --> 00:49:41.320
I want to stack them up
so that-- oh didn't work.
00:49:41.320 --> 00:49:44.430
Going to stack them up
in the following way.
00:49:44.430 --> 00:49:48.280
I want to do it so that
the top one is completely
00:49:48.280 --> 00:49:51.540
to the right of the bottom one.
00:49:51.540 --> 00:49:53.170
That's the question
can I do that?
00:49:53.170 --> 00:49:57.870
Can I get-- Can I build this up?
00:49:57.870 --> 00:50:01.980
So let's see here.
00:50:01.980 --> 00:50:04.590
I just seem to be missing--
but anyway what I'm going to do
00:50:04.590 --> 00:50:05.965
is I'm going to
try to build this
00:50:05.965 --> 00:50:10.880
and we're going to see how far
we can get with this next time.