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HERBERT GROSS: Hi.
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Well, I guess we're in the home
stretch of the course.
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We're down to our last concept,
and it will take us
13
00:00:39,210 --> 00:00:43,150
essentially two lectures to
cover this new concept.
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00:00:43,150 --> 00:00:46,430
Today we'll do the concept in
general and next time we'll
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00:00:46,430 --> 00:00:50,640
apply it specifically to
the concept of series.
16
00:00:50,640 --> 00:00:53,710
But the concept we want to talk
about is something called
17
00:00:53,710 --> 00:00:55,420
'Uniform Convergence'.
18
00:00:55,420 --> 00:00:59,050
Let me say at the outset that
this is a very subtle topic.
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00:00:59,050 --> 00:01:00,510
It is difficult.
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00:01:00,510 --> 00:01:03,620
It seems to be beyond the scope
of our textbook, because
21
00:01:03,620 --> 00:01:04,900
it's not mentioned there.
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00:01:04,900 --> 00:01:08,840
Consequently, I will try to give
the highlights as I speak
23
00:01:08,840 --> 00:01:11,430
with you, but the supplementary
notes will
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00:01:11,430 --> 00:01:15,210
contain a more detailed
explanation of the things that
25
00:01:15,210 --> 00:01:16,770
we're going to talk about.
26
00:01:16,770 --> 00:01:20,690
Now, to set the stage properly
for our discussion of uniform
27
00:01:20,690 --> 00:01:24,980
convergence, I think it's wise
that we at least review the
28
00:01:24,980 --> 00:01:27,670
concept of convergence
in general.
29
00:01:27,670 --> 00:01:30,180
Well, let's take a
look over here.
30
00:01:30,180 --> 00:01:34,320
Recall that when we write that
the limit of ''f sub n' of x'
31
00:01:34,320 --> 00:01:39,360
as 'n' approaches infinity
equals 'f of x' for all 'x' in
32
00:01:39,360 --> 00:01:40,980
the closed interval
from 'a' to 'b'--
33
00:01:40,980 --> 00:01:44,910
when this happens, another way
of saying this is that the
34
00:01:44,910 --> 00:01:48,110
sequence 'f sub n', the sequence
of functions 'f sub
35
00:01:48,110 --> 00:01:52,710
n', converges to the function
'f' on the closed interval
36
00:01:52,710 --> 00:01:54,060
from 'a' to 'b'.
37
00:01:54,060 --> 00:01:57,350
Now, again, these tend to be
words unless you look at a
38
00:01:57,350 --> 00:01:58,520
specific example.
39
00:01:58,520 --> 00:02:00,210
Let's just pick one over here.
40
00:02:00,210 --> 00:02:03,630
Let ''f sub n' of x' be
''n' over '2n plus
41
00:02:03,630 --> 00:02:05,940
1'' times 'x squared'.
42
00:02:05,940 --> 00:02:10,880
Notice, of course, that the
value of this particular
43
00:02:10,880 --> 00:02:14,880
number depends both
on 'x' and 'n'.
44
00:02:14,880 --> 00:02:19,770
At any rate, let's pick a fixed
'x', hold it that way,
45
00:02:19,770 --> 00:02:22,240
and take the limit of
''f sub n' of x' as
46
00:02:22,240 --> 00:02:23,770
'n' approaches infinity.
47
00:02:23,770 --> 00:02:26,300
In other words, as we let 'n'
approach infinity in this
48
00:02:26,300 --> 00:02:29,020
case, notice that the limit
of 'n' over '2n
49
00:02:29,020 --> 00:02:31,260
plus 1' becomes 1/2.
50
00:02:31,260 --> 00:02:33,050
'x' has been chosen
independently of
51
00:02:33,050 --> 00:02:34,150
the choice of 'n'.
52
00:02:34,150 --> 00:02:36,510
Consequently, the limit function
in this case, 'f of
53
00:02:36,510 --> 00:02:39,880
x', is 1/2 'x squared'.
54
00:02:39,880 --> 00:02:43,150
And in this case, what we say
is that the sequence of
55
00:02:43,150 --> 00:02:48,440
functions 'n 'x squared'' over
'2n plus 1' converges to 1/2
56
00:02:48,440 --> 00:02:49,720
'x squared'.
57
00:02:49,720 --> 00:02:52,240
Now what does this mean
more specifically?
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00:02:52,240 --> 00:02:54,250
In other words, let's see if we
can look at a few specific
59
00:02:54,250 --> 00:02:55,450
values of 'x'.
60
00:02:55,450 --> 00:02:59,390
For example, if I choose
'x' to be 2--
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00:02:59,390 --> 00:03:02,360
in other words, if I choose 'x'
to be 2, if we look over
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00:03:02,360 --> 00:03:06,430
here, this says that ''f
sub n' of x' is '4n'
63
00:03:06,430 --> 00:03:08,490
over '2n plus 1'.
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00:03:08,490 --> 00:03:11,550
If I now take the limit as 'n'
approaches infinity, I'm going
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00:03:11,550 --> 00:03:13,800
to wind up with what?
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00:03:13,800 --> 00:03:18,560
'x' is replaced by 2, 2 squared
is 4, 1/2 of 4 is 2.
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00:03:18,560 --> 00:03:22,500
In other words, the limit of ''f
sub n' of 2', as 'n' goes
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00:03:22,500 --> 00:03:24,270
to infinity, is 2.
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00:03:24,270 --> 00:03:29,270
In a similar way, if I replace
'x' by 4, 1/2 'x
70
00:03:29,270 --> 00:03:30,560
squared' becomes 8.
71
00:03:30,560 --> 00:03:33,870
And so what we have is, at the
limit, as 'n' approaches
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00:03:33,870 --> 00:03:36,790
infinity, ''f sub
n' of 4' is 8.
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00:03:36,790 --> 00:03:39,980
The key thing being that once
you choose 'x', notice that
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00:03:39,980 --> 00:03:43,870
for a fixed 'x', ''f sub n' of
x' is a constant and you're
75
00:03:43,870 --> 00:03:47,690
now taking the limit of
a sequence of numbers.
76
00:03:47,690 --> 00:03:49,990
At any rate, here's what
the key point is.
77
00:03:49,990 --> 00:03:53,120
What does this mean by
our basic definition?
78
00:03:53,120 --> 00:03:55,910
By our basic definition, it
means that we can find the
79
00:03:55,910 --> 00:03:59,540
number, capital 'N sub 1', such
that when 'n' is greater
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00:03:59,540 --> 00:04:03,020
than capital 'N sub 1', the
absolute value of ''f sub n'
81
00:04:03,020 --> 00:04:06,260
of 2', minus 2, is less
than epsilon.
82
00:04:06,260 --> 00:04:10,170
In a similar way, this means
that we can find the number
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00:04:10,170 --> 00:04:13,400
capital 'N sub 2' such that
for any 'n' greater than
84
00:04:13,400 --> 00:04:16,426
capital 'N sub 2', the absolute
value of ''f sub n'
85
00:04:16,426 --> 00:04:19,680
of 4' minus 8 is also
less than epsilon.
86
00:04:19,680 --> 00:04:24,060
The key point is that 'N1' and
'N2' can be different.
87
00:04:24,060 --> 00:04:27,590
In other words, you may have to
go out further to make this
88
00:04:27,590 --> 00:04:30,500
difference less than epsilon
than you do to make this
89
00:04:30,500 --> 00:04:32,190
difference less than epsilon.
90
00:04:32,190 --> 00:04:34,900
In other words, you see what's
happening here-- and we're
91
00:04:34,900 --> 00:04:36,830
going to review this in writing
in a few minutes, so
92
00:04:36,830 --> 00:04:38,310
that you'll see it
in front you.
93
00:04:38,310 --> 00:04:42,350
What happens here is that you
see that for different values
94
00:04:42,350 --> 00:04:45,460
of 'x', we get different
values of 'n'.
95
00:04:45,460 --> 00:04:48,910
And since there are infinitely
many values of 'x', it means
96
00:04:48,910 --> 00:04:51,150
that in general, we're going
to be in a little bit of
97
00:04:51,150 --> 00:04:55,430
trouble trying to find one 'n'
that works for everything.
98
00:04:55,430 --> 00:04:56,460
And let me show you
what that means
99
00:04:56,460 --> 00:04:59,230
again, going more slowly.
100
00:04:59,230 --> 00:05:01,720
I simply call this two
basic definitions.
101
00:05:01,720 --> 00:05:04,720
In other words, if we have a
sequence of functions 'f sub
102
00:05:04,720 --> 00:05:07,670
n', each of which is defined
on the closed interval from
103
00:05:07,670 --> 00:05:11,680
'a' to 'b', we say that that
sequence of functions
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00:05:11,680 --> 00:05:13,450
converges point-wise--
105
00:05:13,450 --> 00:05:14,910
that means number by number--
106
00:05:14,910 --> 00:05:17,370
to 'f' on [a, b]
107
00:05:17,370 --> 00:05:21,170
if this limit, ''f sub n' of x'
as 'n' approaches infinity,
108
00:05:21,170 --> 00:05:25,340
equals 'f of x' for
each 'x' in [a,b].
109
00:05:25,340 --> 00:05:27,930
In other words, given epsilon
greater than 0, we can find
110
00:05:27,930 --> 00:05:31,550
'N1' such that 'n' greater than
'N1' implies that the
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00:05:31,550 --> 00:05:35,900
absolute value of ''f sub n' of
x1' minus 'f of x1' is less
112
00:05:35,900 --> 00:05:39,720
than epsilon for a given
'x1' in [a,b].
113
00:05:39,720 --> 00:05:43,860
In general, the choice of 'N1'
depends on the choice of 'x1',
114
00:05:43,860 --> 00:05:47,450
and there are infinitely many
such choices to make in [a,b].
115
00:05:47,450 --> 00:05:50,080
Now the key point is this,
and this is where uniform
116
00:05:50,080 --> 00:05:51,740
convergence comes in.
117
00:05:51,740 --> 00:05:56,130
If we can find one 'N' such that
whenever little 'n' is
118
00:05:56,130 --> 00:06:00,090
greater than that capital 'N',
''f sub n' of x' minus 'f of
119
00:06:00,090 --> 00:06:07,320
x' in absolute value is less
than epsilon for every 'x' in
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00:06:07,320 --> 00:06:10,000
the closed interval, then
we say that the
121
00:06:10,000 --> 00:06:12,530
convergence is uniform.
122
00:06:12,530 --> 00:06:16,220
In other words, if we can find
one 'N' that makes that
123
00:06:16,220 --> 00:06:19,960
difference less than epsilon for
the entire interval, then
124
00:06:19,960 --> 00:06:22,970
we call the convergence
uniform.
125
00:06:22,970 --> 00:06:26,070
Now, you see, convergence in
general is a tough topic.
126
00:06:26,070 --> 00:06:30,240
In particular, uniform
convergence may seem even more
127
00:06:30,240 --> 00:06:33,620
remote, and therefore what I'd
like to do now is-- saving the
128
00:06:33,620 --> 00:06:36,490
formal proofs for the
supplementary notes, let me
129
00:06:36,490 --> 00:06:40,860
show you pictorially just what
the concept of uniform
130
00:06:40,860 --> 00:06:42,200
convergence really is.
131
00:06:42,200 --> 00:06:45,570
132
00:06:45,570 --> 00:06:48,810
So let me give you a pictorial
representation.
133
00:06:48,810 --> 00:06:53,020
Let's suppose I have the curve
'y' equals 'f of x'.
134
00:06:53,020 --> 00:06:57,580
Now, to be within epsilon of 'f
of x' means I retrace this
135
00:06:57,580 --> 00:07:01,650
curve displaced epsilon units
above the original position,
136
00:07:01,650 --> 00:07:03,350
and epsilon units below.
137
00:07:03,350 --> 00:07:06,430
In other words, for a given an
epsilon, I now draw the curve
138
00:07:06,430 --> 00:07:09,710
'y' equals ''f of x' plus
epsilon' and 'y' equals ''f of
139
00:07:09,710 --> 00:07:11,730
x' minus epsilon'.
140
00:07:11,730 --> 00:07:15,390
Now, what uniform convergence
means is this, that for this
141
00:07:15,390 --> 00:07:19,630
given epsilon, I can find a
capital 'N' such that whenever
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00:07:19,630 --> 00:07:23,290
'n' is greater than capital 'N',
the curve 'y' equals ''f
143
00:07:23,290 --> 00:07:26,190
sub n' of x' lies in
this shaded region.
144
00:07:26,190 --> 00:07:29,960
In other words, it can bounce
around all over, but it can't
145
00:07:29,960 --> 00:07:31,180
get outside of this region.
146
00:07:31,180 --> 00:07:35,270
In other words, once I'm far
enough out of my sequence, all
147
00:07:35,270 --> 00:07:41,120
of the curves lie in this
particular region.
148
00:07:41,120 --> 00:07:43,590
Now, of course, the question
is what does this all mean?
149
00:07:43,590 --> 00:07:44,820
And the answer is--
well, look.
150
00:07:44,820 --> 00:07:48,510
Let's take epsilon to be, and I
put this in quotation marks,
151
00:07:48,510 --> 00:07:52,020
"very, very small." Let's take
epsilon, for example, to be so
152
00:07:52,020 --> 00:07:55,290
small that it's within the
thickness of our chalk.
153
00:07:55,290 --> 00:07:56,540
If I now do this--
154
00:07:56,540 --> 00:07:58,900
155
00:07:58,900 --> 00:08:01,540
see, I draw the curve
'y' equals 'f of x'.
156
00:08:01,540 --> 00:08:04,930
Notice now that the thickness of
my curve itself is the band
157
00:08:04,930 --> 00:08:06,710
width 2 epsilon.
158
00:08:06,710 --> 00:08:10,410
All I'm saying is that for this
very, very small epsilon,
159
00:08:10,410 --> 00:08:14,140
when 'n' is sufficiently large,
the curve 'y' equals
160
00:08:14,140 --> 00:08:18,720
''f sub n' of x' appears to
lie inside of this curve.
161
00:08:18,720 --> 00:08:21,590
You see, in other words, what
you're saying is that for a
162
00:08:21,590 --> 00:08:24,930
large enough 'n' and small
enough epsilon--
163
00:08:24,930 --> 00:08:28,330
loosely speaking what you saying
is that 'y' equals ''f
164
00:08:28,330 --> 00:08:31,620
sub n' of x' looks like 'y'
equals 'f of x', for a
165
00:08:31,620 --> 00:08:33,510
sufficiently large
values of 'n'.
166
00:08:33,510 --> 00:08:38,220
In other words, it appears that
we can't really tell the
167
00:08:38,220 --> 00:08:42,750
n-th curve in the sequence
from the limit function.
168
00:08:42,750 --> 00:08:44,430
And I want to make a few
key observations
169
00:08:44,430 --> 00:08:45,850
about what that means.
170
00:08:45,850 --> 00:08:48,370
I've written the whole thing out
on the blackboard so that
171
00:08:48,370 --> 00:08:51,860
you can see this after I say it,
but what I want you to see
172
00:08:51,860 --> 00:08:55,100
is, can you begin to get the
feeling that with this kind of
173
00:08:55,100 --> 00:08:56,970
a condition, for example--
174
00:08:56,970 --> 00:09:01,260
if each 'f sub n' happens to be
continuous, in other words,
175
00:09:01,260 --> 00:09:05,200
if each member of my sequence
is unbroken, then the limit
176
00:09:05,200 --> 00:09:07,430
function itself must
also be unbroken.
177
00:09:07,430 --> 00:09:10,970
Because you see I can squeeze
this thing down to such a
178
00:09:10,970 --> 00:09:14,950
narrow width that there's no
room for a break in here.
179
00:09:14,950 --> 00:09:19,700
Also notice that if the curve
'y' equals ''f sub n' of x' is
180
00:09:19,700 --> 00:09:21,490
caught inside this curve--
181
00:09:21,490 --> 00:09:24,840
if I, for example, were
computing the area of the
182
00:09:24,840 --> 00:09:26,010
region 'R'--
183
00:09:26,010 --> 00:09:28,970
for a large enough 'n', I
couldn't tell the difference
184
00:09:28,970 --> 00:09:33,470
in area if I use 'y' equals 'f
of x' for my top curve, or
185
00:09:33,470 --> 00:09:36,730
whether I use 'y' equals
''f sub n' of x'.
186
00:09:36,730 --> 00:09:40,560
Well, keep that in mind, and all
I'm saying is this, that
187
00:09:40,560 --> 00:09:43,810
from our picture, it should
seem clear that if the
188
00:09:43,810 --> 00:09:47,670
sequence 'f sub n' converges
uniformly to 'f' on [a, b],
189
00:09:47,670 --> 00:09:51,420
and if each 'f sub n' is
continuous on [a,b], then--
190
00:09:51,420 --> 00:09:53,140
and this is a fundamental
result.
191
00:09:53,140 --> 00:09:57,380
Fundamental result one, 'f' is
also continuous on [a, b].
192
00:09:57,380 --> 00:10:01,230
193
00:10:01,230 --> 00:10:02,240
Now you say, look.
194
00:10:02,240 --> 00:10:04,300
That's what you'd expect, isn't
it, if every member of
195
00:10:04,300 --> 00:10:05,620
the sequence is continuous?
196
00:10:05,620 --> 00:10:08,570
Why shouldn't the limit function
also be continuous?
197
00:10:08,570 --> 00:10:12,030
The point is, well, maybe that's
what you expect, but
198
00:10:12,030 --> 00:10:13,500
note this--
199
00:10:13,500 --> 00:10:17,070
in one of our earlier lectures,
we already saw that
200
00:10:17,070 --> 00:10:20,640
if we only had point-wise
convergence, this did not need
201
00:10:20,640 --> 00:10:21,640
to be true.
202
00:10:21,640 --> 00:10:25,240
In particular, recall our
example in which we defined
203
00:10:25,240 --> 00:10:29,480
''f sub n' of x' to be 'x to
the n', where the domain of
204
00:10:29,480 --> 00:10:32,650
'f' was the closed interval
from 0 to 1.
205
00:10:32,650 --> 00:10:34,330
Remember what happened
in that case?
206
00:10:34,330 --> 00:10:36,960
Each of these 'f sub
n's is continuous.
207
00:10:36,960 --> 00:10:42,060
But the limit function, you
may recall, is what?
208
00:10:42,060 --> 00:10:47,990
It's 0 if 'x' is less than
1, and 1 if 'x' equals 1.
209
00:10:47,990 --> 00:10:51,540
In other words, the limit
function was discontinuous at
210
00:10:51,540 --> 00:10:52,890
'x' equals 1.
211
00:10:52,890 --> 00:10:55,360
By the way, you might like to
see what this means from
212
00:10:55,360 --> 00:10:56,870
another point of view.
213
00:10:56,870 --> 00:10:58,750
And let me show you what
it does mean from
214
00:10:58,750 --> 00:10:59,910
another point of view.
215
00:10:59,910 --> 00:11:04,190
Since 'f' continuous at 'x'
equals 'x sub 1' means that
216
00:11:04,190 --> 00:11:08,110
the limit of 'f of x' as 'x'
approaches 'x1' is 'f of x1',
217
00:11:08,110 --> 00:11:13,620
and since 'f of x' itself, by
definition, is the limit of
218
00:11:13,620 --> 00:11:22,360
''f sub n' of x' as 'n'
approaches infinity, we may
219
00:11:22,360 --> 00:11:26,550
rewrite our first condition
in the form-- what?
220
00:11:26,550 --> 00:11:32,100
We may rewrite our first
equation, this equation here,
221
00:11:32,100 --> 00:11:33,320
in what form?
222
00:11:33,320 --> 00:11:38,820
Limit as 'x' approaches 'x1'
of limit 'n' approaches
223
00:11:38,820 --> 00:11:41,270
infinity, ''f sub n' of x'.
224
00:11:41,270 --> 00:11:44,470
And that equals the limit as 'n'
approaches infinity, ''f
225
00:11:44,470 --> 00:11:46,660
sub n' of x1'.
226
00:11:46,660 --> 00:11:50,330
Now keep in mind that since each
'f sub n' is given to be
227
00:11:50,330 --> 00:11:55,510
continuous, by definition of
continuity ''f sub n' of x1'
228
00:11:55,510 --> 00:11:59,280
is the same as saying the limit
as 'x' approaches 'x1',
229
00:11:59,280 --> 00:12:00,690
''f sub n' of x'.
230
00:12:00,690 --> 00:12:05,060
The point I'm making is, if you
now put this together with
231
00:12:05,060 --> 00:12:09,410
this, it says the rather
remarkable thing that unless
232
00:12:09,410 --> 00:12:13,860
you have uniform convergence
when you interchange the order
233
00:12:13,860 --> 00:12:16,770
in which you take the limits
over here, you may very well
234
00:12:16,770 --> 00:12:18,200
get a different answer.
235
00:12:18,200 --> 00:12:20,920
In other words, when you're
dealing with non-uniform
236
00:12:20,920 --> 00:12:25,550
convergence, you must be very,
very careful to perform every
237
00:12:25,550 --> 00:12:27,650
operation in the given order.
238
00:12:27,650 --> 00:12:29,210
What we're saying is what?
239
00:12:29,210 --> 00:12:31,090
This will give you an answer.
240
00:12:31,090 --> 00:12:32,710
This will give you an answer.
241
00:12:32,710 --> 00:12:36,140
But if the convergence is not
uniform, the answers may be
242
00:12:36,140 --> 00:12:39,730
different, and consequently by
changing the order you destroy
243
00:12:39,730 --> 00:12:42,000
the whole physical meaning
of the problem.
244
00:12:42,000 --> 00:12:44,590
Well, again, that's reemphasized
in the
245
00:12:44,590 --> 00:12:45,890
supplementary notes.
246
00:12:45,890 --> 00:12:47,940
Let me continue on here.
247
00:12:47,940 --> 00:12:51,510
Let me tell you another
interesting property of
248
00:12:51,510 --> 00:12:54,510
uniform convergence.
249
00:12:54,510 --> 00:12:58,420
Suppose the sequence ''f sub n'
of x' converges uniformly
250
00:12:58,420 --> 00:13:00,590
to 'f of x' on [a, b].
251
00:13:00,590 --> 00:13:03,220
The point that's rather
interesting is that you can
252
00:13:03,220 --> 00:13:06,530
reverse the order of integration
and taking the
253
00:13:06,530 --> 00:13:08,210
limit in this particular case.
254
00:13:08,210 --> 00:13:12,060
In other words, suppose you want
to compute the integral
255
00:13:12,060 --> 00:13:14,800
of the limit function
from 'a' to 'b'.
256
00:13:14,800 --> 00:13:19,790
What you can do instead is
compute the integral of the
257
00:13:19,790 --> 00:13:23,440
n-th number of your sequence,
and then take the limit as 'n'
258
00:13:23,440 --> 00:13:24,700
goes to infinity.
259
00:13:24,700 --> 00:13:27,820
In other words, rewriting this,
it says that if you have
260
00:13:27,820 --> 00:13:32,080
uniform convergence, you can
take the limit inside the
261
00:13:32,080 --> 00:13:33,470
integral sign.
262
00:13:33,470 --> 00:13:37,740
And again, these results
are proven in our
263
00:13:37,740 --> 00:13:39,400
supplementary notes.
264
00:13:39,400 --> 00:13:43,040
We also say a few words about
corresponding results for
265
00:13:43,040 --> 00:13:45,750
differentiation in our
supplementary notes.
266
00:13:45,750 --> 00:13:50,100
And I should point out that
differentiation is a far more
267
00:13:50,100 --> 00:13:52,000
subtle thing than integration.
268
00:13:52,000 --> 00:13:55,080
See, remember that for
integration, all you need is
269
00:13:55,080 --> 00:13:56,520
continuity.
270
00:13:56,520 --> 00:13:59,340
For differentiation, you
need smoothness.
271
00:13:59,340 --> 00:14:04,420
The point is that as you put a
thin band around the function
272
00:14:04,420 --> 00:14:07,760
'y' equals 'f of x' when you
have uniform convergence,
273
00:14:07,760 --> 00:14:10,410
that's enough to make sure that
the limit function must
274
00:14:10,410 --> 00:14:13,200
be continuous if each of the
members in the sequence is
275
00:14:13,200 --> 00:14:13,960
continuous.
276
00:14:13,960 --> 00:14:16,790
But without going into the
details of this thing, it does
277
00:14:16,790 --> 00:14:20,000
turn out that for the degree of
smoothness that you need,
278
00:14:20,000 --> 00:14:22,510
these things can jump around
enough so that for
279
00:14:22,510 --> 00:14:25,860
differentiation, we do have to
be a little bit more careful.
280
00:14:25,860 --> 00:14:28,650
Rather than to becloud the
issue, I will stick with
281
00:14:28,650 --> 00:14:31,810
integration topics for
our lecture today.
282
00:14:31,810 --> 00:14:34,880
Now, the other thing that I want
to mention is, again, in
283
00:14:34,880 --> 00:14:38,830
terms of doing what comes
naturally, I think we are
284
00:14:38,830 --> 00:14:41,850
tempted to look at something
like this and say, look, all I
285
00:14:41,850 --> 00:14:43,270
did was bring the
limit inside.
286
00:14:43,270 --> 00:14:45,360
Why can't I do that?
287
00:14:45,360 --> 00:14:48,120
And instead of saying, look,
you can't, I think the best
288
00:14:48,120 --> 00:14:50,290
thing to show is that when
you don't have uniform
289
00:14:50,290 --> 00:14:52,570
convergence, you get two
different answers.
290
00:14:52,570 --> 00:14:53,980
Again, the idea being what?
291
00:14:53,980 --> 00:14:56,270
We are not saying that
you can't do this.
292
00:14:56,270 --> 00:14:58,920
We are not saying that you
can't compute this.
293
00:14:58,920 --> 00:15:01,530
All we're saying is that if
the convergence is not
294
00:15:01,530 --> 00:15:08,050
uniform, these two expressions
may very well
295
00:15:08,050 --> 00:15:10,880
name different numbers.
296
00:15:10,880 --> 00:15:13,180
Now, to show you what I have in
mind here, let me give you
297
00:15:13,180 --> 00:15:15,110
an example.
298
00:15:15,110 --> 00:15:18,180
See, to show you that 2 need not
be true if the convergence
299
00:15:18,180 --> 00:15:21,630
is not uniform, consider
the following example.
300
00:15:21,630 --> 00:15:25,290
Now, in the supplementary notes,
I repeat this example
301
00:15:25,290 --> 00:15:28,270
both the way I have it on the
board and also from an
302
00:15:28,270 --> 00:15:30,880
algebraic point of view, without
using the pictures.
303
00:15:30,880 --> 00:15:33,700
But in terms of the picture,
here's what we do.
304
00:15:33,700 --> 00:15:37,020
We define a function on the
closed interval from 0 to 2,
305
00:15:37,020 --> 00:15:40,160
which I'll call 'f sub
n', as follows.
306
00:15:40,160 --> 00:15:45,460
For a given 'n', I will locate
the points '1/n' and '2/n'.
307
00:15:45,460 --> 00:15:49,900
For example, if 'n' happened to
be 50, this would be 1/50
308
00:15:49,900 --> 00:15:52,030
and this would be 2/50.
309
00:15:52,030 --> 00:15:57,570
Now what I do, is at the 'x'
value '1/n', I take as the
310
00:15:57,570 --> 00:16:01,510
corresponding y-value,
'n squared'.
311
00:16:01,510 --> 00:16:05,060
And I draw the straight line
that goes from the origin to
312
00:16:05,060 --> 00:16:08,110
this point, '1/n' comma
'n squared'.
313
00:16:08,110 --> 00:16:12,290
Then I draw the straight line
that comes right back to the
314
00:16:12,290 --> 00:16:17,800
x-intercept, '2/n', and I finish
off the curve by just
315
00:16:17,800 --> 00:16:22,180
letting it hug the x-axis till
we get over to 'x' equals 2.
316
00:16:22,180 --> 00:16:24,660
I'll come back to this on the
next board, to show you why I
317
00:16:24,660 --> 00:16:27,590
chose this, but let's make a few
observations just to make
318
00:16:27,590 --> 00:16:30,860
sure that you understand what
this function looks like.
319
00:16:30,860 --> 00:16:33,630
I'll just make a few arbitrary
remarks about it.
320
00:16:33,630 --> 00:16:36,310
First of all, for each
'n', ''f sub n' of
321
00:16:36,310 --> 00:16:38,640
'1/n'' is 'n squared'.
322
00:16:38,640 --> 00:16:41,010
That's just another way
of indicating a
323
00:16:41,010 --> 00:16:42,260
label for this point.
324
00:16:42,260 --> 00:16:44,610
325
00:16:44,610 --> 00:16:50,990
Secondly, my claim is that for
any number 'x sub 0', if 'x
326
00:16:50,990 --> 00:16:56,580
sub 0' is greater than '2/n',
''f sub n' of x0' must be 0.
327
00:16:56,580 --> 00:16:58,810
And the reason for that
is quite simple.
328
00:16:58,810 --> 00:17:00,760
I'm just trying to show you
how to read this picture.
329
00:17:00,760 --> 00:17:04,410
Namely, notice that as soon as
'x' gets to be as great as
330
00:17:04,410 --> 00:17:10,140
'2/n', the 'f' value is 0,
because the function is
331
00:17:10,140 --> 00:17:11,960
hugging the x-axis.
332
00:17:11,960 --> 00:17:14,579
And just as a final observation,
notice that when
333
00:17:14,579 --> 00:17:19,420
'x sub 0' is 0, ''f sub n' of 0'
is 0 for every 'n', meaning
334
00:17:19,420 --> 00:17:23,660
that every member of my family
of functions goes through this
335
00:17:23,660 --> 00:17:24,520
particular point.
336
00:17:24,520 --> 00:17:26,020
In other words, let me
just label this.
337
00:17:26,020 --> 00:17:31,850
This is 'y' equals
''f sub n' of x'.
338
00:17:31,850 --> 00:17:35,110
Well, by the way if ''f sub n'
of 0' is 0 for each 'n', in
339
00:17:35,110 --> 00:17:37,810
particular the limit of ''f sub
n' of 0' as 'n' approaches
340
00:17:37,810 --> 00:17:39,920
infinity is 0.
341
00:17:39,920 --> 00:17:43,080
What happens if we pick
a non-zero value?
342
00:17:43,080 --> 00:17:47,490
For example, suppose I pick 'x0'
to be greater than 0 but
343
00:17:47,490 --> 00:17:49,190
less than or equal to 2?
344
00:17:49,190 --> 00:17:54,030
The key point is this, that
since the limit of '2/n' as
345
00:17:54,030 --> 00:17:58,490
'n' approaches infinity is 0,
given a value of 'x0' which is
346
00:17:58,490 --> 00:18:02,720
not 0, I can find the capital
'N' such that when 'n' is
347
00:18:02,720 --> 00:18:06,660
greater than capital 'N',
'2/n' is less than 'x0'.
348
00:18:06,660 --> 00:18:10,020
In other words, if 'x0' is
greater than 0, and '2/n'
349
00:18:10,020 --> 00:18:13,860
approaches 0, for large enough
values of 'n', '2/n'
350
00:18:13,860 --> 00:18:15,340
be less than 'x0'.
351
00:18:15,340 --> 00:18:18,210
In particular, when that
happens, if we couple this
352
00:18:18,210 --> 00:18:20,010
with our earlier observation--
353
00:18:20,010 --> 00:18:21,510
what earlier observation?
354
00:18:21,510 --> 00:18:24,780
Well, this one.
355
00:18:24,780 --> 00:18:27,610
If we couple that with our
earlier observation, we see
356
00:18:27,610 --> 00:18:30,150
that when 'n' is greater than
capital 'N', ''f sub
357
00:18:30,150 --> 00:18:32,110
n' of x0' is 0.
358
00:18:32,110 --> 00:18:36,530
Correspondingly, then, the limit
of ''f sub n' of x0' as
359
00:18:36,530 --> 00:18:39,870
'n' approaches infinity,
by definition, is 0.
360
00:18:39,870 --> 00:18:40,260
In other words--
361
00:18:40,260 --> 00:18:42,650
I'm going to reinforce this
later, but notice that the
362
00:18:42,650 --> 00:18:45,650
limit function here is
the function which
363
00:18:45,650 --> 00:18:47,480
is identically 0.
364
00:18:47,480 --> 00:18:51,600
Now, since this may look very
abstract to you, let me take a
365
00:18:51,600 --> 00:18:52,450
few minutes--
366
00:18:52,450 --> 00:18:54,960
and I hope this doesn't insult
your intelligence, but let me
367
00:18:54,960 --> 00:18:58,490
just take a few minutes and
redraw this for a couple of
368
00:18:58,490 --> 00:19:01,200
different values of 'n', just
so that you can see what's
369
00:19:01,200 --> 00:19:02,450
starting to happen here.
370
00:19:02,450 --> 00:19:04,870
371
00:19:04,870 --> 00:19:08,730
Keep that picture in mind, and
now look what this means.
372
00:19:08,730 --> 00:19:15,460
For example, when 'n' is 1,
'1/n' is 1, '2/n' is 2, 'n
373
00:19:15,460 --> 00:19:16,760
squared' is 1.
374
00:19:16,760 --> 00:19:20,920
In other words, the graph 'y'
equals 'f1 of x' is just this
375
00:19:20,920 --> 00:19:23,876
triangular--
376
00:19:23,876 --> 00:19:25,480
just this.
377
00:19:25,480 --> 00:19:27,300
Why give it a name?
378
00:19:27,300 --> 00:19:30,030
Well, let's try a tougher one.
379
00:19:30,030 --> 00:19:34,100
Let's see what the member
'f sub 20' looks like.
380
00:19:34,100 --> 00:19:35,590
Recall how you draw this, now.
381
00:19:35,590 --> 00:19:38,000
With the subscript 20,
what do you do?
382
00:19:38,000 --> 00:19:43,670
You come in to the point
1/20, and at that
383
00:19:43,670 --> 00:19:44,930
point, you do what?
384
00:19:44,930 --> 00:19:49,900
You locate the point 1/20
comma 'n squared'.
385
00:19:49,900 --> 00:19:52,200
In this case, it's 400.
386
00:19:52,200 --> 00:19:54,700
And I have obviously haven't
drawn this to scale, but you
387
00:19:54,700 --> 00:19:55,440
now do what?
388
00:19:55,440 --> 00:19:57,060
Draw the straight line
that goes from the
389
00:19:57,060 --> 00:19:59,120
origin to this point.
390
00:19:59,120 --> 00:20:01,870
Then from this point, you draw
the straight line that comes
391
00:20:01,870 --> 00:20:06,190
back to the x-axis, hitting
it at 'x' equals 1/10.
392
00:20:06,190 --> 00:20:08,640
And then you come across
the x-axis all the way
393
00:20:08,640 --> 00:20:10,050
to 'x' equals 2.
394
00:20:10,050 --> 00:20:13,810
This would be the graph of 'y'
equals ''f sub 20' of x'.
395
00:20:13,810 --> 00:20:17,010
And by the way, do you sense
what's happening over here?
396
00:20:17,010 --> 00:20:23,040
See, notice that as 'n' gets
very, very large, the curve
397
00:20:23,040 --> 00:20:26,650
hugs the x-axis, starting
in closer and
398
00:20:26,650 --> 00:20:28,220
closer to the y-axis.
399
00:20:28,220 --> 00:20:31,580
But what happens is someplace
in here, no matter how close
400
00:20:31,580 --> 00:20:35,560
'x sub 0' is to 0, there
comes a very high peak.
401
00:20:35,560 --> 00:20:37,220
In fact, what is
that high peak?
402
00:20:37,220 --> 00:20:38,830
It's 'n squared'.
403
00:20:38,830 --> 00:20:41,470
In other words, when this number
is very close to the
404
00:20:41,470 --> 00:20:45,220
y-axis, the peak is
very, very high.
405
00:20:45,220 --> 00:20:47,580
In other words, no matter how
you put the squeeze on over
406
00:20:47,580 --> 00:20:52,800
here, this particular
peak jumps out.
407
00:20:52,800 --> 00:20:55,830
This is why this particular
sequence of functions is not
408
00:20:55,830 --> 00:20:57,070
uniformly convergent.
409
00:20:57,070 --> 00:20:59,820
Again, this is done more
slowly in the notes.
410
00:20:59,820 --> 00:21:01,880
But at any rate, let me show you
an interesting thing that
411
00:21:01,880 --> 00:21:03,760
happens over here.
412
00:21:03,760 --> 00:21:07,290
Let me redraw this now
for a general 'n'.
413
00:21:07,290 --> 00:21:09,930
In other words, let me draw 'y'
equals ''f sub n' of x'
414
00:21:09,930 --> 00:21:11,430
for any old 'n'.
415
00:21:11,430 --> 00:21:14,140
Recall what our definition
was, now, especially
416
00:21:14,140 --> 00:21:16,000
with this as review.
417
00:21:16,000 --> 00:21:18,960
We locate the point '1/n'
comma 'n squared'.
418
00:21:18,960 --> 00:21:20,710
We then draw the line
that goes from the
419
00:21:20,710 --> 00:21:22,180
origin to that point.
420
00:21:22,180 --> 00:21:26,220
Then we draw the line that goes
from that point back to
421
00:21:26,220 --> 00:21:29,410
the x-axis at the point '2/n'.
422
00:21:29,410 --> 00:21:32,300
And then we come across
to 'x' equals 2.
423
00:21:32,300 --> 00:21:37,770
Let's try to visualize what the
integral from 0 to 2, ''f
424
00:21:37,770 --> 00:21:39,790
sub n' of x', 'dx', means.
425
00:21:39,790 --> 00:21:43,090
After all, in a case of a
continuous curve, which this
426
00:21:43,090 --> 00:21:46,250
is, isn't the definite integral
interpreted just as
427
00:21:46,250 --> 00:21:47,790
the area under the curve?
428
00:21:47,790 --> 00:21:51,370
Well, you see, the curve
coincides with the x-axis from
429
00:21:51,370 --> 00:21:53,380
'2/n' on to 2.
430
00:21:53,380 --> 00:21:57,860
Consequently, this triangular
region which I call 'R' is the
431
00:21:57,860 --> 00:21:59,210
area under the curve.
432
00:21:59,210 --> 00:22:02,440
In other words, the integral
from 0 to 2, ''f sub n' of x',
433
00:22:02,440 --> 00:22:05,720
'dx', is the area of
the region 'R'.
434
00:22:05,720 --> 00:22:06,750
But here's the point.
435
00:22:06,750 --> 00:22:09,870
We can compute the area of the
region 'R' very easily.
436
00:22:09,870 --> 00:22:11,780
It's a triangle, right?
437
00:22:11,780 --> 00:22:13,300
What is the area
of a triangle?
438
00:22:13,300 --> 00:22:17,040
Well, it's 1/2 times
the base--
439
00:22:17,040 --> 00:22:19,410
but the base is just '2/n'--
440
00:22:19,410 --> 00:22:20,790
times the height.
441
00:22:20,790 --> 00:22:22,900
The height is 'n squared'.
442
00:22:22,900 --> 00:22:27,150
In other words, the area of the
region 'R' simply is 'n'.
443
00:22:27,150 --> 00:22:28,530
And that's rather interesting.
444
00:22:28,530 --> 00:22:32,730
In other words, for each 'n',
this particular integral just
445
00:22:32,730 --> 00:22:34,770
turns out to be 'n' itself.
446
00:22:34,770 --> 00:22:37,220
That's what's interesting about
this particular diagram.
447
00:22:37,220 --> 00:22:40,620
In other words, this thing rises
so high that even though
448
00:22:40,620 --> 00:22:44,800
the base gets very, very small
as 'n' gets large, the height
449
00:22:44,800 --> 00:22:47,980
increases so rapidly that the
area under this curve,
450
00:22:47,980 --> 00:22:52,270
numerically, is always
equal to 'n'.
451
00:22:52,270 --> 00:22:54,300
In fact, we can check
that if you'd like.
452
00:22:54,300 --> 00:22:56,160
Come back to this
particular case.
453
00:22:56,160 --> 00:22:58,280
Look at this particular
triangle.
454
00:22:58,280 --> 00:23:00,510
The base is 2, the
height is 1.
455
00:23:00,510 --> 00:23:04,560
The area is 1 unit.
456
00:23:04,560 --> 00:23:07,490
Look at this particular
triangle.
457
00:23:07,490 --> 00:23:12,520
The base is 1/10, the
height is 400.
458
00:23:12,520 --> 00:23:17,540
400 times 1/10 is 40, and
half of that is 20.
459
00:23:17,540 --> 00:23:20,470
The area of this triangle
is 20, which exactly
460
00:23:20,470 --> 00:23:21,590
matches this subscript.
461
00:23:21,590 --> 00:23:23,870
That's what's going to happen
here all the time.
462
00:23:23,870 --> 00:23:27,640
In particular, then, if we
compute the integral from 0 to
463
00:23:27,640 --> 00:23:33,810
2, 'f of n', 'x dx', and then
let the limit as 'n' goes to
464
00:23:33,810 --> 00:23:36,430
infinity be computed, what
do we get for an answer?
465
00:23:36,430 --> 00:23:40,450
We get that this limit is the
limit of 'n' as 'n' approaches
466
00:23:40,450 --> 00:23:43,690
infinity, and that of
course is infinity.
467
00:23:43,690 --> 00:23:47,050
On the other hand, suppose we
bring the limit inside?
468
00:23:47,050 --> 00:23:49,960
In other words, suppose
we compute this.
469
00:23:49,960 --> 00:23:53,620
Well, the point is that we have
already shown that this
470
00:23:53,620 --> 00:23:57,370
is identically 0 for all 'x'.
471
00:23:57,370 --> 00:24:01,800
Consequently, this integral is
the integral from 0 to 2, 0
472
00:24:01,800 --> 00:24:03,990
'dx', which is 0.
473
00:24:03,990 --> 00:24:06,760
In other words, if you first
take the limit and then
474
00:24:06,760 --> 00:24:09,530
integrate, you get 0.
475
00:24:09,530 --> 00:24:12,490
On the other hand, if you first
integrate and then take
476
00:24:12,490 --> 00:24:14,490
the limit, you get infinity.
477
00:24:14,490 --> 00:24:17,820
And this should be a glaring
example to show you that the
478
00:24:17,820 --> 00:24:20,990
answer that you get indeed does
depend on the order in
479
00:24:20,990 --> 00:24:22,620
which you do the operation.
480
00:24:22,620 --> 00:24:24,260
Again, let me emphasize--
481
00:24:24,260 --> 00:24:26,600
which of these two is wrong?
482
00:24:26,600 --> 00:24:29,210
The answer is, neither
is wrong.
483
00:24:29,210 --> 00:24:32,150
All we're saying is that if
you were supposed to solve
484
00:24:32,150 --> 00:24:35,710
this problem and by mistake you
solve this one, you are
485
00:24:35,710 --> 00:24:39,150
going to get a drastically
different answer.
486
00:24:39,150 --> 00:24:40,230
OK.
487
00:24:40,230 --> 00:24:41,670
Let's not beat this to death.
488
00:24:41,670 --> 00:24:43,170
So far, so good.
489
00:24:43,170 --> 00:24:47,340
Let me make one more remark,
namely, what does all of this
490
00:24:47,340 --> 00:24:49,550
have to do with the
study of series?
491
00:24:49,550 --> 00:24:53,710
See, now we're just talking
about sequences of functions.
492
00:24:53,710 --> 00:24:56,630
And you see, the answer to this
question is essentially
493
00:24:56,630 --> 00:25:02,020
going to be our last lecture
of the course.
494
00:25:02,020 --> 00:25:04,140
But for now, what I'd like
to do is to give you
495
00:25:04,140 --> 00:25:05,420
a preview of that.
496
00:25:05,420 --> 00:25:08,580
Namely, the application of
uniform convergence to series
497
00:25:08,580 --> 00:25:09,750
is the following.
498
00:25:09,750 --> 00:25:13,300
Recall that when we write
summation 'n' goes from 0 to
499
00:25:13,300 --> 00:25:16,310
infinity, 'a sub n', 'x
to the n', that's an
500
00:25:16,310 --> 00:25:18,070
abbreviation for what?
501
00:25:18,070 --> 00:25:23,860
A polynomial, 'k' goes from 0 to
'n', 'a sub k', 'x sub k',
502
00:25:23,860 --> 00:25:25,550
as 'n' goes to infinity.
503
00:25:25,550 --> 00:25:30,390
In other words, recall that
the sum of the series is a
504
00:25:30,390 --> 00:25:34,240
limit of a sequence of partial
sums, and this is the n-th
505
00:25:34,240 --> 00:25:37,020
member of that sequence
of partial sums.
506
00:25:37,020 --> 00:25:40,560
Again, if the sigma notation is
throwing you off, all I'm
507
00:25:40,560 --> 00:25:46,800
saying is to observe that 'a0'
plus 'a1 x' plus 'a2 'x
508
00:25:46,800 --> 00:25:50,160
squared'' plus-- et cetera, et
cetera, et cetera, forever,
509
00:25:50,160 --> 00:25:53,500
just represents the limit of
the following sequence.
510
00:25:53,500 --> 00:25:57,940
'a0', next member is
'a0 plus a1 x'.
511
00:25:57,940 --> 00:26:02,090
Next member is 'a0' plus 'a1
x' plus 'a2 'x squared''.
512
00:26:02,090 --> 00:26:06,310
The next member is 'a0' plus 'a1
x' plus 'a2 'x squared''
513
00:26:06,310 --> 00:26:07,960
plus 'a3 'x cubed''.
514
00:26:07,960 --> 00:26:10,860
By the way, what is each
member of the sequence?
515
00:26:10,860 --> 00:26:13,290
It's a polynomial.
516
00:26:13,290 --> 00:26:16,570
And polynomials have very
nice properties,
517
00:26:16,570 --> 00:26:17,960
among which are what?
518
00:26:17,960 --> 00:26:20,780
Well, a polynomial is a
continuous function.
519
00:26:20,780 --> 00:26:23,850
A polynomial is an integral
function, et cetera.
520
00:26:23,850 --> 00:26:27,670
The idea, therefore, is that
if this sequence of partial
521
00:26:27,670 --> 00:26:35,840
sums converges uniformly to the
limit function, then, for
522
00:26:35,840 --> 00:26:39,800
example, the limit function,
namely the power series, must
523
00:26:39,800 --> 00:26:44,640
be continuous since each partial
sum that makes up the
524
00:26:44,640 --> 00:26:47,570
sequence of partial sums
is also continuous.
525
00:26:47,570 --> 00:26:50,100
Namely, every polynomial
is continuous.
526
00:26:50,100 --> 00:26:53,670
Also, if, for some reason or
other, you want to integrate
527
00:26:53,670 --> 00:26:57,260
that power series from 'a' to
'b', if the convergence is
528
00:26:57,260 --> 00:26:59,800
uniform, I can then do what?
529
00:26:59,800 --> 00:27:03,880
I can then take the summation
sign outside, integrate the
530
00:27:03,880 --> 00:27:08,100
n-th partial sum, and
add these all up.
531
00:27:08,100 --> 00:27:09,610
You see, the idea being what?
532
00:27:09,610 --> 00:27:13,350
That the n-th partial sum is a
polynomial, and a polynomial
533
00:27:13,350 --> 00:27:16,560
is a particularly simple
thing to integrate.
534
00:27:16,560 --> 00:27:18,400
That's one of the easiest
functions to
535
00:27:18,400 --> 00:27:20,150
integrate, in fact.
536
00:27:20,150 --> 00:27:21,380
OK.
537
00:27:21,380 --> 00:27:23,120
Now, here's the wrap up, then.
538
00:27:23,120 --> 00:27:26,760
What we shall show next time is
that within the interval of
539
00:27:26,760 --> 00:27:30,780
absolute convergence, the
sequence of partial sums,
540
00:27:30,780 --> 00:27:36,620
which we already know converges
absolutely to the
541
00:27:36,620 --> 00:27:39,840
limit function, also converges
uniformly.
542
00:27:39,840 --> 00:27:43,040
In other words, within the
radius of convergence, the
543
00:27:43,040 --> 00:27:43,670
power series--
544
00:27:43,670 --> 00:27:46,530
and I don't know how to say
this other than to say, it
545
00:27:46,530 --> 00:27:50,750
enjoys the usual polynomial
properties associated with a
546
00:27:50,750 --> 00:27:54,610
polynomial such as summation 'k'
goes from 0 to 'n', 'a sub
547
00:27:54,610 --> 00:27:57,090
k', 'x to the k'.
548
00:27:57,090 --> 00:28:00,820
In other words, then, this
about wraps up what our
549
00:28:00,820 --> 00:28:03,780
introductory lecture for today
wanted to be, namely the
550
00:28:03,780 --> 00:28:06,250
concept of uniform
convergence.
551
00:28:06,250 --> 00:28:09,320
What I would like you to do now
is to study this material
552
00:28:09,320 --> 00:28:12,040
very carefully in the
supplementary notes, go over
553
00:28:12,040 --> 00:28:15,260
the learning exercises so that
you become familiar with this.
554
00:28:15,260 --> 00:28:18,080
Then we will wrap up our course
in our next lecture,
555
00:28:18,080 --> 00:28:23,480
when we show what a very, very
powerful tool this particular
556
00:28:23,480 --> 00:28:28,480
concept of absolute convergence
is in the study of
557
00:28:28,480 --> 00:28:31,110
the mathematical concept
of convergence.
558
00:28:31,110 --> 00:28:33,970
At any rate, until next
time, then, goodbye.
559
00:28:33,970 --> 00:28:36,550
560
00:28:36,550 --> 00:28:39,750
Funding for the publication of
this video was provided by the
561
00:28:39,750 --> 00:28:43,810
Gabriella and Paul Rosenbaum
Foundation.
562
00:28:43,810 --> 00:28:47,970
Help OCW continue to provide
free and open access to MIT
563
00:28:47,970 --> 00:28:52,170
courses by making a donation
at ocw.mit.edu/donate.
564
00:28:52,170 --> 00:28:56,916