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HERBERT GROSS: Hi.
00:00:33.210 --> 00:00:35.210
Well, I guess we're in the home
stretch of the course.
00:00:35.210 --> 00:00:39.210
We're down to our last concept,
and it will take us
00:00:39.210 --> 00:00:43.150
essentially two lectures to
cover this new concept.
00:00:43.150 --> 00:00:46.430
Today we'll do the concept in
general and next time we'll
00:00:46.430 --> 00:00:50.640
apply it specifically to
the concept of series.
00:00:50.640 --> 00:00:53.710
But the concept we want to talk
about is something called
00:00:53.710 --> 00:00:55.420
'Uniform Convergence'.
00:00:55.420 --> 00:00:59.050
Let me say at the outset that
this is a very subtle topic.
00:00:59.050 --> 00:01:00.510
It is difficult.
00:01:00.510 --> 00:01:03.620
It seems to be beyond the scope
of our textbook, because
00:01:03.620 --> 00:01:04.900
it's not mentioned there.
00:01:04.900 --> 00:01:08.840
Consequently, I will try to give
the highlights as I speak
00:01:08.840 --> 00:01:11.430
with you, but the supplementary
notes will
00:01:11.430 --> 00:01:15.210
contain a more detailed
explanation of the things that
00:01:15.210 --> 00:01:16.770
we're going to talk about.
00:01:16.770 --> 00:01:20.690
Now, to set the stage properly
for our discussion of uniform
00:01:20.690 --> 00:01:24.980
convergence, I think it's wise
that we at least review the
00:01:24.980 --> 00:01:27.670
concept of convergence
in general.
00:01:27.670 --> 00:01:30.180
Well, let's take a
look over here.
00:01:30.180 --> 00:01:34.320
Recall that when we write that
the limit of ''f sub n' of x'
00:01:34.320 --> 00:01:39.360
as 'n' approaches infinity
equals 'f of x' for all 'x' in
00:01:39.360 --> 00:01:40.980
the closed interval
from 'a' to 'b'--
00:01:40.980 --> 00:01:44.910
when this happens, another way
of saying this is that the
00:01:44.910 --> 00:01:48.110
sequence 'f sub n', the sequence
of functions 'f sub
00:01:48.110 --> 00:01:52.710
n', converges to the function
'f' on the closed interval
00:01:52.710 --> 00:01:54.060
from 'a' to 'b'.
00:01:54.060 --> 00:01:57.350
Now, again, these tend to be
words unless you look at a
00:01:57.350 --> 00:01:58.520
specific example.
00:01:58.520 --> 00:02:00.210
Let's just pick one over here.
00:02:00.210 --> 00:02:03.630
Let ''f sub n' of x' be
''n' over '2n plus
00:02:03.630 --> 00:02:05.940
1'' times 'x squared'.
00:02:05.940 --> 00:02:10.880
Notice, of course, that the
value of this particular
00:02:10.880 --> 00:02:14.880
number depends both
on 'x' and 'n'.
00:02:14.880 --> 00:02:19.770
At any rate, let's pick a fixed
'x', hold it that way,
00:02:19.770 --> 00:02:22.240
and take the limit of
''f sub n' of x' as
00:02:22.240 --> 00:02:23.770
'n' approaches infinity.
00:02:23.770 --> 00:02:26.300
In other words, as we let 'n'
approach infinity in this
00:02:26.300 --> 00:02:29.020
case, notice that the limit
of 'n' over '2n
00:02:29.020 --> 00:02:31.260
plus 1' becomes 1/2.
00:02:31.260 --> 00:02:33.050
'x' has been chosen
independently of
00:02:33.050 --> 00:02:34.150
the choice of 'n'.
00:02:34.150 --> 00:02:36.510
Consequently, the limit function
in this case, 'f of
00:02:36.510 --> 00:02:39.880
x', is 1/2 'x squared'.
00:02:39.880 --> 00:02:43.150
And in this case, what we say
is that the sequence of
00:02:43.150 --> 00:02:48.440
functions 'n 'x squared'' over
'2n plus 1' converges to 1/2
00:02:48.440 --> 00:02:49.720
'x squared'.
00:02:49.720 --> 00:02:52.240
Now what does this mean
more specifically?
00:02:52.240 --> 00:02:54.250
In other words, let's see if we
can look at a few specific
00:02:54.250 --> 00:02:55.450
values of 'x'.
00:02:55.450 --> 00:02:59.390
For example, if I choose
'x' to be 2--
00:02:59.390 --> 00:03:02.360
in other words, if I choose 'x'
to be 2, if we look over
00:03:02.360 --> 00:03:06.430
here, this says that ''f
sub n' of x' is '4n'
00:03:06.430 --> 00:03:08.490
over '2n plus 1'.
00:03:08.490 --> 00:03:11.550
If I now take the limit as 'n'
approaches infinity, I'm going
00:03:11.550 --> 00:03:13.800
to wind up with what?
00:03:13.800 --> 00:03:18.560
'x' is replaced by 2, 2 squared
is 4, 1/2 of 4 is 2.
00:03:18.560 --> 00:03:22.500
In other words, the limit of ''f
sub n' of 2', as 'n' goes
00:03:22.500 --> 00:03:24.270
to infinity, is 2.
00:03:24.270 --> 00:03:29.270
In a similar way, if I replace
'x' by 4, 1/2 'x
00:03:29.270 --> 00:03:30.560
squared' becomes 8.
00:03:30.560 --> 00:03:33.870
And so what we have is, at the
limit, as 'n' approaches
00:03:33.870 --> 00:03:36.790
infinity, ''f sub
n' of 4' is 8.
00:03:36.790 --> 00:03:39.980
The key thing being that once
you choose 'x', notice that
00:03:39.980 --> 00:03:43.870
for a fixed 'x', ''f sub n' of
x' is a constant and you're
00:03:43.870 --> 00:03:47.690
now taking the limit of
a sequence of numbers.
00:03:47.690 --> 00:03:49.990
At any rate, here's what
the key point is.
00:03:49.990 --> 00:03:53.120
What does this mean by
our basic definition?
00:03:53.120 --> 00:03:55.910
By our basic definition, it
means that we can find the
00:03:55.910 --> 00:03:59.540
number, capital 'N sub 1', such
that when 'n' is greater
00:03:59.540 --> 00:04:03.020
than capital 'N sub 1', the
absolute value of ''f sub n'
00:04:03.020 --> 00:04:06.260
of 2', minus 2, is less
than epsilon.
00:04:06.260 --> 00:04:10.170
In a similar way, this means
that we can find the number
00:04:10.170 --> 00:04:13.400
capital 'N sub 2' such that
for any 'n' greater than
00:04:13.400 --> 00:04:16.426
capital 'N sub 2', the absolute
value of ''f sub n'
00:04:16.426 --> 00:04:19.680
of 4' minus 8 is also
less than epsilon.
00:04:19.680 --> 00:04:24.060
The key point is that 'N1' and
'N2' can be different.
00:04:24.060 --> 00:04:27.590
In other words, you may have to
go out further to make this
00:04:27.590 --> 00:04:30.500
difference less than epsilon
than you do to make this
00:04:30.500 --> 00:04:32.190
difference less than epsilon.
00:04:32.190 --> 00:04:34.900
In other words, you see what's
happening here-- and we're
00:04:34.900 --> 00:04:36.830
going to review this in writing
in a few minutes, so
00:04:36.830 --> 00:04:38.310
that you'll see it
in front you.
00:04:38.310 --> 00:04:42.350
What happens here is that you
see that for different values
00:04:42.350 --> 00:04:45.460
of 'x', we get different
values of 'n'.
00:04:45.460 --> 00:04:48.910
And since there are infinitely
many values of 'x', it means
00:04:48.910 --> 00:04:51.150
that in general, we're going
to be in a little bit of
00:04:51.150 --> 00:04:55.430
trouble trying to find one 'n'
that works for everything.
00:04:55.430 --> 00:04:56.460
And let me show you
what that means
00:04:56.460 --> 00:04:59.230
again, going more slowly.
00:04:59.230 --> 00:05:01.720
I simply call this two
basic definitions.
00:05:01.720 --> 00:05:04.720
In other words, if we have a
sequence of functions 'f sub
00:05:04.720 --> 00:05:07.670
n', each of which is defined
on the closed interval from
00:05:07.670 --> 00:05:11.680
'a' to 'b', we say that that
sequence of functions
00:05:11.680 --> 00:05:13.450
converges point-wise--
00:05:13.450 --> 00:05:14.910
that means number by number--
00:05:14.910 --> 00:05:17.370
to 'f' on [a, b]
00:05:17.370 --> 00:05:21.170
if this limit, ''f sub n' of x'
as 'n' approaches infinity,
00:05:21.170 --> 00:05:25.340
equals 'f of x' for
each 'x' in [a,b].
00:05:25.340 --> 00:05:27.930
In other words, given epsilon
greater than 0, we can find
00:05:27.930 --> 00:05:31.550
'N1' such that 'n' greater than
'N1' implies that the
00:05:31.550 --> 00:05:35.900
absolute value of ''f sub n' of
x1' minus 'f of x1' is less
00:05:35.900 --> 00:05:39.720
than epsilon for a given
'x1' in [a,b].
00:05:39.720 --> 00:05:43.860
In general, the choice of 'N1'
depends on the choice of 'x1',
00:05:43.860 --> 00:05:47.450
and there are infinitely many
such choices to make in [a,b].
00:05:47.450 --> 00:05:50.080
Now the key point is this,
and this is where uniform
00:05:50.080 --> 00:05:51.740
convergence comes in.
00:05:51.740 --> 00:05:56.130
If we can find one 'N' such that
whenever little 'n' is
00:05:56.130 --> 00:06:00.090
greater than that capital 'N',
''f sub n' of x' minus 'f of
00:06:00.090 --> 00:06:07.320
x' in absolute value is less
than epsilon for every 'x' in
00:06:07.320 --> 00:06:10.000
the closed interval, then
we say that the
00:06:10.000 --> 00:06:12.530
convergence is uniform.
00:06:12.530 --> 00:06:16.220
In other words, if we can find
one 'N' that makes that
00:06:16.220 --> 00:06:19.960
difference less than epsilon for
the entire interval, then
00:06:19.960 --> 00:06:22.970
we call the convergence
uniform.
00:06:22.970 --> 00:06:26.070
Now, you see, convergence in
general is a tough topic.
00:06:26.070 --> 00:06:30.240
In particular, uniform
convergence may seem even more
00:06:30.240 --> 00:06:33.620
remote, and therefore what I'd
like to do now is-- saving the
00:06:33.620 --> 00:06:36.490
formal proofs for the
supplementary notes, let me
00:06:36.490 --> 00:06:40.860
show you pictorially just what
the concept of uniform
00:06:40.860 --> 00:06:42.200
convergence really is.
00:06:45.570 --> 00:06:48.810
So let me give you a pictorial
representation.
00:06:48.810 --> 00:06:53.020
Let's suppose I have the curve
'y' equals 'f of x'.
00:06:53.020 --> 00:06:57.580
Now, to be within epsilon of 'f
of x' means I retrace this
00:06:57.580 --> 00:07:01.650
curve displaced epsilon units
above the original position,
00:07:01.650 --> 00:07:03.350
and epsilon units below.
00:07:03.350 --> 00:07:06.430
In other words, for a given an
epsilon, I now draw the curve
00:07:06.430 --> 00:07:09.710
'y' equals ''f of x' plus
epsilon' and 'y' equals ''f of
00:07:09.710 --> 00:07:11.730
x' minus epsilon'.
00:07:11.730 --> 00:07:15.390
Now, what uniform convergence
means is this, that for this
00:07:15.390 --> 00:07:19.630
given epsilon, I can find a
capital 'N' such that whenever
00:07:19.630 --> 00:07:23.290
'n' is greater than capital 'N',
the curve 'y' equals ''f
00:07:23.290 --> 00:07:26.190
sub n' of x' lies in
this shaded region.
00:07:26.190 --> 00:07:29.960
In other words, it can bounce
around all over, but it can't
00:07:29.960 --> 00:07:31.180
get outside of this region.
00:07:31.180 --> 00:07:35.270
In other words, once I'm far
enough out of my sequence, all
00:07:35.270 --> 00:07:41.120
of the curves lie in this
particular region.
00:07:41.120 --> 00:07:43.590
Now, of course, the question
is what does this all mean?
00:07:43.590 --> 00:07:44.820
And the answer is--
well, look.
00:07:44.820 --> 00:07:48.510
Let's take epsilon to be, and I
put this in quotation marks,
00:07:48.510 --> 00:07:52.020
"very, very small." Let's take
epsilon, for example, to be so
00:07:52.020 --> 00:07:55.290
small that it's within the
thickness of our chalk.
00:07:55.290 --> 00:07:56.540
If I now do this--
00:07:58.900 --> 00:08:01.540
see, I draw the curve
'y' equals 'f of x'.
00:08:01.540 --> 00:08:04.930
Notice now that the thickness of
my curve itself is the band
00:08:04.930 --> 00:08:06.710
width 2 epsilon.
00:08:06.710 --> 00:08:10.410
All I'm saying is that for this
very, very small epsilon,
00:08:10.410 --> 00:08:14.140
when 'n' is sufficiently large,
the curve 'y' equals
00:08:14.140 --> 00:08:18.720
''f sub n' of x' appears to
lie inside of this curve.
00:08:18.720 --> 00:08:21.590
You see, in other words, what
you're saying is that for a
00:08:21.590 --> 00:08:24.930
large enough 'n' and small
enough epsilon--
00:08:24.930 --> 00:08:28.330
loosely speaking what you saying
is that 'y' equals ''f
00:08:28.330 --> 00:08:31.620
sub n' of x' looks like 'y'
equals 'f of x', for a
00:08:31.620 --> 00:08:33.510
sufficiently large
values of 'n'.
00:08:33.510 --> 00:08:38.220
In other words, it appears that
we can't really tell the
00:08:38.220 --> 00:08:42.750
n-th curve in the sequence
from the limit function.
00:08:42.750 --> 00:08:44.430
And I want to make a few
key observations
00:08:44.430 --> 00:08:45.850
about what that means.
00:08:45.850 --> 00:08:48.370
I've written the whole thing out
on the blackboard so that
00:08:48.370 --> 00:08:51.860
you can see this after I say it,
but what I want you to see
00:08:51.860 --> 00:08:55.100
is, can you begin to get the
feeling that with this kind of
00:08:55.100 --> 00:08:56.970
a condition, for example--
00:08:56.970 --> 00:09:01.260
if each 'f sub n' happens to be
continuous, in other words,
00:09:01.260 --> 00:09:05.200
if each member of my sequence
is unbroken, then the limit
00:09:05.200 --> 00:09:07.430
function itself must
also be unbroken.
00:09:07.430 --> 00:09:10.970
Because you see I can squeeze
this thing down to such a
00:09:10.970 --> 00:09:14.950
narrow width that there's no
room for a break in here.
00:09:14.950 --> 00:09:19.700
Also notice that if the curve
'y' equals ''f sub n' of x' is
00:09:19.700 --> 00:09:21.490
caught inside this curve--
00:09:21.490 --> 00:09:24.840
if I, for example, were
computing the area of the
00:09:24.840 --> 00:09:26.010
region 'R'--
00:09:26.010 --> 00:09:28.970
for a large enough 'n', I
couldn't tell the difference
00:09:28.970 --> 00:09:33.470
in area if I use 'y' equals 'f
of x' for my top curve, or
00:09:33.470 --> 00:09:36.730
whether I use 'y' equals
''f sub n' of x'.
00:09:36.730 --> 00:09:40.560
Well, keep that in mind, and all
I'm saying is this, that
00:09:40.560 --> 00:09:43.810
from our picture, it should
seem clear that if the
00:09:43.810 --> 00:09:47.670
sequence 'f sub n' converges
uniformly to 'f' on [a, b],
00:09:47.670 --> 00:09:51.420
and if each 'f sub n' is
continuous on [a,b], then--
00:09:51.420 --> 00:09:53.140
and this is a fundamental
result.
00:09:53.140 --> 00:09:57.380
Fundamental result one, 'f' is
also continuous on [a, b].
00:10:01.230 --> 00:10:02.240
Now you say, look.
00:10:02.240 --> 00:10:04.300
That's what you'd expect, isn't
it, if every member of
00:10:04.300 --> 00:10:05.620
the sequence is continuous?
00:10:05.620 --> 00:10:08.570
Why shouldn't the limit function
also be continuous?
00:10:08.570 --> 00:10:12.030
The point is, well, maybe that's
what you expect, but
00:10:12.030 --> 00:10:13.500
note this--
00:10:13.500 --> 00:10:17.070
in one of our earlier lectures,
we already saw that
00:10:17.070 --> 00:10:20.640
if we only had point-wise
convergence, this did not need
00:10:20.640 --> 00:10:21.640
to be true.
00:10:21.640 --> 00:10:25.240
In particular, recall our
example in which we defined
00:10:25.240 --> 00:10:29.480
''f sub n' of x' to be 'x to
the n', where the domain of
00:10:29.480 --> 00:10:32.650
'f' was the closed interval
from 0 to 1.
00:10:32.650 --> 00:10:34.330
Remember what happened
in that case?
00:10:34.330 --> 00:10:36.960
Each of these 'f sub
n's is continuous.
00:10:36.960 --> 00:10:42.060
But the limit function, you
may recall, is what?
00:10:42.060 --> 00:10:47.990
It's 0 if 'x' is less than
1, and 1 if 'x' equals 1.
00:10:47.990 --> 00:10:51.540
In other words, the limit
function was discontinuous at
00:10:51.540 --> 00:10:52.890
'x' equals 1.
00:10:52.890 --> 00:10:55.360
By the way, you might like to
see what this means from
00:10:55.360 --> 00:10:56.870
another point of view.
00:10:56.870 --> 00:10:58.750
And let me show you what
it does mean from
00:10:58.750 --> 00:10:59.910
another point of view.
00:10:59.910 --> 00:11:04.190
Since 'f' continuous at 'x'
equals 'x sub 1' means that
00:11:04.190 --> 00:11:08.110
the limit of 'f of x' as 'x'
approaches 'x1' is 'f of x1',
00:11:08.110 --> 00:11:13.620
and since 'f of x' itself, by
definition, is the limit of
00:11:13.620 --> 00:11:22.360
''f sub n' of x' as 'n'
approaches infinity, we may
00:11:22.360 --> 00:11:26.550
rewrite our first condition
in the form-- what?
00:11:26.550 --> 00:11:32.100
We may rewrite our first
equation, this equation here,
00:11:32.100 --> 00:11:33.320
in what form?
00:11:33.320 --> 00:11:38.820
Limit as 'x' approaches 'x1'
of limit 'n' approaches
00:11:38.820 --> 00:11:41.270
infinity, ''f sub n' of x'.
00:11:41.270 --> 00:11:44.470
And that equals the limit as 'n'
approaches infinity, ''f
00:11:44.470 --> 00:11:46.660
sub n' of x1'.
00:11:46.660 --> 00:11:50.330
Now keep in mind that since each
'f sub n' is given to be
00:11:50.330 --> 00:11:55.510
continuous, by definition of
continuity ''f sub n' of x1'
00:11:55.510 --> 00:11:59.280
is the same as saying the limit
as 'x' approaches 'x1',
00:11:59.280 --> 00:12:00.690
''f sub n' of x'.
00:12:00.690 --> 00:12:05.060
The point I'm making is, if you
now put this together with
00:12:05.060 --> 00:12:09.410
this, it says the rather
remarkable thing that unless
00:12:09.410 --> 00:12:13.860
you have uniform convergence
when you interchange the order
00:12:13.860 --> 00:12:16.770
in which you take the limits
over here, you may very well
00:12:16.770 --> 00:12:18.200
get a different answer.
00:12:18.200 --> 00:12:20.920
In other words, when you're
dealing with non-uniform
00:12:20.920 --> 00:12:25.550
convergence, you must be very,
very careful to perform every
00:12:25.550 --> 00:12:27.650
operation in the given order.
00:12:27.650 --> 00:12:29.210
What we're saying is what?
00:12:29.210 --> 00:12:31.090
This will give you an answer.
00:12:31.090 --> 00:12:32.710
This will give you an answer.
00:12:32.710 --> 00:12:36.140
But if the convergence is not
uniform, the answers may be
00:12:36.140 --> 00:12:39.730
different, and consequently by
changing the order you destroy
00:12:39.730 --> 00:12:42.000
the whole physical meaning
of the problem.
00:12:42.000 --> 00:12:44.590
Well, again, that's reemphasized
in the
00:12:44.590 --> 00:12:45.890
supplementary notes.
00:12:45.890 --> 00:12:47.940
Let me continue on here.
00:12:47.940 --> 00:12:51.510
Let me tell you another
interesting property of
00:12:51.510 --> 00:12:54.510
uniform convergence.
00:12:54.510 --> 00:12:58.420
Suppose the sequence ''f sub n'
of x' converges uniformly
00:12:58.420 --> 00:13:00.590
to 'f of x' on [a, b].
00:13:00.590 --> 00:13:03.220
The point that's rather
interesting is that you can
00:13:03.220 --> 00:13:06.530
reverse the order of integration
and taking the
00:13:06.530 --> 00:13:08.210
limit in this particular case.
00:13:08.210 --> 00:13:12.060
In other words, suppose you want
to compute the integral
00:13:12.060 --> 00:13:14.800
of the limit function
from 'a' to 'b'.
00:13:14.800 --> 00:13:19.790
What you can do instead is
compute the integral of the
00:13:19.790 --> 00:13:23.440
n-th number of your sequence,
and then take the limit as 'n'
00:13:23.440 --> 00:13:24.700
goes to infinity.
00:13:24.700 --> 00:13:27.820
In other words, rewriting this,
it says that if you have
00:13:27.820 --> 00:13:32.080
uniform convergence, you can
take the limit inside the
00:13:32.080 --> 00:13:33.470
integral sign.
00:13:33.470 --> 00:13:37.740
And again, these results
are proven in our
00:13:37.740 --> 00:13:39.400
supplementary notes.
00:13:39.400 --> 00:13:43.040
We also say a few words about
corresponding results for
00:13:43.040 --> 00:13:45.750
differentiation in our
supplementary notes.
00:13:45.750 --> 00:13:50.100
And I should point out that
differentiation is a far more
00:13:50.100 --> 00:13:52.000
subtle thing than integration.
00:13:52.000 --> 00:13:55.080
See, remember that for
integration, all you need is
00:13:55.080 --> 00:13:56.520
continuity.
00:13:56.520 --> 00:13:59.340
For differentiation, you
need smoothness.
00:13:59.340 --> 00:14:04.420
The point is that as you put a
thin band around the function
00:14:04.420 --> 00:14:07.760
'y' equals 'f of x' when you
have uniform convergence,
00:14:07.760 --> 00:14:10.410
that's enough to make sure that
the limit function must
00:14:10.410 --> 00:14:13.200
be continuous if each of the
members in the sequence is
00:14:13.200 --> 00:14:13.960
continuous.
00:14:13.960 --> 00:14:16.790
But without going into the
details of this thing, it does
00:14:16.790 --> 00:14:20.000
turn out that for the degree of
smoothness that you need,
00:14:20.000 --> 00:14:22.510
these things can jump around
enough so that for
00:14:22.510 --> 00:14:25.860
differentiation, we do have to
be a little bit more careful.
00:14:25.860 --> 00:14:28.650
Rather than to becloud the
issue, I will stick with
00:14:28.650 --> 00:14:31.810
integration topics for
our lecture today.
00:14:31.810 --> 00:14:34.880
Now, the other thing that I want
to mention is, again, in
00:14:34.880 --> 00:14:38.830
terms of doing what comes
naturally, I think we are
00:14:38.830 --> 00:14:41.850
tempted to look at something
like this and say, look, all I
00:14:41.850 --> 00:14:43.270
did was bring the
limit inside.
00:14:43.270 --> 00:14:45.360
Why can't I do that?
00:14:45.360 --> 00:14:48.120
And instead of saying, look,
you can't, I think the best
00:14:48.120 --> 00:14:50.290
thing to show is that when
you don't have uniform
00:14:50.290 --> 00:14:52.570
convergence, you get two
different answers.
00:14:52.570 --> 00:14:53.980
Again, the idea being what?
00:14:53.980 --> 00:14:56.270
We are not saying that
you can't do this.
00:14:56.270 --> 00:14:58.920
We are not saying that you
can't compute this.
00:14:58.920 --> 00:15:01.530
All we're saying is that if
the convergence is not
00:15:01.530 --> 00:15:08.050
uniform, these two expressions
may very well
00:15:08.050 --> 00:15:10.880
name different numbers.
00:15:10.880 --> 00:15:13.180
Now, to show you what I have in
mind here, let me give you
00:15:13.180 --> 00:15:15.110
an example.
00:15:15.110 --> 00:15:18.180
See, to show you that 2 need not
be true if the convergence
00:15:18.180 --> 00:15:21.630
is not uniform, consider
the following example.
00:15:21.630 --> 00:15:25.290
Now, in the supplementary notes,
I repeat this example
00:15:25.290 --> 00:15:28.270
both the way I have it on the
board and also from an
00:15:28.270 --> 00:15:30.880
algebraic point of view, without
using the pictures.
00:15:30.880 --> 00:15:33.700
But in terms of the picture,
here's what we do.
00:15:33.700 --> 00:15:37.020
We define a function on the
closed interval from 0 to 2,
00:15:37.020 --> 00:15:40.160
which I'll call 'f sub
n', as follows.
00:15:40.160 --> 00:15:45.460
For a given 'n', I will locate
the points '1/n' and '2/n'.
00:15:45.460 --> 00:15:49.900
For example, if 'n' happened to
be 50, this would be 1/50
00:15:49.900 --> 00:15:52.030
and this would be 2/50.
00:15:52.030 --> 00:15:57.570
Now what I do, is at the 'x'
value '1/n', I take as the
00:15:57.570 --> 00:16:01.510
corresponding y-value,
'n squared'.
00:16:01.510 --> 00:16:05.060
And I draw the straight line
that goes from the origin to
00:16:05.060 --> 00:16:08.110
this point, '1/n' comma
'n squared'.
00:16:08.110 --> 00:16:12.290
Then I draw the straight line
that comes right back to the
00:16:12.290 --> 00:16:17.800
x-intercept, '2/n', and I finish
off the curve by just
00:16:17.800 --> 00:16:22.180
letting it hug the x-axis till
we get over to 'x' equals 2.
00:16:22.180 --> 00:16:24.660
I'll come back to this on the
next board, to show you why I
00:16:24.660 --> 00:16:27.590
chose this, but let's make a few
observations just to make
00:16:27.590 --> 00:16:30.860
sure that you understand what
this function looks like.
00:16:30.860 --> 00:16:33.630
I'll just make a few arbitrary
remarks about it.
00:16:33.630 --> 00:16:36.310
First of all, for each
'n', ''f sub n' of
00:16:36.310 --> 00:16:38.640
'1/n'' is 'n squared'.
00:16:38.640 --> 00:16:41.010
That's just another way
of indicating a
00:16:41.010 --> 00:16:42.260
label for this point.
00:16:44.610 --> 00:16:50.990
Secondly, my claim is that for
any number 'x sub 0', if 'x
00:16:50.990 --> 00:16:56.580
sub 0' is greater than '2/n',
''f sub n' of x0' must be 0.
00:16:56.580 --> 00:16:58.810
And the reason for that
is quite simple.
00:16:58.810 --> 00:17:00.760
I'm just trying to show you
how to read this picture.
00:17:00.760 --> 00:17:04.410
Namely, notice that as soon as
'x' gets to be as great as
00:17:04.410 --> 00:17:10.140
'2/n', the 'f' value is 0,
because the function is
00:17:10.140 --> 00:17:11.960
hugging the x-axis.
00:17:11.960 --> 00:17:14.579
And just as a final observation,
notice that when
00:17:14.579 --> 00:17:19.420
'x sub 0' is 0, ''f sub n' of 0'
is 0 for every 'n', meaning
00:17:19.420 --> 00:17:23.660
that every member of my family
of functions goes through this
00:17:23.660 --> 00:17:24.520
particular point.
00:17:24.520 --> 00:17:26.020
In other words, let me
just label this.
00:17:26.020 --> 00:17:31.850
This is 'y' equals
''f sub n' of x'.
00:17:31.850 --> 00:17:35.110
Well, by the way if ''f sub n'
of 0' is 0 for each 'n', in
00:17:35.110 --> 00:17:37.810
particular the limit of ''f sub
n' of 0' as 'n' approaches
00:17:37.810 --> 00:17:39.920
infinity is 0.
00:17:39.920 --> 00:17:43.080
What happens if we pick
a non-zero value?
00:17:43.080 --> 00:17:47.490
For example, suppose I pick 'x0'
to be greater than 0 but
00:17:47.490 --> 00:17:49.190
less than or equal to 2?
00:17:49.190 --> 00:17:54.030
The key point is this, that
since the limit of '2/n' as
00:17:54.030 --> 00:17:58.490
'n' approaches infinity is 0,
given a value of 'x0' which is
00:17:58.490 --> 00:18:02.720
not 0, I can find the capital
'N' such that when 'n' is
00:18:02.720 --> 00:18:06.660
greater than capital 'N',
'2/n' is less than 'x0'.
00:18:06.660 --> 00:18:10.020
In other words, if 'x0' is
greater than 0, and '2/n'
00:18:10.020 --> 00:18:13.860
approaches 0, for large enough
values of 'n', '2/n'
00:18:13.860 --> 00:18:15.340
be less than 'x0'.
00:18:15.340 --> 00:18:18.210
In particular, when that
happens, if we couple this
00:18:18.210 --> 00:18:20.010
with our earlier observation--
00:18:20.010 --> 00:18:21.510
what earlier observation?
00:18:21.510 --> 00:18:24.780
Well, this one.
00:18:24.780 --> 00:18:27.610
If we couple that with our
earlier observation, we see
00:18:27.610 --> 00:18:30.150
that when 'n' is greater than
capital 'N', ''f sub
00:18:30.150 --> 00:18:32.110
n' of x0' is 0.
00:18:32.110 --> 00:18:36.530
Correspondingly, then, the limit
of ''f sub n' of x0' as
00:18:36.530 --> 00:18:39.870
'n' approaches infinity,
by definition, is 0.
00:18:39.870 --> 00:18:40.260
In other words--
00:18:40.260 --> 00:18:42.650
I'm going to reinforce this
later, but notice that the
00:18:42.650 --> 00:18:45.650
limit function here is
the function which
00:18:45.650 --> 00:18:47.480
is identically 0.
00:18:47.480 --> 00:18:51.600
Now, since this may look very
abstract to you, let me take a
00:18:51.600 --> 00:18:52.450
few minutes--
00:18:52.450 --> 00:18:54.960
and I hope this doesn't insult
your intelligence, but let me
00:18:54.960 --> 00:18:58.490
just take a few minutes and
redraw this for a couple of
00:18:58.490 --> 00:19:01.200
different values of 'n', just
so that you can see what's
00:19:01.200 --> 00:19:02.450
starting to happen here.
00:19:04.870 --> 00:19:08.730
Keep that picture in mind, and
now look what this means.
00:19:08.730 --> 00:19:15.460
For example, when 'n' is 1,
'1/n' is 1, '2/n' is 2, 'n
00:19:15.460 --> 00:19:16.760
squared' is 1.
00:19:16.760 --> 00:19:20.920
In other words, the graph 'y'
equals 'f1 of x' is just this
00:19:20.920 --> 00:19:23.876
triangular--
00:19:23.876 --> 00:19:25.480
just this.
00:19:25.480 --> 00:19:27.300
Why give it a name?
00:19:27.300 --> 00:19:30.030
Well, let's try a tougher one.
00:19:30.030 --> 00:19:34.100
Let's see what the member
'f sub 20' looks like.
00:19:34.100 --> 00:19:35.590
Recall how you draw this, now.
00:19:35.590 --> 00:19:38.000
With the subscript 20,
what do you do?
00:19:38.000 --> 00:19:43.670
You come in to the point
1/20, and at that
00:19:43.670 --> 00:19:44.930
point, you do what?
00:19:44.930 --> 00:19:49.900
You locate the point 1/20
comma 'n squared'.
00:19:49.900 --> 00:19:52.200
In this case, it's 400.
00:19:52.200 --> 00:19:54.700
And I have obviously haven't
drawn this to scale, but you
00:19:54.700 --> 00:19:55.440
now do what?
00:19:55.440 --> 00:19:57.060
Draw the straight line
that goes from the
00:19:57.060 --> 00:19:59.120
origin to this point.
00:19:59.120 --> 00:20:01.870
Then from this point, you draw
the straight line that comes
00:20:01.870 --> 00:20:06.190
back to the x-axis, hitting
it at 'x' equals 1/10.
00:20:06.190 --> 00:20:08.640
And then you come across
the x-axis all the way
00:20:08.640 --> 00:20:10.050
to 'x' equals 2.
00:20:10.050 --> 00:20:13.810
This would be the graph of 'y'
equals ''f sub 20' of x'.
00:20:13.810 --> 00:20:17.010
And by the way, do you sense
what's happening over here?
00:20:17.010 --> 00:20:23.040
See, notice that as 'n' gets
very, very large, the curve
00:20:23.040 --> 00:20:26.650
hugs the x-axis, starting
in closer and
00:20:26.650 --> 00:20:28.220
closer to the y-axis.
00:20:28.220 --> 00:20:31.580
But what happens is someplace
in here, no matter how close
00:20:31.580 --> 00:20:35.560
'x sub 0' is to 0, there
comes a very high peak.
00:20:35.560 --> 00:20:37.220
In fact, what is
that high peak?
00:20:37.220 --> 00:20:38.830
It's 'n squared'.
00:20:38.830 --> 00:20:41.470
In other words, when this number
is very close to the
00:20:41.470 --> 00:20:45.220
y-axis, the peak is
very, very high.
00:20:45.220 --> 00:20:47.580
In other words, no matter how
you put the squeeze on over
00:20:47.580 --> 00:20:52.800
here, this particular
peak jumps out.
00:20:52.800 --> 00:20:55.830
This is why this particular
sequence of functions is not
00:20:55.830 --> 00:20:57.070
uniformly convergent.
00:20:57.070 --> 00:20:59.820
Again, this is done more
slowly in the notes.
00:20:59.820 --> 00:21:01.880
But at any rate, let me show you
an interesting thing that
00:21:01.880 --> 00:21:03.760
happens over here.
00:21:03.760 --> 00:21:07.290
Let me redraw this now
for a general 'n'.
00:21:07.290 --> 00:21:09.930
In other words, let me draw 'y'
equals ''f sub n' of x'
00:21:09.930 --> 00:21:11.430
for any old 'n'.
00:21:11.430 --> 00:21:14.140
Recall what our definition
was, now, especially
00:21:14.140 --> 00:21:16.000
with this as review.
00:21:16.000 --> 00:21:18.960
We locate the point '1/n'
comma 'n squared'.
00:21:18.960 --> 00:21:20.710
We then draw the line
that goes from the
00:21:20.710 --> 00:21:22.180
origin to that point.
00:21:22.180 --> 00:21:26.220
Then we draw the line that goes
from that point back to
00:21:26.220 --> 00:21:29.410
the x-axis at the point '2/n'.
00:21:29.410 --> 00:21:32.300
And then we come across
to 'x' equals 2.
00:21:32.300 --> 00:21:37.770
Let's try to visualize what the
integral from 0 to 2, ''f
00:21:37.770 --> 00:21:39.790
sub n' of x', 'dx', means.
00:21:39.790 --> 00:21:43.090
After all, in a case of a
continuous curve, which this
00:21:43.090 --> 00:21:46.250
is, isn't the definite integral
interpreted just as
00:21:46.250 --> 00:21:47.790
the area under the curve?
00:21:47.790 --> 00:21:51.370
Well, you see, the curve
coincides with the x-axis from
00:21:51.370 --> 00:21:53.380
'2/n' on to 2.
00:21:53.380 --> 00:21:57.860
Consequently, this triangular
region which I call 'R' is the
00:21:57.860 --> 00:21:59.210
area under the curve.
00:21:59.210 --> 00:22:02.440
In other words, the integral
from 0 to 2, ''f sub n' of x',
00:22:02.440 --> 00:22:05.720
'dx', is the area of
the region 'R'.
00:22:05.720 --> 00:22:06.750
But here's the point.
00:22:06.750 --> 00:22:09.870
We can compute the area of the
region 'R' very easily.
00:22:09.870 --> 00:22:11.780
It's a triangle, right?
00:22:11.780 --> 00:22:13.300
What is the area
of a triangle?
00:22:13.300 --> 00:22:17.040
Well, it's 1/2 times
the base--
00:22:17.040 --> 00:22:19.410
but the base is just '2/n'--
00:22:19.410 --> 00:22:20.790
times the height.
00:22:20.790 --> 00:22:22.900
The height is 'n squared'.
00:22:22.900 --> 00:22:27.150
In other words, the area of the
region 'R' simply is 'n'.
00:22:27.150 --> 00:22:28.530
And that's rather interesting.
00:22:28.530 --> 00:22:32.730
In other words, for each 'n',
this particular integral just
00:22:32.730 --> 00:22:34.770
turns out to be 'n' itself.
00:22:34.770 --> 00:22:37.220
That's what's interesting about
this particular diagram.
00:22:37.220 --> 00:22:40.620
In other words, this thing rises
so high that even though
00:22:40.620 --> 00:22:44.800
the base gets very, very small
as 'n' gets large, the height
00:22:44.800 --> 00:22:47.980
increases so rapidly that the
area under this curve,
00:22:47.980 --> 00:22:52.270
numerically, is always
equal to 'n'.
00:22:52.270 --> 00:22:54.300
In fact, we can check
that if you'd like.
00:22:54.300 --> 00:22:56.160
Come back to this
particular case.
00:22:56.160 --> 00:22:58.280
Look at this particular
triangle.
00:22:58.280 --> 00:23:00.510
The base is 2, the
height is 1.
00:23:00.510 --> 00:23:04.560
The area is 1 unit.
00:23:04.560 --> 00:23:07.490
Look at this particular
triangle.
00:23:07.490 --> 00:23:12.520
The base is 1/10, the
height is 400.
00:23:12.520 --> 00:23:17.540
400 times 1/10 is 40, and
half of that is 20.
00:23:17.540 --> 00:23:20.470
The area of this triangle
is 20, which exactly
00:23:20.470 --> 00:23:21.590
matches this subscript.
00:23:21.590 --> 00:23:23.870
That's what's going to happen
here all the time.
00:23:23.870 --> 00:23:27.640
In particular, then, if we
compute the integral from 0 to
00:23:27.640 --> 00:23:33.810
2, 'f of n', 'x dx', and then
let the limit as 'n' goes to
00:23:33.810 --> 00:23:36.430
infinity be computed, what
do we get for an answer?
00:23:36.430 --> 00:23:40.450
We get that this limit is the
limit of 'n' as 'n' approaches
00:23:40.450 --> 00:23:43.690
infinity, and that of
course is infinity.
00:23:43.690 --> 00:23:47.050
On the other hand, suppose we
bring the limit inside?
00:23:47.050 --> 00:23:49.960
In other words, suppose
we compute this.
00:23:49.960 --> 00:23:53.620
Well, the point is that we have
already shown that this
00:23:53.620 --> 00:23:57.370
is identically 0 for all 'x'.
00:23:57.370 --> 00:24:01.800
Consequently, this integral is
the integral from 0 to 2, 0
00:24:01.800 --> 00:24:03.990
'dx', which is 0.
00:24:03.990 --> 00:24:06.760
In other words, if you first
take the limit and then
00:24:06.760 --> 00:24:09.530
integrate, you get 0.
00:24:09.530 --> 00:24:12.490
On the other hand, if you first
integrate and then take
00:24:12.490 --> 00:24:14.490
the limit, you get infinity.
00:24:14.490 --> 00:24:17.820
And this should be a glaring
example to show you that the
00:24:17.820 --> 00:24:20.990
answer that you get indeed does
depend on the order in
00:24:20.990 --> 00:24:22.620
which you do the operation.
00:24:22.620 --> 00:24:24.260
Again, let me emphasize--
00:24:24.260 --> 00:24:26.600
which of these two is wrong?
00:24:26.600 --> 00:24:29.210
The answer is, neither
is wrong.
00:24:29.210 --> 00:24:32.150
All we're saying is that if
you were supposed to solve
00:24:32.150 --> 00:24:35.710
this problem and by mistake you
solve this one, you are
00:24:35.710 --> 00:24:39.150
going to get a drastically
different answer.
00:24:39.150 --> 00:24:40.230
OK.
00:24:40.230 --> 00:24:41.670
Let's not beat this to death.
00:24:41.670 --> 00:24:43.170
So far, so good.
00:24:43.170 --> 00:24:47.340
Let me make one more remark,
namely, what does all of this
00:24:47.340 --> 00:24:49.550
have to do with the
study of series?
00:24:49.550 --> 00:24:53.710
See, now we're just talking
about sequences of functions.
00:24:53.710 --> 00:24:56.630
And you see, the answer to this
question is essentially
00:24:56.630 --> 00:25:02.020
going to be our last lecture
of the course.
00:25:02.020 --> 00:25:04.140
But for now, what I'd like
to do is to give you
00:25:04.140 --> 00:25:05.420
a preview of that.
00:25:05.420 --> 00:25:08.580
Namely, the application of
uniform convergence to series
00:25:08.580 --> 00:25:09.750
is the following.
00:25:09.750 --> 00:25:13.300
Recall that when we write
summation 'n' goes from 0 to
00:25:13.300 --> 00:25:16.310
infinity, 'a sub n', 'x
to the n', that's an
00:25:16.310 --> 00:25:18.070
abbreviation for what?
00:25:18.070 --> 00:25:23.860
A polynomial, 'k' goes from 0 to
'n', 'a sub k', 'x sub k',
00:25:23.860 --> 00:25:25.550
as 'n' goes to infinity.
00:25:25.550 --> 00:25:30.390
In other words, recall that
the sum of the series is a
00:25:30.390 --> 00:25:34.240
limit of a sequence of partial
sums, and this is the n-th
00:25:34.240 --> 00:25:37.020
member of that sequence
of partial sums.
00:25:37.020 --> 00:25:40.560
Again, if the sigma notation is
throwing you off, all I'm
00:25:40.560 --> 00:25:46.800
saying is to observe that 'a0'
plus 'a1 x' plus 'a2 'x
00:25:46.800 --> 00:25:50.160
squared'' plus-- et cetera, et
cetera, et cetera, forever,
00:25:50.160 --> 00:25:53.500
just represents the limit of
the following sequence.
00:25:53.500 --> 00:25:57.940
'a0', next member is
'a0 plus a1 x'.
00:25:57.940 --> 00:26:02.090
Next member is 'a0' plus 'a1
x' plus 'a2 'x squared''.
00:26:02.090 --> 00:26:06.310
The next member is 'a0' plus 'a1
x' plus 'a2 'x squared''
00:26:06.310 --> 00:26:07.960
plus 'a3 'x cubed''.
00:26:07.960 --> 00:26:10.860
By the way, what is each
member of the sequence?
00:26:10.860 --> 00:26:13.290
It's a polynomial.
00:26:13.290 --> 00:26:16.570
And polynomials have very
nice properties,
00:26:16.570 --> 00:26:17.960
among which are what?
00:26:17.960 --> 00:26:20.780
Well, a polynomial is a
continuous function.
00:26:20.780 --> 00:26:23.850
A polynomial is an integral
function, et cetera.
00:26:23.850 --> 00:26:27.670
The idea, therefore, is that
if this sequence of partial
00:26:27.670 --> 00:26:35.840
sums converges uniformly to the
limit function, then, for
00:26:35.840 --> 00:26:39.800
example, the limit function,
namely the power series, must
00:26:39.800 --> 00:26:44.640
be continuous since each partial
sum that makes up the
00:26:44.640 --> 00:26:47.570
sequence of partial sums
is also continuous.
00:26:47.570 --> 00:26:50.100
Namely, every polynomial
is continuous.
00:26:50.100 --> 00:26:53.670
Also, if, for some reason or
other, you want to integrate
00:26:53.670 --> 00:26:57.260
that power series from 'a' to
'b', if the convergence is
00:26:57.260 --> 00:26:59.800
uniform, I can then do what?
00:26:59.800 --> 00:27:03.880
I can then take the summation
sign outside, integrate the
00:27:03.880 --> 00:27:08.100
n-th partial sum, and
add these all up.
00:27:08.100 --> 00:27:09.610
You see, the idea being what?
00:27:09.610 --> 00:27:13.350
That the n-th partial sum is a
polynomial, and a polynomial
00:27:13.350 --> 00:27:16.560
is a particularly simple
thing to integrate.
00:27:16.560 --> 00:27:18.400
That's one of the easiest
functions to
00:27:18.400 --> 00:27:20.150
integrate, in fact.
00:27:20.150 --> 00:27:21.380
OK.
00:27:21.380 --> 00:27:23.120
Now, here's the wrap up, then.
00:27:23.120 --> 00:27:26.760
What we shall show next time is
that within the interval of
00:27:26.760 --> 00:27:30.780
absolute convergence, the
sequence of partial sums,
00:27:30.780 --> 00:27:36.620
which we already know converges
absolutely to the
00:27:36.620 --> 00:27:39.840
limit function, also converges
uniformly.
00:27:39.840 --> 00:27:43.040
In other words, within the
radius of convergence, the
00:27:43.040 --> 00:27:43.670
power series--
00:27:43.670 --> 00:27:46.530
and I don't know how to say
this other than to say, it
00:27:46.530 --> 00:27:50.750
enjoys the usual polynomial
properties associated with a
00:27:50.750 --> 00:27:54.610
polynomial such as summation 'k'
goes from 0 to 'n', 'a sub
00:27:54.610 --> 00:27:57.090
k', 'x to the k'.
00:27:57.090 --> 00:28:00.820
In other words, then, this
about wraps up what our
00:28:00.820 --> 00:28:03.780
introductory lecture for today
wanted to be, namely the
00:28:03.780 --> 00:28:06.250
concept of uniform
convergence.
00:28:06.250 --> 00:28:09.320
What I would like you to do now
is to study this material
00:28:09.320 --> 00:28:12.040
very carefully in the
supplementary notes, go over
00:28:12.040 --> 00:28:15.260
the learning exercises so that
you become familiar with this.
00:28:15.260 --> 00:28:18.080
Then we will wrap up our course
in our next lecture,
00:28:18.080 --> 00:28:23.480
when we show what a very, very
powerful tool this particular
00:28:23.480 --> 00:28:28.480
concept of absolute convergence
is in the study of
00:28:28.480 --> 00:28:31.110
the mathematical concept
of convergence.
00:28:31.110 --> 00:28:33.970
At any rate, until next
time, then, goodbye.
00:28:36.550 --> 00:28:39.750
Funding for the publication of
this video was provided by the
00:28:39.750 --> 00:28:43.810
Gabriella and Paul Rosenbaum
Foundation.
00:28:43.810 --> 00:28:47.970
Help OCW continue to provide
free and open access to MIT
00:28:47.970 --> 00:28:52.170
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