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PROFESSOR: So, Professor Jerison
is relaxing in sunny
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00:00:27,700 --> 00:00:32,130
London, Ontario today and sent
me in as his substitute again.
11
00:00:32,130 --> 00:00:33,760
I'm glad to the here and
see you all again.
12
00:00:33,760 --> 00:00:39,190
13
00:00:39,190 --> 00:00:41,220
So our agenda today -- he said
that he'd already talked about
14
00:00:41,220 --> 00:00:48,820
power series and Taylor's
formula I gues on last week
15
00:00:48,820 --> 00:00:50,850
right, on Friday?
16
00:00:50,850 --> 00:00:53,990
So I'm going to go a little
further with that and show you
17
00:00:53,990 --> 00:00:58,250
some examples, show you some
applications, and then I have
18
00:00:58,250 --> 00:01:02,580
this course evaluation survey
that I'll hand out in the last
19
00:01:02,580 --> 00:01:06,890
10 minutes or so of the class.
20
00:01:06,890 --> 00:01:10,860
I also have this handout that he
made that says 18.01 end of
21
00:01:10,860 --> 00:01:12,520
term, 2007.
22
00:01:12,520 --> 00:01:15,690
If you didn't pick this up
coming in, grab it going out.
23
00:01:15,690 --> 00:01:18,850
People tend not to pick it up
when they walk in, I see.
24
00:01:18,850 --> 00:01:22,010
So grab this when you're
going out.
25
00:01:22,010 --> 00:01:23,390
There's some things
missing from it.
26
00:01:23,390 --> 00:01:27,760
He has not decided when his
office hours will be at the
27
00:01:27,760 --> 00:01:28,520
end of term.
28
00:01:28,520 --> 00:01:31,280
He will have them, just
hasn't decided when.
29
00:01:31,280 --> 00:01:34,575
So, check the website for
that information.
30
00:01:34,575 --> 00:01:38,000
31
00:01:38,000 --> 00:01:42,820
And we're looking forward to the
final exam, which is uh --
32
00:01:42,820 --> 00:01:44,070
aren't we?
33
00:01:44,070 --> 00:01:47,030
34
00:01:47,030 --> 00:01:49,590
Any questions about this
technical stuff?
35
00:01:49,590 --> 00:01:52,900
36
00:01:52,900 --> 00:01:56,860
All right, let's talk about
power series for a little bit.
37
00:01:56,860 --> 00:02:00,800
So I thought I should review
for you what the story with
38
00:02:00,800 --> 00:02:02,050
power series is.
39
00:02:02,050 --> 00:02:21,790
40
00:02:21,790 --> 00:02:23,220
OK, could I have your
attention please?
41
00:02:23,220 --> 00:02:26,920
42
00:02:26,920 --> 00:02:31,810
So, power series is a way of
writing a function as a sum of
43
00:02:31,810 --> 00:02:33,560
integral powers of x.
44
00:02:33,560 --> 00:02:38,090
These a0 , a1, and so
on, are numbers.
45
00:02:38,090 --> 00:02:43,060
An example of a power series
is a polynomial.
46
00:02:43,060 --> 00:02:48,050
47
00:02:48,050 --> 00:02:55,350
Not to be forgotten, one type
of power series is one which
48
00:02:55,350 --> 00:02:59,510
goes on for a finite number of
terms and then ends, so that
49
00:02:59,510 --> 00:03:03,650
all of the other, all the higher
a sub i's are all 0.
50
00:03:03,650 --> 00:03:06,280
This is a perfectly good example
of a power series it's
51
00:03:06,280 --> 00:03:09,040
a very special kind
of power series.
52
00:03:09,040 --> 00:03:11,500
And part of what I want to tell
you today is that power
53
00:03:11,500 --> 00:03:14,970
series behave, almost exactly
like, polynomials.
54
00:03:14,970 --> 00:03:17,540
There's just one thing that you
have to be careful about
55
00:03:17,540 --> 00:03:21,540
when you're using power series
that isn't a concern for
56
00:03:21,540 --> 00:03:24,540
polynomials, and I'll show you
what that is in a minute.
57
00:03:24,540 --> 00:03:29,320
So, you should think of them
as generalized polynomials.
58
00:03:29,320 --> 00:03:46,010
The one thing that you have to
be careful about is that there
59
00:03:46,010 --> 00:03:46,596
iss a number, so one caution.
60
00:03:46,596 --> 00:03:54,730
There's a number which I'll call
R, where r can be between
61
00:03:54,730 --> 00:03:57,350
0 and it can also be infinity.
62
00:03:57,350 --> 00:04:03,410
It's a number between 0 and
infinity, inclusive, so that
63
00:04:03,410 --> 00:04:07,880
when the absolute value of x is
less than R. So when x is
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00:04:07,880 --> 00:04:11,260
smaller than R in size,
the sum converges.
65
00:04:11,260 --> 00:04:17,220
66
00:04:17,220 --> 00:04:21,170
This sum -- that sum converges
to a finite value.
67
00:04:21,170 --> 00:04:24,320
And when x is bigger
than R in absolute
68
00:04:24,320 --> 00:04:26,050
value, the sum diverges.
69
00:04:26,050 --> 00:04:30,260
70
00:04:30,260 --> 00:04:32,165
This R is called the radius
of convergence.
71
00:04:32,165 --> 00:04:42,960
72
00:04:42,960 --> 00:04:45,750
So we'll see some examples of
what the radius of convergence
73
00:04:45,750 --> 00:04:48,450
is in various powers
series as well, and
74
00:04:48,450 --> 00:04:49,700
how you find it also.
75
00:04:49,700 --> 00:04:55,890
76
00:04:55,890 --> 00:04:59,420
But, let me go on and give you
a few more of the properties
77
00:04:59,420 --> 00:05:02,980
about power series which I think
that professor Jerison
78
00:05:02,980 --> 00:05:05,410
talked about earlier.
79
00:05:05,410 --> 00:05:08,740
So one of them is there's
a radius of convergence.
80
00:05:08,740 --> 00:05:10,200
Here's another one.
81
00:05:10,200 --> 00:05:15,760
82
00:05:15,760 --> 00:05:21,090
If you're inside of the radius
convergence, then the function
83
00:05:21,090 --> 00:05:33,200
has all it's derivatives, has
all it's derivatives, just
84
00:05:33,200 --> 00:05:34,530
like a polynomial does.
85
00:05:34,530 --> 00:05:37,210
You can differentiate it
over and over again.
86
00:05:37,210 --> 00:05:43,060
And in terms of those
derivatives, the number a sub
87
00:05:43,060 --> 00:05:48,300
n in the power series can be
expressed in terms of the
88
00:05:48,300 --> 00:05:52,000
value of the derivative at 0.
89
00:05:52,000 --> 00:05:53,700
And this is called
Taylor's formula.
90
00:05:53,700 --> 00:05:58,540
91
00:05:58,540 --> 00:06:02,390
So I'm saying that inside of
this radius of convergence,
92
00:06:02,390 --> 00:06:05,430
the function that we're looking
at, this f of x, can
93
00:06:05,430 --> 00:06:10,710
be written as the value of the
function at 0, that's a sub 0,
94
00:06:10,710 --> 00:06:13,830
plus the value of
the derivative.
95
00:06:13,830 --> 00:06:17,630
This bracket n means you take
the derivative n times.
96
00:06:17,630 --> 00:06:20,940
So, when n is one, you take
the derivative once at 0
97
00:06:20,940 --> 00:06:23,590
divided by one factorial,
which is one, and
98
00:06:23,590 --> 00:06:25,100
multiply it by x.
99
00:06:25,100 --> 00:06:27,630
That's the linear term
in the power series.
100
00:06:27,630 --> 00:06:29,750
And then the qaudratic term
is you take the second
101
00:06:29,750 --> 00:06:30,260
derivative.
102
00:06:30,260 --> 00:06:33,740
Remember to divide by 2
factorial which is 2.
103
00:06:33,740 --> 00:06:39,730
Multiply that by x squared
and so on out.
104
00:06:39,730 --> 00:06:42,320
So, in terms, so the
coefficients in the power
105
00:06:42,320 --> 00:06:46,000
series just record the values
of the derivatives of the
106
00:06:46,000 --> 00:06:48,080
function at x equals 0.
107
00:06:48,080 --> 00:06:52,270
They can be computed
that way also.
108
00:06:52,270 --> 00:06:53,160
Let's see.
109
00:06:53,160 --> 00:06:56,300
I think that's a end of my
summary of things that he
110
00:06:56,300 --> 00:06:57,070
talked about.
111
00:06:57,070 --> 00:07:02,570
I think he did one example, and
I'll repeat that example
112
00:07:02,570 --> 00:07:04,680
of a power series.
113
00:07:04,680 --> 00:07:07,060
This example wasn't due to David
Jerison; it was due to
114
00:07:07,060 --> 00:07:08,310
Leonard Euler.
115
00:07:08,310 --> 00:07:11,590
116
00:07:11,590 --> 00:07:15,490
It's the example of where the
function is the exponential
117
00:07:15,490 --> 00:07:16,740
function e to the x.
118
00:07:16,740 --> 00:07:19,280
119
00:07:19,280 --> 00:07:22,980
So, let's see.
120
00:07:22,980 --> 00:07:23,140
Let's compute what --
121
00:07:23,140 --> 00:07:26,040
I will just repeat for you the
computation of the power
122
00:07:26,040 --> 00:07:28,590
series for e to the x, just
because it's such an important
123
00:07:28,590 --> 00:07:30,460
thing to do.
124
00:07:30,460 --> 00:07:32,270
So, in order to do that,
I have to know what the
125
00:07:32,270 --> 00:07:34,750
derivative of e to the x is, and
what the second derivative
126
00:07:34,750 --> 00:07:38,910
of e to the x is, and so on,
because that comes into the
127
00:07:38,910 --> 00:07:42,690
Taylor formula for
the coefficients.
128
00:07:42,690 --> 00:07:44,940
But we know what the derivative
of e to the x is,
129
00:07:44,940 --> 00:07:47,870
it's just e to the x again,
and it's that
130
00:07:47,870 --> 00:07:49,310
way all the way down.
131
00:07:49,310 --> 00:07:53,310
All the derivatives are e to
the x over and over again.
132
00:07:53,310 --> 00:07:57,310
So when I evaluate this at x
equal 0, well, the value of e
133
00:07:57,310 --> 00:08:00,390
to the x is one, the value
of e to the x is one
134
00:08:00,390 --> 00:08:01,910
and x equals 0.
135
00:08:01,910 --> 00:08:05,180
You get a value of one
all the way down.
136
00:08:05,180 --> 00:08:10,090
So all these derivatives at
0 have the value one.
137
00:08:10,090 --> 00:08:16,490
And now, when I plug into this
formula, I find e to the x is
138
00:08:16,490 --> 00:08:24,620
1, plus 1 times x, plus 1 over
2 factorial times x squared
139
00:08:24,620 --> 00:08:32,250
plus 1 over 3 factorial times
x cubed plus and so on.
140
00:08:32,250 --> 00:08:35,380
So all of these numbers are one,
and all you wind up with
141
00:08:35,380 --> 00:08:37,130
is the factorials and
the denominators.
142
00:08:37,130 --> 00:08:38,870
That's the power series
for e to the x.
143
00:08:38,870 --> 00:08:40,280
This was a discovery
of Leonhard
144
00:08:40,280 --> 00:08:42,050
Euler in 1740 or something.
145
00:08:42,050 --> 00:08:42,842
Yes Ma'am.
146
00:08:42,842 --> 00:08:45,403
AUDIENCE: When your writing out
the power series, how far
147
00:08:45,403 --> 00:08:46,910
do you have to write it out?
148
00:08:46,910 --> 00:08:48,700
PROFESSOR: How far do you have
to write the power series
149
00:08:48,700 --> 00:08:51,100
before it becomes
well defined?
150
00:08:51,100 --> 00:08:54,420
Before its a satisfactory
solution to an exam problem, I
151
00:08:54,420 --> 00:08:58,100
suppose, is another way to
phrase the question.
152
00:08:58,100 --> 00:09:00,920
Until you can see what
the pattern is.
153
00:09:00,920 --> 00:09:02,230
I can see what the pattern is.
154
00:09:02,230 --> 00:09:03,940
Is there anyone who's in
doubt about what the
155
00:09:03,940 --> 00:09:07,890
next term might be?
156
00:09:07,890 --> 00:09:09,900
Some people would tell you that
you have to write the
157
00:09:09,900 --> 00:09:11,900
summation convention thing.
158
00:09:11,900 --> 00:09:13,910
Don't believe them.
159
00:09:13,910 --> 00:09:15,610
If you right out enough
terms to make it
160
00:09:15,610 --> 00:09:17,020
clear, that's good enough.
161
00:09:17,020 --> 00:09:19,240
OK?
162
00:09:19,240 --> 00:09:20,070
Is that an answer for you?
163
00:09:20,070 --> 00:09:22,990
AUDIENCE: Yes, Thank you.
164
00:09:22,990 --> 00:09:25,980
PROFESSOR: OK, so that's
a basic example.
165
00:09:25,980 --> 00:09:28,960
Let's do another basic example
of a powers series.
166
00:09:28,960 --> 00:09:31,560
Oh yes, and by the way, whenever
you write out a power
167
00:09:31,560 --> 00:09:35,240
series, you should say what the
readius of convergence is.
168
00:09:35,240 --> 00:09:37,830
And for now, I will just to tell
you that the radius of
169
00:09:37,830 --> 00:09:41,140
convergence of this power series
is infiinity; that is,
170
00:09:41,140 --> 00:09:46,330
this sum always convergence
for any value of x.
171
00:09:46,330 --> 00:09:47,890
I'll say a little more about
that in a few minutes.
172
00:09:47,890 --> 00:09:49,780
Yeah?
173
00:09:49,780 --> 00:09:50,480
AUDIENCE: So which
functions can be
174
00:09:50,480 --> 00:09:52,880
written as power series?
175
00:09:52,880 --> 00:09:57,060
PROFESSOR: Which functions can
be written as power series?
176
00:09:57,060 --> 00:10:00,030
That's an excellent question.
177
00:10:00,030 --> 00:10:06,380
Any function that has a
reasonable expression can be
178
00:10:06,380 --> 00:10:08,840
written as a power series.
179
00:10:08,840 --> 00:10:11,220
I'm not giving you a very good
answer because the true answer
180
00:10:11,220 --> 00:10:12,730
is a little bit complicated.
181
00:10:12,730 --> 00:10:16,050
But any of the functions that
occur in calculus like sines,
182
00:10:16,050 --> 00:10:18,730
cosines, tangents, they
all have power
183
00:10:18,730 --> 00:10:21,740
series expansions, OK?
184
00:10:21,740 --> 00:10:22,990
We'll see more examples.
185
00:10:22,990 --> 00:10:25,830
186
00:10:25,830 --> 00:10:27,070
Let's do another example.
187
00:10:27,070 --> 00:10:30,130
Here's another example.
188
00:10:30,130 --> 00:10:31,520
I guess this was example one.
189
00:10:31,520 --> 00:10:35,520
190
00:10:35,520 --> 00:10:42,140
So, this example, I think, was
due to Newton, not Euler.
191
00:10:42,140 --> 00:10:46,530
Let's find the power series
expansion of this function, 1
192
00:10:46,530 --> 00:10:48,200
over 1 plus x.
193
00:10:48,200 --> 00:10:52,430
Well, I think that somewhere
along the line, you learned
194
00:10:52,430 --> 00:10:56,930
about the geometric series which
tells you that, which
195
00:10:56,930 --> 00:10:58,480
tells you what the answer
to this is, and I'll
196
00:10:58,480 --> 00:11:00,190
just write it out.
197
00:11:00,190 --> 00:11:12,490
The geometric series tells you
that this function can be
198
00:11:12,490 --> 00:11:16,460
written as an alternating
sum of powers of x.
199
00:11:16,460 --> 00:11:18,810
You may wonder where these
minuses came from.
200
00:11:18,810 --> 00:11:21,210
Well, if you really think about
the geometric series, as
201
00:11:21,210 --> 00:11:24,630
you probably remembered, there
was a minus sign here, and
202
00:11:24,630 --> 00:11:28,420
that gets replaced by
these minus signs.
203
00:11:28,420 --> 00:11:31,810
I think maybe Jerison talked
about this also.
204
00:11:31,810 --> 00:11:34,640
Anyway, here's another
basic example.
205
00:11:34,640 --> 00:11:37,070
Remember what the graph of this
function looks like when
206
00:11:37,070 --> 00:11:41,620
x is equal to minus one.
207
00:11:41,620 --> 00:11:43,860
Then there's a little problem
here because the denominator
208
00:11:43,860 --> 00:11:47,600
becomes 0, so the graph
has a poll there.
209
00:11:47,600 --> 00:11:53,700
It goes up to infinity at x
equals minus one, and that's
210
00:11:53,700 --> 00:11:56,650
an indication that
the radius of
211
00:11:56,650 --> 00:11:59,150
convergence is not infinity.
212
00:11:59,150 --> 00:12:01,390
Because if you try to converge
to this infinite number by
213
00:12:01,390 --> 00:12:03,550
putting in x equals
minus one here,
214
00:12:03,550 --> 00:12:04,930
you'll have a big problem.
215
00:12:04,930 --> 00:12:07,320
In fact, you see when you put
in x equals minus one, you
216
00:12:07,320 --> 00:12:09,530
keep getting one in every term,
and it gets bigger and
217
00:12:09,530 --> 00:12:11,390
bigger and does not converge.
218
00:12:11,390 --> 00:12:14,940
In this example, the radius
of convergence is one.
219
00:12:14,940 --> 00:12:18,570
220
00:12:18,570 --> 00:12:22,210
OK, so, let's do a
new example now.
221
00:12:22,210 --> 00:12:25,410
Oh, and by the way, I should
say you can calculate these
222
00:12:25,410 --> 00:12:27,770
numbers using Taylor's
formula.
223
00:12:27,770 --> 00:12:29,940
If you haven't seen
it, check it out.
224
00:12:29,940 --> 00:12:36,580
Calculate the iterated
derivatives of this function
225
00:12:36,580 --> 00:12:40,350
and plug in x equals 0 and see
that you get plus one, minus
226
00:12:40,350 --> 00:12:41,930
one, plus one, minus
one, and so on.
227
00:12:41,930 --> 00:12:42,706
Yes sir.
228
00:12:42,706 --> 00:12:46,290
AUDIENCE: For the radius of
convergence as stated, if you
229
00:12:46,290 --> 00:12:48,090
do minus one it'll fall out.
230
00:12:48,090 --> 00:12:50,740
If you put in one though, seems
like it would be fine.
231
00:12:50,740 --> 00:12:53,520
PROFESSOR: The questions is I
can see that there's a problem
232
00:12:53,520 --> 00:12:56,710
at x equals minus one, why is
there also a problem at x
233
00:12:56,710 --> 00:12:59,870
equals one where the graph is
perfectly smooth and innocuous
234
00:12:59,870 --> 00:13:00,760
and finite.
235
00:13:00,760 --> 00:13:04,490
That's another excellent
question.
236
00:13:04,490 --> 00:13:08,280
The problem is, that if you go
off to a radius of one in any
237
00:13:08,280 --> 00:13:11,530
direction and there's a
problem, that's it.
238
00:13:11,530 --> 00:13:13,530
That's what the radius
of convergence is.
239
00:13:13,530 --> 00:13:18,070
Here, what does happen if I
put an x equals plus one?
240
00:13:18,070 --> 00:13:21,930
So, let' look at the partial
sums. Do x equals plus one in
241
00:13:21,930 --> 00:13:23,060
your mind here.
242
00:13:23,060 --> 00:13:28,030
So I'll get a partial sum one,
then 0, and then one, and then
243
00:13:28,030 --> 00:13:29,990
0, and then one.
244
00:13:29,990 --> 00:13:32,270
So even though it doesn't go up
to infinity, it still does
245
00:13:32,270 --> 00:13:32,900
not converge.
246
00:13:32,900 --> 00:13:35,500
AUDIENCE: And anything
in between?
247
00:13:35,500 --> 00:13:38,470
PROFESSOR: Any of these other
things will also fail to
248
00:13:38,470 --> 00:13:41,330
converge in this example.
249
00:13:41,330 --> 00:13:43,685
Well, that's the only two real
numbers at the edge.
250
00:13:43,685 --> 00:13:44,935
Right?
251
00:13:44,935 --> 00:13:46,940
252
00:13:46,940 --> 00:13:49,050
OK, let's do a different
example now.
253
00:13:49,050 --> 00:13:50,398
How about a trig function?
254
00:13:50,398 --> 00:13:51,648
The sine of x.
255
00:13:51,648 --> 00:13:55,422
256
00:13:55,422 --> 00:14:00,380
I'm going to compute the power
series expansion for the sine
257
00:14:00,380 --> 00:14:04,400
of x, and I'm going to do it
using Taylor's formula.
258
00:14:04,400 --> 00:14:07,170
So Taylor's formula says that
I have to start computing
259
00:14:07,170 --> 00:14:09,585
derivatives of the sine of x.
260
00:14:09,585 --> 00:14:22,100
261
00:14:22,100 --> 00:14:25,280
Sounds like it's going
to be a lot of work.
262
00:14:25,280 --> 00:14:28,005
Let's see, the derivative of
the sine is the cosine.
263
00:14:28,005 --> 00:14:30,870
264
00:14:30,870 --> 00:14:33,330
and the derivative of the
cosine, that's the second
265
00:14:33,330 --> 00:14:36,530
derivative of the
sine, is what?
266
00:14:36,530 --> 00:14:40,270
Remember the minus, it's
minus sine of x.
267
00:14:40,270 --> 00:14:42,940
OK, now I want to take the third
derivative of the sine,
268
00:14:42,940 --> 00:14:46,020
which is the derivative of sine,
prime, prime, so it's
269
00:14:46,020 --> 00:14:47,760
the derivative of this.
270
00:14:47,760 --> 00:14:50,720
And we just decided the
derivative of sine is cosine,
271
00:14:50,720 --> 00:14:53,730
so I get cosine, but I have
this minus sign in front.
272
00:14:53,730 --> 00:14:56,660
273
00:14:56,660 --> 00:14:59,110
And now I want to differentiate
again, so the
274
00:14:59,110 --> 00:15:03,730
cosine becomes a minus sine,
and that minus sine cancels
275
00:15:03,730 --> 00:15:08,520
with this minus sine to
give me sine of x.
276
00:15:08,520 --> 00:15:10,102
You follow that?
277
00:15:10,102 --> 00:15:13,660
It's a lot of minus one's
canceling out there.
278
00:15:13,660 --> 00:15:17,500
So, all of a sudden, I'm right
back where I started; these
279
00:15:17,500 --> 00:15:21,160
two are the same and the
pattern will now repeat
280
00:15:21,160 --> 00:15:22,780
forever and ever.
281
00:15:22,780 --> 00:15:25,150
Higher and higher derivatives
of sines are just plus or
282
00:15:25,150 --> 00:15:28,830
minus sines and cosines.
283
00:15:28,830 --> 00:15:33,790
Now Taylor's formula says I
should now substitute x equals
284
00:15:33,790 --> 00:15:37,580
0 into this and see what
happens, so let's do that.
285
00:15:37,580 --> 00:15:40,842
When x is equals to 0,
the sine is 0 and
286
00:15:40,842 --> 00:15:43,240
the cosine is one.
287
00:15:43,240 --> 00:15:47,410
The sine is 0, so minus
0 is also 0.
288
00:15:47,410 --> 00:15:51,460
The cosine is one, but now
there's a minus one, and now
289
00:15:51,460 --> 00:15:58,760
I'm back where I started, and
so the pattern will repeat.
290
00:15:58,760 --> 00:16:02,690
OK, so the values of the
derivatives are all zeros and
291
00:16:02,690 --> 00:16:05,980
plus and minus ones and they
go through that pattern,
292
00:16:05,980 --> 00:16:09,670
four-fold periodicity,
over and over again.
293
00:16:09,670 --> 00:16:14,530
And si we can write out what
the sine of x is using
294
00:16:14,530 --> 00:16:15,780
Taylor's formula, using
this formula.
295
00:16:15,780 --> 00:16:18,000
296
00:16:18,000 --> 00:16:22,220
So I put in the value at 0 which
is 0, then I put in the
297
00:16:22,220 --> 00:16:27,620
derivative which is
1 multiplied by x.
298
00:16:27,620 --> 00:16:31,520
Then, I have the second
derivative divided by 2
299
00:16:31,520 --> 00:16:35,150
factorial, but the second
derivatve at 0 is 0.
300
00:16:35,150 --> 00:16:38,280
So I'm going to drop
that term out.
301
00:16:38,280 --> 00:16:41,365
Now I have the third derivative
which is minus one.
302
00:16:41,365 --> 00:16:43,930
303
00:16:43,930 --> 00:16:46,790
And remember the 3 factorial
in the denominator.
304
00:16:46,790 --> 00:16:50,050
That's the coefficient
of x cubed.
305
00:16:50,050 --> 00:16:51,680
What's the fourth derivative?
306
00:16:51,680 --> 00:16:54,400
Well, here we are, it's
on the board, it's 0.
307
00:16:54,400 --> 00:16:58,660
So I drop that term out go up
to the fifth term, the fifth
308
00:16:58,660 --> 00:16:59,830
power of x.
309
00:16:59,830 --> 00:17:02,260
It's derivative is now one.
310
00:17:02,260 --> 00:17:05,060
We've gone through the pattern,
we're back at plus
311
00:17:05,060 --> 00:17:08,590
one as the value of the iterated
derivative, so now I
312
00:17:08,590 --> 00:17:13,180
get 1/5 factorial times
x to the fifth.
313
00:17:13,180 --> 00:17:16,010
Now, you tell me, have we done
enough terms to see what the
314
00:17:16,010 --> 00:17:17,900
pattern is?
315
00:17:17,900 --> 00:17:22,620
I guess the next term will be a
minus 1/7 factorial x to the
316
00:17:22,620 --> 00:17:23,750
seventh and so on.
317
00:17:23,750 --> 00:17:28,160
Let me write this out again
just so we have it.
318
00:17:28,160 --> 00:17:30,250
X cubed over 3 factorial, so
it's x minus x cubed over 3
319
00:17:30,250 --> 00:17:34,580
factorial plus x to the fifth
over 5 factorial.
320
00:17:34,580 --> 00:17:38,740
You guessed it, and so on.
321
00:17:38,740 --> 00:17:40,390
That's the power series
expansion for
322
00:17:40,390 --> 00:17:43,935
the sine of x, OK?
323
00:17:43,935 --> 00:17:46,950
324
00:17:46,950 --> 00:17:49,910
And so, the sines alternate,
and these denominators get
325
00:17:49,910 --> 00:17:52,250
very big, don't they?
326
00:17:52,250 --> 00:17:55,830
Exponentials grow very fast.
Let me make a remark.
327
00:17:55,830 --> 00:17:58,800
R is infinity here.
328
00:17:58,800 --> 00:18:02,070
The radius of convergence of
this power series again is
329
00:18:02,070 --> 00:18:03,424
infinity, and let
me just say why.
330
00:18:03,424 --> 00:18:13,020
The reason is that the general
term is going to be like x to
331
00:18:13,020 --> 00:18:18,740
the 2n plus 1 divided by 2n
plus one factorial, an odd
332
00:18:18,740 --> 00:18:21,900
number I can write
as 2n plus 1.
333
00:18:21,900 --> 00:18:25,340
And what I want to say is that
the size of this, what happens
334
00:18:25,340 --> 00:18:34,000
to the size of this as
n goes to infinity.
335
00:18:34,000 --> 00:18:35,280
So let's just think
about this.
336
00:18:35,280 --> 00:18:38,270
For a fixed x, let's fix
the number of x.
337
00:18:38,270 --> 00:18:41,570
Look at powers of x and think
about the size of this
338
00:18:41,570 --> 00:18:45,930
expression when n gets
to be large.
339
00:18:45,930 --> 00:18:47,140
So let's just do that
for a second.
340
00:18:47,140 --> 00:18:53,410
So, for x to the 2n plus 1 over
2n plus 1 factorial, I
341
00:18:53,410 --> 00:18:54,420
can write out like this.
342
00:18:54,420 --> 00:19:00,890
It's x over one times x over 2
times-- sorry-- times x over
343
00:19:00,890 --> 00:19:09,250
3, times x over 2n plus 1.
344
00:19:09,250 --> 00:19:13,220
I've multiplied x by itself 2n
plus 1 times in the numerator,
345
00:19:13,220 --> 00:19:16,340
and I've multiplied the numbers
1, 2, 3, 4, and so on,
346
00:19:16,340 --> 00:19:19,210
by each other in the
denominator, and that gives me
347
00:19:19,210 --> 00:19:19,810
the factorial.
348
00:19:19,810 --> 00:19:22,330
So I've just written
this out like this.
349
00:19:22,330 --> 00:19:26,820
Now x is fixed, so maybe
it's a million, OK?
350
00:19:26,820 --> 00:19:28,700
It's big, but fixed.
351
00:19:28,700 --> 00:19:30,650
What happens to these numbers?
352
00:19:30,650 --> 00:19:32,050
Well at first, they're
pretty big.
353
00:19:32,050 --> 00:19:34,560
This is a million over 2, this
is a million over 3.
354
00:19:34,560 --> 00:19:39,240
But when n gets to be maybe if
n is a million, then this is
355
00:19:39,240 --> 00:19:41,180
about one half.
356
00:19:41,180 --> 00:19:48,320
If n is a billion, then this
is about 1/2,000, right?
357
00:19:48,320 --> 00:19:50,830
The denominators keep getting
bigger and bigger, but the
358
00:19:50,830 --> 00:19:54,470
numerators stay the same;
they're always x.
359
00:19:54,470 --> 00:19:57,875
So when I take the product, if I
go far enough out, I'm going
360
00:19:57,875 --> 00:20:00,550
to be multiplying, by very, very
small numbers and more
361
00:20:00,550 --> 00:20:01,730
and more of them.
362
00:20:01,730 --> 00:20:05,980
And so no matter what
x is, these numbers
363
00:20:05,980 --> 00:20:07,220
will converge to 0.
364
00:20:07,220 --> 00:20:08,985
They'll get smaller
and smaller as
365
00:20:08,985 --> 00:20:11,550
x gets to be bigger.
366
00:20:11,550 --> 00:20:15,610
That's the sign that x is
inside of the radius of
367
00:20:15,610 --> 00:20:16,590
convergence.
368
00:20:16,590 --> 00:20:22,520
This is the sign for you that
this series converges for that
369
00:20:22,520 --> 00:20:23,780
value of x.
370
00:20:23,780 --> 00:20:33,600
And because I could do this
for any x, this works.
371
00:20:33,600 --> 00:20:40,260
This convergence to
0 for any fixed x.
372
00:20:40,260 --> 00:20:43,910
That's what tells you that you
can take, that the raidus of
373
00:20:43,910 --> 00:20:47,500
convergence is infinity because
in the formula--
374
00:20:47,500 --> 00:20:52,530
375
00:20:52,530 --> 00:20:54,570
in the fact the radius of
convergence talks about.
376
00:20:54,570 --> 00:20:58,160
If R is equal infinity, this
is no condition on x.
377
00:20:58,160 --> 00:21:02,250
Every number is less than
infinity in absolute value.
378
00:21:02,250 --> 00:21:06,590
So if this covergence to 0 of
the general term works for
379
00:21:06,590 --> 00:21:10,320
every x, then radius of
convergence is infinity.
380
00:21:10,320 --> 00:21:12,330
Well that was kind of fast, but
I think that you've heard
381
00:21:12,330 --> 00:21:16,020
something about that
earlier as well.
382
00:21:16,020 --> 00:21:19,330
Anyway, so we've got the sine
function, a new function with
383
00:21:19,330 --> 00:21:20,880
its own power series.
384
00:21:20,880 --> 00:21:23,730
It's a way of computing
sine of x.
385
00:21:23,730 --> 00:21:27,790
If you take enough terms you'll
get a good evaluation
386
00:21:27,790 --> 00:21:30,000
of the sine of x for any x.
387
00:21:30,000 --> 00:21:32,760
This tells you a lot about the
function sine of x but not
388
00:21:32,760 --> 00:21:33,750
everything at all.
389
00:21:33,750 --> 00:21:37,690
For example, from this formula,
it's very hard to see
390
00:21:37,690 --> 00:21:39,745
that the sine of
x is periodic.
391
00:21:39,745 --> 00:21:41,930
It's not obvious at all.
392
00:21:41,930 --> 00:21:45,620
Somewhere hidden away in this
expression is the number pi,
393
00:21:45,620 --> 00:21:47,400
the half of the period.
394
00:21:47,400 --> 00:21:51,100
But that's not clear from
the power series at all.
395
00:21:51,100 --> 00:21:53,620
So the power series are very
good for some things, but they
396
00:21:53,620 --> 00:21:55,510
hide other properties
of functions.
397
00:21:55,510 --> 00:21:58,150
398
00:21:58,150 --> 00:22:02,070
Well, so I want to spend a few
minutes telling you about what
399
00:22:02,070 --> 00:22:06,100
you can do with a power series
once you have one to get new
400
00:22:06,100 --> 00:22:08,530
power series, and new power
series from old.
401
00:22:08,530 --> 00:22:18,300
402
00:22:18,300 --> 00:22:25,490
And this is also called
operations on power series.
403
00:22:25,490 --> 00:22:28,270
So what are the things that we
can do to a power series?
404
00:22:28,270 --> 00:22:29,905
Well one of the things you
can do is multiply.
405
00:22:29,905 --> 00:22:33,990
406
00:22:33,990 --> 00:22:37,710
So, for example, what if I want
to compute a power series
407
00:22:37,710 --> 00:22:40,970
for x times the sine of x.
408
00:22:40,970 --> 00:22:44,160
Well I have a power series for
the sine of x, I just did it.
409
00:22:44,160 --> 00:22:45,920
How about a power
series for x?
410
00:22:45,920 --> 00:22:48,910
411
00:22:48,910 --> 00:22:51,480
Actually, I did that here too.
412
00:22:51,480 --> 00:22:55,120
The function x is a very
simple polynomial.
413
00:22:55,120 --> 00:22:59,080
It's a polynomial where, about
0, a1 is 1, and all the other
414
00:22:59,080 --> 00:23:00,930
coefficients are 0.
415
00:23:00,930 --> 00:23:04,090
So x itself is a power series,
a very simple one.
416
00:23:04,090 --> 00:23:08,440
The sine of x is a
powers series.
417
00:23:08,440 --> 00:23:10,660
And what I want to encourage
you to do is treat power
418
00:23:10,660 --> 00:23:14,330
series just like polynomials
and multiply them together.
419
00:23:14,330 --> 00:23:16,960
We'll see other operations
too.
420
00:23:16,960 --> 00:23:19,900
So, to compute the power series
for x times the sine of
421
00:23:19,900 --> 00:23:23,500
x, I just take this one
and multiply it by x.
422
00:23:23,500 --> 00:23:26,650
423
00:23:26,650 --> 00:23:26,929
So let's see if I can
do that right.
424
00:23:26,929 --> 00:23:31,040
It distributes through x squared
minus x to the fourth
425
00:23:31,040 --> 00:23:35,180
over 3 factorial plus x
to the sixth over 5
426
00:23:35,180 --> 00:23:42,190
factorial, and so on.
427
00:23:42,190 --> 00:23:44,790
And again, the radius of
convergence is going to be the
428
00:23:44,790 --> 00:23:48,220
smaller of the two radii
of convergence here.
429
00:23:48,220 --> 00:23:51,840
So it's R equals infinity
in this case.
430
00:23:51,840 --> 00:23:54,010
OK, you can multiply power
series together.
431
00:23:54,010 --> 00:23:57,990
It can be a pain if the power
series are very long, but if
432
00:23:57,990 --> 00:24:01,800
one of them is x, it's
pretty simple.
433
00:24:01,800 --> 00:24:06,040
OK, that's one thing I can do.
434
00:24:06,040 --> 00:24:08,910
Notice something by the way.
435
00:24:08,910 --> 00:24:10,175
You know that even
an odd functions?
436
00:24:10,175 --> 00:24:13,180
437
00:24:13,180 --> 00:24:17,490
So, sine is an odd function,
x is an odd function, the
438
00:24:17,490 --> 00:24:20,570
product of two large functions
is an even function.
439
00:24:20,570 --> 00:24:23,550
And that's reflected in the fact
that all the powers that
440
00:24:23,550 --> 00:24:26,790
occur in the power
series are even.
441
00:24:26,790 --> 00:24:30,370
For an odd function, like the
sine, all the powers that
442
00:24:30,370 --> 00:24:32,600
occur are odd powers of x.
443
00:24:32,600 --> 00:24:33,850
That's always true.
444
00:24:33,850 --> 00:24:37,510
445
00:24:37,510 --> 00:24:39,000
OK, we can multiply.
446
00:24:39,000 --> 00:24:40,250
I can also differentiate.
447
00:24:40,250 --> 00:24:48,660
448
00:24:48,660 --> 00:24:57,660
so let's just do a case of that
and use the process of
449
00:24:57,660 --> 00:25:00,410
differentiation to find out what
the power series for the
450
00:25:00,410 --> 00:25:05,690
cosine of x is by writing the
cosine of x as the derivative
451
00:25:05,690 --> 00:25:09,010
of the sine and differentiating
term by term.
452
00:25:09,010 --> 00:25:11,970
So, I'll take this expression
for the power series of the
453
00:25:11,970 --> 00:25:14,750
sine and differentiate it term
by term, and I'll get the
454
00:25:14,750 --> 00:25:18,210
power series for cosine.
455
00:25:18,210 --> 00:25:19,030
So, let's see.
456
00:25:19,030 --> 00:25:22,510
The derivative of x is one.
457
00:25:22,510 --> 00:25:25,940
Now, the derivative of x cubed
is 3x squared, and then
458
00:25:25,940 --> 00:25:28,910
there's a 3 factorial
in the denominator.
459
00:25:28,910 --> 00:25:33,030
And the derivative of x to the
fifth is 5x to the fourth, and
460
00:25:33,030 --> 00:25:35,830
there's a 5 factorial in
the denominator, and
461
00:25:35,830 --> 00:25:38,680
so on and so on.
462
00:25:38,680 --> 00:25:40,950
And now some cancellation
happens.
463
00:25:40,950 --> 00:25:45,500
So this is one minus, well, the
3 cancels with the last
464
00:25:45,500 --> 00:25:48,995
factor in this 3 factorial and
leaves you with 2 factorial.
465
00:25:48,995 --> 00:25:52,460
466
00:25:52,460 --> 00:25:56,040
And the 5 cancels with the last
factor in the 5 factorial
467
00:25:56,040 --> 00:25:58,920
and leaves you with a 4
factorial in the denominator.
468
00:25:58,920 --> 00:26:01,570
469
00:26:01,570 --> 00:26:05,160
and so there you go, there's the
power series expansion for
470
00:26:05,160 --> 00:26:05,980
the cosine.
471
00:26:05,980 --> 00:26:07,970
It's got all even powers of x.
472
00:26:07,970 --> 00:26:12,720
They alternate, and you have
factorials in the denominator.
473
00:26:12,720 --> 00:26:15,650
And of course, you could derive
that expression by
474
00:26:15,650 --> 00:26:19,210
using Taylor's formula by the
same kind of calculation you
475
00:26:19,210 --> 00:26:21,720
did here, taking higher
and higher
476
00:26:21,720 --> 00:26:22,970
derivatives of the cosine.
477
00:26:22,970 --> 00:26:26,520
You get the same periodic
pattern of derivatives and
478
00:26:26,520 --> 00:26:30,080
values of derivatives
at x equals 0.
479
00:26:30,080 --> 00:26:33,200
But here's a cleaner way to do
it, simpler way to do it
480
00:26:33,200 --> 00:26:36,830
because we already knew the
derivative of the sine.
481
00:26:36,830 --> 00:26:39,210
When you differentiate, you
keep the same radius of
482
00:26:39,210 --> 00:26:40,460
convergence.
483
00:26:40,460 --> 00:26:44,420
484
00:26:44,420 --> 00:26:50,060
OK, so we can multiply, I can
add too and multiply that
485
00:26:50,060 --> 00:26:52,400
constant, things like that.
486
00:26:52,400 --> 00:26:54,280
How about integrating?
487
00:26:54,280 --> 00:26:56,580
That's what half of this course
was about isn't it?
488
00:26:56,580 --> 00:26:58,550
So, let's integrate something.
489
00:26:58,550 --> 00:27:07,210
490
00:27:07,210 --> 00:27:16,100
So, the integration I'm going
to do is this one: the
491
00:27:16,100 --> 00:27:20,160
integral from 0 to x of
dt over one plus x.
492
00:27:20,160 --> 00:27:21,980
What is that integral
as a function?
493
00:27:21,980 --> 00:27:28,360
494
00:27:28,360 --> 00:27:32,350
So, when I find the
anti-derivative of this, I get
495
00:27:32,350 --> 00:27:37,890
the natural log of 1 plus t, and
then when I evaluate that
496
00:27:37,890 --> 00:27:42,600
at t equals x, I get the natural
log of 1 plus x.
497
00:27:42,600 --> 00:27:47,060
And when I evaluate the natural
log at 0, I get the
498
00:27:47,060 --> 00:27:55,070
natural log of 1, which is 0,
so this is what you get, OK?
499
00:27:55,070 --> 00:28:02,860
This is really valid,
by the way, for x
500
00:28:02,860 --> 00:28:05,690
bigger than minus 1.
501
00:28:05,690 --> 00:28:09,060
But you don't want to think
about this quite like this
502
00:28:09,060 --> 00:28:10,310
when x is smaller than that.
503
00:28:10,310 --> 00:28:13,770
504
00:28:13,770 --> 00:28:19,005
Now, I'm going to try to apply
power series methods here and
505
00:28:19,005 --> 00:28:22,900
find, use this integral to find
a power series for the
506
00:28:22,900 --> 00:28:29,280
natural log, and I'll do it by
plugging in to this expression
507
00:28:29,280 --> 00:28:34,760
what the power series for
1 over 1 plus t was.
508
00:28:34,760 --> 00:28:36,570
And I know what that is because
I wrote it down on the
509
00:28:36,570 --> 00:28:38,330
board up here.
510
00:28:38,330 --> 00:28:42,320
Change the variable from x to t
there, and so 1 over 1 plus
511
00:28:42,320 --> 00:28:47,660
t is 1 minus t, plus
t squared, minust t
512
00:28:47,660 --> 00:28:49,410
cubed, and so on.
513
00:28:49,410 --> 00:28:52,620
514
00:28:52,620 --> 00:28:55,196
So that's the thing in the in
the inside of the integral,
515
00:28:55,196 --> 00:29:01,710
and now it's legal to integrate
that term by term,
516
00:29:01,710 --> 00:29:03,820
so let's do that.
517
00:29:03,820 --> 00:29:06,270
I'm going to get something which
I will then evaluate at
518
00:29:06,270 --> 00:29:09,230
x and at 0.
519
00:29:09,230 --> 00:29:12,610
So, when I integrate one
I get x, and when they
520
00:29:12,610 --> 00:29:14,593
integrate t, I get t.
521
00:29:14,593 --> 00:29:15,843
I'm sorry.
522
00:29:15,843 --> 00:29:19,150
523
00:29:19,150 --> 00:29:24,190
When I intergrate t, I get t
squared over 2, and t squared
524
00:29:24,190 --> 00:29:29,970
gives me t cubed over 3,
and so on and so on.
525
00:29:29,970 --> 00:29:32,520
526
00:29:32,520 --> 00:29:37,100
And then, when I put in t equals
x, while I just replace
527
00:29:37,100 --> 00:29:41,780
all the t's by x's, and when I
put in t equals 0, I get 0.
528
00:29:41,780 --> 00:29:43,950
So this equals x.
529
00:29:43,950 --> 00:29:50,430
So, I've discovered that the
natural log of 1 plus x is x
530
00:29:50,430 --> 00:29:57,010
minus x squared over 2, plus x
cubed over 3 minus x to the
531
00:29:57,010 --> 00:30:02,020
fourth over 4, and
so on and so on.
532
00:30:02,020 --> 00:30:04,320
There's the power series
expansion for the natural log
533
00:30:04,320 --> 00:30:05,570
of 1 plus x.
534
00:30:05,570 --> 00:30:07,800
535
00:30:07,800 --> 00:30:11,060
And because I began with a power
series who's radius of
536
00:30:11,060 --> 00:30:15,570
convergence was just 1, I began
with this power series,
537
00:30:15,570 --> 00:30:18,150
the radius of convergence of
this is also going to be 1.
538
00:30:18,150 --> 00:30:22,200
539
00:30:22,200 --> 00:30:25,830
Also, because this function,
as I just pointed out, this
540
00:30:25,830 --> 00:30:29,870
function goes bad when x becomes
less than minus 1, so
541
00:30:29,870 --> 00:30:32,660
some problem happens, and that's
reflected in the radius
542
00:30:32,660 --> 00:30:35,590
of convergence.
543
00:30:35,590 --> 00:30:36,750
Cool.
544
00:30:36,750 --> 00:30:41,110
So, you can integrate.
545
00:30:41,110 --> 00:30:44,250
That is the correct power
series expansion for the
546
00:30:44,250 --> 00:30:48,060
natural log of 1 plus x, and
another victory of Euler's was
547
00:30:48,060 --> 00:30:50,850
to use this kind of power series
expansion to calculate
548
00:30:50,850 --> 00:30:53,560
natural logarithms in a much
more efficient way than people
549
00:30:53,560 --> 00:30:54,810
had done before.
550
00:30:54,810 --> 00:30:57,850
551
00:30:57,850 --> 00:31:08,380
OK, one more property,
I think.
552
00:31:08,380 --> 00:31:12,920
553
00:31:12,920 --> 00:31:17,080
What are we at here, 3?
554
00:31:17,080 --> 00:31:18,690
4.
555
00:31:18,690 --> 00:31:19,940
Substitute.
556
00:31:19,940 --> 00:31:25,410
557
00:31:25,410 --> 00:31:29,420
Very appropriate for me as a
substitute teacher to tell you
558
00:31:29,420 --> 00:31:30,670
about substitution.
559
00:31:30,670 --> 00:31:32,810
560
00:31:32,810 --> 00:31:35,590
So I'm going to try to find the
power series expansion of
561
00:31:35,590 --> 00:31:36,727
e to the minus t squared.
562
00:31:36,727 --> 00:31:37,977
OK?
563
00:31:37,977 --> 00:31:41,740
564
00:31:41,740 --> 00:31:45,190
And the way I'll do that is
by taking the power series
565
00:31:45,190 --> 00:31:50,450
expansion for e to the x, which
we have up there, and
566
00:31:50,450 --> 00:32:00,310
make the substitution x equals
minus t squared in the
567
00:32:00,310 --> 00:32:01,630
expansion for e to the x.
568
00:32:01,630 --> 00:32:03,272
Did you have a question?
569
00:32:03,272 --> 00:32:05,955
AUDIENCE: Well, it's just
concerning the radius of
570
00:32:05,955 --> 00:32:07,624
convergence.
571
00:32:07,624 --> 00:32:11,740
You can't define x so that is
always positive, and if so, it
572
00:32:11,740 --> 00:32:14,360
wouldn't have a radius of
convergence, right?
573
00:32:14,360 --> 00:32:19,370
PROFESSOR: Like I say, again the
worry is this natural log
574
00:32:19,370 --> 00:32:21,510
of 1 plus x function
is perfectly well
575
00:32:21,510 --> 00:32:24,120
behaved for large x.
576
00:32:24,120 --> 00:32:27,720
Why does the power series fail
to converge for large x?
577
00:32:27,720 --> 00:32:31,110
Well suppose that x is bigger
than one, then here you get
578
00:32:31,110 --> 00:32:35,160
bigger and bigger powers of x,
which will grow to infinity,
579
00:32:35,160 --> 00:32:38,190
and they grow large
faster than the
580
00:32:38,190 --> 00:32:40,330
numbers 2, 3, 4, 5, 6.
581
00:32:40,330 --> 00:32:45,850
They grow exponentially, and
these just grow linearly.
582
00:32:45,850 --> 00:32:49,530
So, again, the general term,
when x is bigger than one, the
583
00:32:49,530 --> 00:32:52,340
general term will go off to
infinity, even though the
584
00:32:52,340 --> 00:32:55,460
function that you're talking
about, log of net of 1 plus x
585
00:32:55,460 --> 00:32:56,750
is perfectly good.
586
00:32:56,750 --> 00:33:01,960
So the power series is not good
outside of the radius of
587
00:33:01,960 --> 00:33:02,240
convergence.
588
00:33:02,240 --> 00:33:04,720
It's just a fact of life.
589
00:33:04,720 --> 00:33:05,150
Yes?
590
00:33:05,150 --> 00:33:06,400
AUDIENCE: [INAUDIBLE]
591
00:33:06,400 --> 00:33:18,490
592
00:33:18,490 --> 00:33:20,290
PROFESSOR: I'd rather-- talk
to me after class.
593
00:33:20,290 --> 00:33:23,150
The question is why is it the
smaller of the two radii of
594
00:33:23,150 --> 00:33:24,110
convergence?
595
00:33:24,110 --> 00:33:31,130
The basic answer is, well, you
can't expect it to be bigger
596
00:33:31,130 --> 00:33:34,100
than that smaller one, because
the power series only gives
597
00:33:34,100 --> 00:33:37,450
you information inside of that
range about the function, so.
598
00:33:37,450 --> 00:33:38,700
AUDIENCE: [INAUDIBLE]
599
00:33:38,700 --> 00:33:41,010
600
00:33:41,010 --> 00:33:43,580
PROFESSOR: Well, in this case,
both of the radii of
601
00:33:43,580 --> 00:33:44,890
convergence are infinity.
602
00:33:44,890 --> 00:33:48,620
x has radius of convergence
infinity for sure, and sine of
603
00:33:48,620 --> 00:33:49,260
x does too.
604
00:33:49,260 --> 00:33:52,140
So you get infinity
in that case, OK?
605
00:33:52,140 --> 00:33:54,850
606
00:33:54,850 --> 00:33:59,000
OK, let's just do this, and then
I'm going to integrate
607
00:33:59,000 --> 00:34:03,830
this and that'll be the end of
what I have time for today.
608
00:34:03,830 --> 00:34:05,880
So what's the power series
expansion for this?
609
00:34:05,880 --> 00:34:08,770
The power series expansion of
this is going to be a function
610
00:34:08,770 --> 00:34:13,530
of t, right, because the
variable here is t.
611
00:34:13,530 --> 00:34:17,060
I get it by taking my expansion
for e to the x and
612
00:34:17,060 --> 00:34:20,202
putting in what x is
in terms of t.
613
00:34:20,202 --> 00:34:21,452
And so on and so on.
614
00:34:21,452 --> 00:34:36,700
615
00:34:36,700 --> 00:34:40,950
I just put in minus t squared
in place of x there in the
616
00:34:40,950 --> 00:34:44,120
series expansion
for e to the x.
617
00:34:44,120 --> 00:34:47,370
I can work this out a
little bit better.
618
00:34:47,370 --> 00:34:49,030
minus t squared is what it is.
619
00:34:49,030 --> 00:34:52,950
This is going to give me a t
to the fourth and the minus
620
00:34:52,950 --> 00:34:56,110
squared is going to give me a
plus, so I get t to the fourth
621
00:34:56,110 --> 00:34:58,730
over 2 factorial.
622
00:34:58,730 --> 00:35:03,520
Then I get minus t quantity
cubed, so there'll be a minus
623
00:35:03,520 --> 00:35:08,190
sign and a t to the sixth and
the denominator 3 factorial.
624
00:35:08,190 --> 00:35:11,360
So the signs are going to
alternate, the powers are all
625
00:35:11,360 --> 00:35:15,380
even, and the denominators
are these factorials.
626
00:35:15,380 --> 00:35:20,160
627
00:35:20,160 --> 00:35:24,570
Several times as this course
has gone on, the error
628
00:35:24,570 --> 00:35:27,216
function has made
an appearance.
629
00:35:27,216 --> 00:35:31,420
The error function was, I guess
it gets normalized by
630
00:35:31,420 --> 00:35:41,870
putting a 2 over the square root
of pi in front, and it's
631
00:35:41,870 --> 00:35:46,830
the integral of e to the minus
t squared d t from 0 to x.
632
00:35:46,830 --> 00:35:58,090
And this normalization is here
because as x gets to be large
633
00:35:58,090 --> 00:36:01,300
the value becomes one.
634
00:36:01,300 --> 00:36:04,440
So this error function is very
important in the theory of
635
00:36:04,440 --> 00:36:06,080
probability.
636
00:36:06,080 --> 00:36:08,540
And I think you calculated
this fact at some
637
00:36:08,540 --> 00:36:12,236
point in the course.
638
00:36:12,236 --> 00:36:14,860
So the standard definition of
the error function, you put a
639
00:36:14,860 --> 00:36:16,460
2 over the square root
of pi in front.
640
00:36:16,460 --> 00:36:18,425
Let's calculate it's power
series expansion.
641
00:36:18,425 --> 00:36:21,320
642
00:36:21,320 --> 00:36:25,660
So there's a 2 over the square
root of pi that hurts nobody
643
00:36:25,660 --> 00:36:27,280
here in the front.
644
00:36:27,280 --> 00:36:30,530
And now I want to integrate
either the minus t squared,
645
00:36:30,530 --> 00:36:33,190
and I'm going to use this power
series expansion for
646
00:36:33,190 --> 00:36:36,350
that to see what you get.
647
00:36:36,350 --> 00:36:38,800
So I'm just going to write
this out I think.
648
00:36:38,800 --> 00:36:41,560
I did it out carefully in
another example over there so
649
00:36:41,560 --> 00:36:43,100
I'll do it a little
quicker now.
650
00:36:43,100 --> 00:36:46,690
intergrate this terms by term,
you're just integrating powers
651
00:36:46,690 --> 00:36:49,710
of t so it's pretty
simple, so I get--
652
00:36:49,710 --> 00:36:51,830
and then I'm evaluating
at and then 0.
653
00:36:51,830 --> 00:37:01,240
So I get x minus x cubed over 3,
plus x to the fifth over 5
654
00:37:01,240 --> 00:37:05,820
times 2 factorial, 5 from
integrating the t's of the
655
00:37:05,820 --> 00:37:09,070
fourth, and the 2 factorial from
this denominator that we
656
00:37:09,070 --> 00:37:11,390
already had.
657
00:37:11,390 --> 00:37:15,840
And then there's an a minus x to
the seventh over 7 times 3
658
00:37:15,840 --> 00:37:20,250
factorial, and plus, and so on,
and you can imagine how
659
00:37:20,250 --> 00:37:21,620
they go on from there.
660
00:37:21,620 --> 00:37:24,490
661
00:37:24,490 --> 00:37:29,430
I guess to get this exactly in
the form that we began talking
662
00:37:29,430 --> 00:37:32,510
about, I should multiply
through.
663
00:37:32,510 --> 00:37:35,810
So the coeifficient of x is 2
over the square root of pi,
664
00:37:35,810 --> 00:37:39,360
and the coefficient of x cubed
is minus 2/3 times the square
665
00:37:39,360 --> 00:37:41,090
root of pi, and so on.
666
00:37:41,090 --> 00:37:43,470
But this is a perfectly good way
to write this power series
667
00:37:43,470 --> 00:37:45,630
expansion as well.
668
00:37:45,630 --> 00:37:48,810
And, this is a very good way
to compute the value of the
669
00:37:48,810 --> 00:37:49,570
error function.
670
00:37:49,570 --> 00:37:53,210
It's a new function
in our experience.
671
00:37:53,210 --> 00:37:56,050
Your calculator probably
calculates it, and your
672
00:37:56,050 --> 00:37:58,710
calculator probably does
it by this method.
673
00:37:58,710 --> 00:38:01,270
674
00:38:01,270 --> 00:38:07,900
OK, so that's my sermon on
examples of things you can do
675
00:38:07,900 --> 00:38:10,260
with power series.
676
00:38:10,260 --> 00:38:13,870
So, we're going to do the CEG
thing in just a minute.
677
00:38:13,870 --> 00:38:17,740
Professor Jerison wanted me
to make an ad for 18.02.
678
00:38:17,740 --> 00:38:20,620
Just in case you were thinking
of not taking it next term,
679
00:38:20,620 --> 00:38:21,980
you really should take it.
680
00:38:21,980 --> 00:38:25,070
It will put a lot of things
in this course into
681
00:38:25,070 --> 00:38:26,720
context, for one thing.
682
00:38:26,720 --> 00:38:29,030
It's about vector calculus
and so on.
683
00:38:29,030 --> 00:38:32,190
So you'll learn about vectors
and things like that.
684
00:38:32,190 --> 00:38:35,440
But it comes back and explains
some things in this course
685
00:38:35,440 --> 00:38:37,760
that might have been a little
bit strange, like these
686
00:38:37,760 --> 00:38:44,600
strange formulas for the product
rule and the quotient
687
00:38:44,600 --> 00:38:48,710
rule and the sort of
random formulas.
688
00:38:48,710 --> 00:38:51,250
Well, one of the things you
learn in 18.02 is that they're
689
00:38:51,250 --> 00:38:54,560
all special cases of
the chain rule.
690
00:38:54,560 --> 00:38:59,390
And just to drive that point
home, he wanted me to show you
691
00:38:59,390 --> 00:39:04,470
this poem of his that really
drives the points home
692
00:39:04,470 --> 00:39:05,720
forcefully, I think.
693
00:39:05,720 --> 00:39:13,220