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PROFESSOR: So,
Professor Jerison is

9
00:00:26,290 --> 00:00:29,840
relaxing in sunny
London, Ontario today

10
00:00:29,840 --> 00:00:31,969
and sent me in as
his substitute again.

11
00:00:31,969 --> 00:00:33,760
I'm glad to the here
and see you all again.

12
00:00:38,364 --> 00:00:40,280
So our agenda today: he
said that he'd already

13
00:00:40,280 --> 00:00:45,200
talked about power series
and Taylor's formula,

14
00:00:45,200 --> 00:00:50,850
I guess on last week
right, on Friday?

15
00:00:50,850 --> 00:00:53,265
So I'm going to go a
little further with that

16
00:00:53,265 --> 00:00:57,540
and show you some examples,
show you some applications,

17
00:00:57,540 --> 00:00:59,900
and then I have this
course evaluation survey

18
00:00:59,900 --> 00:01:03,805
that I'll hand out in the last
10 minutes or so of the class.

19
00:01:06,890 --> 00:01:09,630
I also have this handout
that he made that says

20
00:01:09,630 --> 00:01:12,520
18.01 end of term 2007.

21
00:01:12,520 --> 00:01:15,690
If you didn't pick this up
coming in, grab it going out.

22
00:01:15,690 --> 00:01:18,850
People tend not to pick it
up when they walk in, I see.

23
00:01:18,850 --> 00:01:21,890
So grab this when
you're going out.

24
00:01:21,890 --> 00:01:23,390
There's some things
missing from it.

25
00:01:23,390 --> 00:01:27,210
He has not decided
when his office hours

26
00:01:27,210 --> 00:01:28,520
will be at the end of term.

27
00:01:28,520 --> 00:01:31,280
He will have them, just
hasn't decided when.

28
00:01:31,280 --> 00:01:34,575
So, check the website
for that information.

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00:01:38,000 --> 00:01:42,640
And we're looking forward to
the final exam, which is uh --

30
00:01:42,640 --> 00:01:43,140
aren't we?

31
00:01:47,030 --> 00:01:49,590
Any questions about
this technical stuff?

32
00:01:52,900 --> 00:01:56,860
All right, let's talk about
power series for a little bit.

33
00:01:56,860 --> 00:02:00,000
So I thought I should
review for you what

34
00:02:00,000 --> 00:02:01,980
the story with power series is.

35
00:02:21,595 --> 00:02:23,220
OK, could I have your
attention please?

36
00:02:26,920 --> 00:02:31,640
So, power series is a way of
writing a function as a sum

37
00:02:31,640 --> 00:02:33,560
of integral powers of x.

38
00:02:33,560 --> 00:02:38,090
These a_0, a_1, and
so on, are numbers.

39
00:02:38,090 --> 00:02:43,060
An example of a power
series is a polynomial.

40
00:02:48,050 --> 00:02:51,250
Not to be forgotten,
one type of power series

41
00:02:51,250 --> 00:02:57,560
is one which goes on for
a finite number of terms

42
00:02:57,560 --> 00:03:02,090
and then ends, so that all of
the other, all the higher a_i's

43
00:03:02,090 --> 00:03:03,650
are all 0.

44
00:03:03,650 --> 00:03:06,150
This is a perfectly good
example of a power series;

45
00:03:06,150 --> 00:03:08,862
it's a very special
kind of power series.

46
00:03:08,862 --> 00:03:10,570
And part of what I
want to tell you today

47
00:03:10,570 --> 00:03:14,290
is that power series
behave, almost exactly like,

48
00:03:14,290 --> 00:03:14,970
polynomials.

49
00:03:14,970 --> 00:03:16,490
There's just one
thing that you have

50
00:03:16,490 --> 00:03:20,800
to be careful about when you're
using power series that isn't

51
00:03:20,800 --> 00:03:22,810
a concern for polynomials,
and I'll show you

52
00:03:22,810 --> 00:03:24,540
what that is in a minute.

53
00:03:24,540 --> 00:03:29,320
So, you should think of them
as generalized polynomials.

54
00:03:29,320 --> 00:03:32,140
The one thing that you
have to be careful about

55
00:03:32,140 --> 00:03:46,596
is that there is a
number-- So one caution.

56
00:03:46,596 --> 00:03:55,490
There's a number which I'll
call R, where R can be between 0

57
00:03:55,490 --> 00:03:57,350
and it can also be infinity.

58
00:03:57,350 --> 00:04:00,710
It's a number between 0
and infinity, inclusive,

59
00:04:00,710 --> 00:04:06,830
so that when the absolute
value of x is less than R.

60
00:04:06,830 --> 00:04:11,260
So when x is smaller than R
in size, the sum converges.

61
00:04:17,220 --> 00:04:21,170
This sum-- that sum
converges to a finite value.

62
00:04:21,170 --> 00:04:25,120
And when x is bigger
than R in absolute value,

63
00:04:25,120 --> 00:04:26,050
the sum diverges.

64
00:04:30,260 --> 00:04:32,165
This R is called the
radius of convergence.

65
00:04:42,960 --> 00:04:45,750
So we'll see some examples of
what the radius of convergence

66
00:04:45,750 --> 00:04:49,625
is in various powers series as
well, and how you find it also.

67
00:04:55,890 --> 00:04:57,710
But, let me go on and
give you a few more

68
00:04:57,710 --> 00:05:01,320
of the properties
about power series

69
00:05:01,320 --> 00:05:05,410
which I think that professor
Jerison talked about earlier.

70
00:05:05,410 --> 00:05:08,740
So one of them is there's
a radius of convergence.

71
00:05:08,740 --> 00:05:10,200
Here's another one.

72
00:05:15,760 --> 00:05:18,510
If you're inside of
the radius convergence,

73
00:05:18,510 --> 00:05:23,460
then the function has
all its derivatives,

74
00:05:23,460 --> 00:05:34,530
has all its derivatives,
just like a polynomial does.

75
00:05:34,530 --> 00:05:37,210
You can differentiate
it over and over again.

76
00:05:37,210 --> 00:05:41,300
And in terms of
those derivatives,

77
00:05:41,300 --> 00:05:46,880
the number a_n in
the power series

78
00:05:46,880 --> 00:05:52,000
can be expressed in terms of the
value of the derivative at 0.

79
00:05:52,000 --> 00:05:53,700
And this is called
Taylor's formula.

80
00:05:58,540 --> 00:06:02,390
So I'm saying that inside of
this radius of convergence,

81
00:06:02,390 --> 00:06:05,200
the function that we're
looking at, this f(x),

82
00:06:05,200 --> 00:06:10,710
can be written as the value of
the function at 0, that's a_0,

83
00:06:10,710 --> 00:06:13,830
plus the value of
the derivative.

84
00:06:13,830 --> 00:06:17,630
This bracket n means you
take the derivative n times.

85
00:06:17,630 --> 00:06:20,940
So when n is 1, you take
the derivative once at 0,

86
00:06:20,940 --> 00:06:25,100
divided by 1!, which is
!, and multiply it by x.

87
00:06:25,100 --> 00:06:27,630
That's the linear term
in the power series.

88
00:06:27,630 --> 00:06:30,260
And then the quadratic term is
you take the second derivative.

89
00:06:30,260 --> 00:06:33,740
Remember to divide
by 2!, which is 2.

90
00:06:33,740 --> 00:06:39,730
Multiply that by
x^2 and so on out.

91
00:06:39,730 --> 00:06:41,870
So, in terms-- So
the coefficients

92
00:06:41,870 --> 00:06:45,800
in the power series just record
the values of the derivatives

93
00:06:45,800 --> 00:06:48,080
of the function at x = 0.

94
00:06:48,080 --> 00:06:52,270
They can be computed
that way also.

95
00:06:52,270 --> 00:06:53,160
Let's see.

96
00:06:53,160 --> 00:06:55,760
I think that's the end
of my summary of things

97
00:06:55,760 --> 00:06:57,070
that he talked about.

98
00:06:57,070 --> 00:06:59,110
I think he did one
example, and I'll repeat

99
00:06:59,110 --> 00:07:04,680
that example of a power series.

100
00:07:04,680 --> 00:07:06,610
This example wasn't
due to David Jerison;

101
00:07:06,610 --> 00:07:07,860
it was due to Leonard Euler.

102
00:07:11,590 --> 00:07:15,490
It's the example of where the
function is the exponential

103
00:07:15,490 --> 00:07:16,210
function e^x.

104
00:07:19,280 --> 00:07:22,695
So, let's see.

105
00:07:22,695 --> 00:07:25,320
Let's compute what-- I will just
repeat for you the computation

106
00:07:25,320 --> 00:07:28,050
of the power series for
e^x, just because it's such

107
00:07:28,050 --> 00:07:30,142
an important thing to do.

108
00:07:30,142 --> 00:07:32,600
So, in order to do that, I have
to know what the derivative

109
00:07:32,600 --> 00:07:36,750
of e^x is, and what the
second derivative of e^x is,

110
00:07:36,750 --> 00:07:41,250
and so on, because that
comes into the Taylor formula

111
00:07:41,250 --> 00:07:42,690
for the coefficients.

112
00:07:42,690 --> 00:07:46,720
But we know what the derivative
of e^x is, it's just e^x again,

113
00:07:46,720 --> 00:07:49,310
and it's that way
all the way down.

114
00:07:49,310 --> 00:07:53,310
All the derivatives are
e^x over and over again.

115
00:07:53,310 --> 00:07:58,505
So when I evaluate this at x =
0, well, the value of e^x is 1,

116
00:07:58,505 --> 00:08:01,910
the value of e^x is 1 at x = 0.

117
00:08:01,910 --> 00:08:05,180
You get a value of
1 all the way down.

118
00:08:05,180 --> 00:08:10,090
So all these derivatives
at 0 have the value 1.

119
00:08:10,090 --> 00:08:12,790
And now, when I plug
into this formula,

120
00:08:12,790 --> 00:08:28,645
I find e^x is 1 plus 1*x
plus 1/2! x^2 plus 1/3! x^3,

121
00:08:28,645 --> 00:08:32,250
plus and so on.

122
00:08:32,250 --> 00:08:34,660
So all of these
numbers are 1, and all

123
00:08:34,660 --> 00:08:37,130
you wind up with is the
factorials in the denominators.

124
00:08:37,130 --> 00:08:38,870
That's the power series for e^x.

125
00:08:38,870 --> 00:08:42,050
This was a discovery of Leonhard
Euler in 1740 or something.

126
00:08:42,050 --> 00:08:42,842
Yes, Ma'am.

127
00:08:42,842 --> 00:08:45,102
AUDIENCE: When you're
writing out the power series,

128
00:08:45,102 --> 00:08:46,910
how far do you have
to write it out?

129
00:08:46,910 --> 00:08:49,243
PROFESSOR: How far do you
have to write the power series

130
00:08:49,243 --> 00:08:51,100
before it becomes well defined?

131
00:08:51,100 --> 00:08:54,340
Before it's a satisfactory
solution to an exam problem,

132
00:08:54,340 --> 00:08:58,100
I suppose, is another way
to phrase the question.

133
00:08:58,100 --> 00:09:00,920
Until you can see
what the pattern is.

134
00:09:00,920 --> 00:09:02,230
I can see what the pattern is.

135
00:09:02,230 --> 00:09:03,938
Is there anyone who's
in doubt about what

136
00:09:03,938 --> 00:09:07,662
the next term might be?

137
00:09:07,662 --> 00:09:09,120
Some people would
tell you that you

138
00:09:09,120 --> 00:09:11,900
have to write the
summation convention thing.

139
00:09:11,900 --> 00:09:13,862
Don't believe them.

140
00:09:13,862 --> 00:09:15,820
If you right out enough
terms to make it clear,

141
00:09:15,820 --> 00:09:17,020
that's good enough.

142
00:09:17,020 --> 00:09:18,987
OK?

143
00:09:18,987 --> 00:09:20,070
Is that an answer for you?

144
00:09:20,070 --> 00:09:22,990
AUDIENCE: Yes, Thank you.

145
00:09:22,990 --> 00:09:25,980
PROFESSOR: OK, so
that's a basic example.

146
00:09:25,980 --> 00:09:28,960
Let's do another basic
example of a power series.

147
00:09:28,960 --> 00:09:32,240
Oh yes, and by the way, whenever
you write out a power series,

148
00:09:32,240 --> 00:09:35,240
you should say what the
radius of convergence is.

149
00:09:35,240 --> 00:09:37,110
And for now, I will
just to tell you

150
00:09:37,110 --> 00:09:39,440
that the radius of convergence
of this power series

151
00:09:39,440 --> 00:09:42,870
is infinity; that
is, this sum always

152
00:09:42,870 --> 00:09:45,765
converges for any value of x.

153
00:09:45,765 --> 00:09:47,890
I'll say a little more
about that in a few minutes.

154
00:09:47,890 --> 00:09:49,780
Yeah?

155
00:09:49,780 --> 00:09:52,880
AUDIENCE: So which functions
can be written as power series?

156
00:09:52,880 --> 00:09:57,060
PROFESSOR: Which functions can
be written as power series?

157
00:09:57,060 --> 00:10:00,030
That's an excellent question.

158
00:10:00,030 --> 00:10:05,930
Any function that has
a reasonable expression

159
00:10:05,930 --> 00:10:08,679
can be written as
a power series.

160
00:10:08,679 --> 00:10:11,220
I'm not giving you a very good
answer because the true answer

161
00:10:11,220 --> 00:10:12,730
is a little bit complicated.

162
00:10:12,730 --> 00:10:14,470
But any of the
functions that occur

163
00:10:14,470 --> 00:10:18,420
in calculus like sines,
cosines, tangents, they all have

164
00:10:18,420 --> 00:10:21,495
power series expansions, OK?

165
00:10:21,495 --> 00:10:22,495
We'll see more examples.

166
00:10:25,830 --> 00:10:27,070
Let's do another example.

167
00:10:27,070 --> 00:10:30,130
Here's another example.

168
00:10:30,130 --> 00:10:31,520
I guess this was example one.

169
00:10:35,520 --> 00:10:42,140
So, this example, I think,
was due to Newton, not Euler.

170
00:10:42,140 --> 00:10:46,730
Let's find the power series
expansion of this function:

171
00:10:46,730 --> 00:10:48,200
1/(1+x).

172
00:10:48,200 --> 00:10:51,740
Well, I think that
somewhere along the line,

173
00:10:51,740 --> 00:10:56,137
you learned about the geometric
series which tells you

174
00:10:56,137 --> 00:10:58,220
that-- which tells you
what the answer to this is,

175
00:10:58,220 --> 00:11:00,190
and I'll just write it out.

176
00:11:00,190 --> 00:11:12,240
The geometric series tells
you that this function

177
00:11:12,240 --> 00:11:16,460
can be written as an
alternating sum of powers of x.

178
00:11:16,460 --> 00:11:18,810
You may wonder where
these minuses came from.

179
00:11:18,810 --> 00:11:21,100
Well, if you really think
about the geometric series,

180
00:11:21,100 --> 00:11:24,430
as you probably remembered,
there was a minus sign here,

181
00:11:24,430 --> 00:11:28,420
and that gets replaced
by these minus signs.

182
00:11:28,420 --> 00:11:31,810
I think maybe Jerison
talked about this also.

183
00:11:31,810 --> 00:11:34,640
Anyway, here's
another basic example.

184
00:11:34,640 --> 00:11:36,460
Remember what the
graph of this function

185
00:11:36,460 --> 00:11:41,404
looks like when x = -1.

186
00:11:41,404 --> 00:11:42,820
Then there's a
little problem here

187
00:11:42,820 --> 00:11:45,080
because the
denominator becomes 0,

188
00:11:45,080 --> 00:11:47,600
so the graph has a pole there.

189
00:11:47,600 --> 00:11:52,110
It goes up to
infinity at x = -1,

190
00:11:52,110 --> 00:11:57,620
and that's an indication that
the radius of convergence

191
00:11:57,620 --> 00:11:58,990
is not infinity.

192
00:11:58,990 --> 00:12:01,240
Because if you try to converge
to this infinite number

193
00:12:01,240 --> 00:12:04,930
by putting in x = -1, here,
you'll have a big problem.

194
00:12:04,930 --> 00:12:07,122
In fact, you see when
you put in x = -1,

195
00:12:07,122 --> 00:12:08,497
you keep getting
1 in every term,

196
00:12:08,497 --> 00:12:11,390
and it gets bigger and
bigger and does not converge.

197
00:12:11,390 --> 00:12:14,940
In this example, the
radius of convergence is 1.

198
00:12:18,570 --> 00:12:22,210
OK, so, let's do
a new example now.

199
00:12:22,210 --> 00:12:24,250
Oh, and by the way,
I should say you

200
00:12:24,250 --> 00:12:27,770
can calculate these numbers
using Taylor's formula.

201
00:12:27,770 --> 00:12:29,940
If you haven't seen
it, check it out.

202
00:12:29,940 --> 00:12:36,580
Calculate the iterated
derivatives of this function

203
00:12:36,580 --> 00:12:41,410
and plug in x = 0 and see
that you get +1, -1, +1, -1,

204
00:12:41,410 --> 00:12:41,930
and so on.

205
00:12:41,930 --> 00:12:42,706
Yes sir.

206
00:12:42,706 --> 00:12:44,636
AUDIENCE: For the
radius of convergence

207
00:12:44,636 --> 00:12:48,090
I see that if you do
-1 it'll blow out.

208
00:12:48,090 --> 00:12:50,740
If you put in 1 though, it
seems like it would be fine.

209
00:12:50,740 --> 00:12:52,550
PROFESSOR: The
questions is I can

210
00:12:52,550 --> 00:12:54,870
see that there's a
problem at x = -1,

211
00:12:54,870 --> 00:12:57,280
why is there also
a problem at x = 1

212
00:12:57,280 --> 00:12:59,090
where the graph is
perfectly smooth

213
00:12:59,090 --> 00:13:00,760
and innocuous and finite.

214
00:13:00,760 --> 00:13:04,490
That's another
excellent question.

215
00:13:04,490 --> 00:13:07,650
The problem is that if you
go off to a radius of 1

216
00:13:07,650 --> 00:13:11,530
in any direction and there's
a problem, that's it.

217
00:13:11,530 --> 00:13:13,530
That's what the radius
of convergence is.

218
00:13:13,530 --> 00:13:18,070
Here, what does happen
if I put an x = +1?

219
00:13:18,070 --> 00:13:20,490
So, let's look at
the partial sums.

220
00:13:20,490 --> 00:13:23,060
Do x = +1 in your mind here.

221
00:13:23,060 --> 00:13:29,190
So I'll get a partial sum 1,
then 0, and then 1, and then 0,

222
00:13:29,190 --> 00:13:29,907
and then 1.

223
00:13:29,907 --> 00:13:31,740
So even though it doesn't
go up to infinity,

224
00:13:31,740 --> 00:13:32,900
it still does not converge.

225
00:13:32,900 --> 00:13:35,500
AUDIENCE: And
anything in between?

226
00:13:35,500 --> 00:13:37,510
PROFESSOR: Any of
these other things

227
00:13:37,510 --> 00:13:41,330
will also fail to
converge in this example.

228
00:13:41,330 --> 00:13:43,685
Well, that's the only two
real numbers at the edge.

229
00:13:43,685 --> 00:13:44,185
Right?

230
00:13:46,940 --> 00:13:49,050
OK, let's do a
different example now.

231
00:13:49,050 --> 00:13:50,210
How about a trig function?

232
00:13:50,210 --> 00:13:50,793
The sine of x.

233
00:13:55,422 --> 00:14:01,810
I'm going to compute the power
series expansion for sin(x).

234
00:14:01,810 --> 00:14:04,400
and I'm going to do it
using Taylor's formula.

235
00:14:04,400 --> 00:14:06,310
So Taylor's formula
says that I have

236
00:14:06,310 --> 00:14:09,452
to start computing
derivatives of sin(x).

237
00:14:22,100 --> 00:14:25,280
Sounds like it's going
to be a lot of work.

238
00:14:25,280 --> 00:14:28,005
Let's see, the derivative
of the sine is the cosine.

239
00:14:30,870 --> 00:14:32,910
And the derivative
of the cosine,

240
00:14:32,910 --> 00:14:36,530
that's the second derivative
of the sine, is what?

241
00:14:36,530 --> 00:14:40,270
Remember the minus,
it's -sin(x).

242
00:14:40,270 --> 00:14:43,180
OK, now I want to take the third
derivative of the sine, which

243
00:14:43,180 --> 00:14:45,680
is the derivative
of sine prime prime,

244
00:14:45,680 --> 00:14:47,760
so it's the derivative of this.

245
00:14:47,760 --> 00:14:49,840
And we just decided
the derivative of sine

246
00:14:49,840 --> 00:14:52,270
is cosine, so I
get cosine, but I

247
00:14:52,270 --> 00:14:53,730
have this minus sign in front.

248
00:14:56,660 --> 00:14:58,710
And now I want to
differentiate again,

249
00:14:58,710 --> 00:15:01,640
so the cosine
becomes a minus sine,

250
00:15:01,640 --> 00:15:08,520
and that sign cancels with this
minus sign to give me sin(x).

251
00:15:08,520 --> 00:15:10,102
You follow that?

252
00:15:10,102 --> 00:15:13,660
It's a lot of -1's
canceling out there.

253
00:15:13,660 --> 00:15:17,290
So, all of a sudden, I'm
right back where I started;

254
00:15:17,290 --> 00:15:21,610
these two are the same and the
pattern will now repeat forever

255
00:15:21,610 --> 00:15:22,780
and ever.

256
00:15:22,780 --> 00:15:24,440
Higher and higher
derivatives of sines

257
00:15:24,440 --> 00:15:28,830
are just plus or minus
sines and cosines.

258
00:15:28,830 --> 00:15:34,300
Now Taylor's formula says I
should now substitute x = 0

259
00:15:34,300 --> 00:15:37,580
into this and see what
happens, so let's do that.

260
00:15:37,580 --> 00:15:43,240
When x is equals to 0, the
sine is 0 and the cosine is 1.

261
00:15:43,240 --> 00:15:47,410
The sine is 0, so
minus 0 is also 0.

262
00:15:47,410 --> 00:15:51,070
The cosine is 1, but
now there's a minus one,

263
00:15:51,070 --> 00:15:53,720
and now I'm back
where I started,

264
00:15:53,720 --> 00:15:58,760
and so the pattern will repeat.

265
00:15:58,760 --> 00:16:00,670
OK, so the values
of the derivatives

266
00:16:00,670 --> 00:16:03,630
are all zeros and
plus and minus ones

267
00:16:03,630 --> 00:16:07,420
and they go through that
pattern, four-fold periodicity,

268
00:16:07,420 --> 00:16:09,670
over and over again.

269
00:16:09,670 --> 00:16:13,277
And so we can write
out what sin(x)

270
00:16:13,277 --> 00:16:15,410
is using Taylor's formula,
using this formula.

271
00:16:18,000 --> 00:16:21,770
So I put in the value
at 0 which is 0, then

272
00:16:21,770 --> 00:16:27,620
I put in the derivative
which is 1, multiplied by x.

273
00:16:27,620 --> 00:16:32,180
Then, I have the second
derivative divided by 2!,

274
00:16:32,180 --> 00:16:35,150
but the second
derivative at 0 is 0.

275
00:16:35,150 --> 00:16:38,280
So I'm going to
drop that term out.

276
00:16:38,280 --> 00:16:41,365
Now I have the third
derivative which is -1.

277
00:16:43,930 --> 00:16:45,550
And remember the 3!

278
00:16:45,550 --> 00:16:46,790
in the denominator.

279
00:16:46,790 --> 00:16:50,050
That's the coefficient of x^3.

280
00:16:50,050 --> 00:16:51,680
What's the fourth derivative?

281
00:16:51,680 --> 00:16:54,400
Well, here we are, it's
on the board, it's 0.

282
00:16:54,400 --> 00:16:58,150
So I drop that term out
go up to the fifth term,

283
00:16:58,150 --> 00:16:59,830
the fifth power of x.

284
00:16:59,830 --> 00:17:02,260
Its derivative is now 1.

285
00:17:02,260 --> 00:17:06,750
We've gone through the pattern,
we're back at +1 as the value

286
00:17:06,750 --> 00:17:13,180
of the iterated derivative,
so now I get 1/5! x^5.

287
00:17:13,180 --> 00:17:15,720
Now, you tell me, have we
done enough terms to see

288
00:17:15,720 --> 00:17:17,900
what the pattern is?

289
00:17:17,900 --> 00:17:22,170
I guess the next
term will be a -1/7!

290
00:17:22,170 --> 00:17:23,750
x^7, and so on.

291
00:17:23,750 --> 00:17:28,160
Let me write this out
again just so we have it.

292
00:17:28,160 --> 00:17:30,830
x^3 / 3!-- So it's
x minus x^3 / 3!

293
00:17:30,830 --> 00:17:31,680
plus x^5 / 5!.

294
00:17:34,580 --> 00:17:38,740
You guessed it, and so on.

295
00:17:38,740 --> 00:17:40,270
That's the power
series expansion

296
00:17:40,270 --> 00:17:43,935
for the sine of x, OK?

297
00:17:46,950 --> 00:17:49,700
And so, the sign alternate,
and these denominators

298
00:17:49,700 --> 00:17:52,250
get very big, don't they?

299
00:17:52,250 --> 00:17:54,410
Exponentials grow very fast.

300
00:17:54,410 --> 00:17:55,830
Let me make a remark.

301
00:17:55,830 --> 00:17:58,800
R is infinity here.

302
00:17:58,800 --> 00:18:01,580
The radius of convergence
of this power series

303
00:18:01,580 --> 00:18:03,424
again is infinity, and
let me just say why.

304
00:18:03,424 --> 00:18:13,110
The reason is that the general
term is going to be like

305
00:18:13,110 --> 00:18:14,100
x^(2n+1) / (2n+1)!.

306
00:18:18,330 --> 00:18:21,900
An odd number I can
write as 2n + 1.

307
00:18:21,900 --> 00:18:24,608
And what I want to
say is that the size

308
00:18:24,608 --> 00:18:30,480
of this, what happens
to the size of this as n

309
00:18:30,480 --> 00:18:33,989
goes to infinity?

310
00:18:33,989 --> 00:18:35,280
So let's just think about this.

311
00:18:35,280 --> 00:18:38,270
For a fixed x, let's
fix the number x.

312
00:18:38,270 --> 00:18:41,260
Look at powers of x and
think about the size

313
00:18:41,260 --> 00:18:45,682
of this expression when
n gets to be large.

314
00:18:45,682 --> 00:18:47,140
So let's just do
that for a second.

315
00:18:47,140 --> 00:18:54,420
So, x^(2n+1) / (2n+1)!, I
can write out like this.

316
00:18:54,420 --> 00:19:03,010
It's x / 1 times x / 2
-- sorry -- times x / 3,

317
00:19:03,010 --> 00:19:09,250
times x / (2n+1).

318
00:19:09,250 --> 00:19:13,220
I've multiplied x by itself
2n+1 times in the numerator,

319
00:19:13,220 --> 00:19:15,680
and I've multiplied
the numbers 1, 2, 3, 4,

320
00:19:15,680 --> 00:19:18,477
and so on, by each other
in the denominator,

321
00:19:18,477 --> 00:19:19,810
and that gives me the factorial.

322
00:19:19,810 --> 00:19:22,330
So I've just written
this out like this.

323
00:19:22,330 --> 00:19:26,820
Now x is fixed, so maybe
it's a million, OK?

324
00:19:26,820 --> 00:19:28,700
It's big, but fixed.

325
00:19:28,700 --> 00:19:30,634
What happens to these numbers?

326
00:19:30,634 --> 00:19:32,050
Well at first,
they're pretty big.

327
00:19:32,050 --> 00:19:34,560
This is 1,000,000 / 2,
this is 1,000,000 / 3.

328
00:19:34,560 --> 00:19:38,820
But when n gets to be--
Maybe if n is 1,000,000,

329
00:19:38,820 --> 00:19:41,180
then this is about 1/2.

330
00:19:41,180 --> 00:19:48,320
If n is a billion, then this
is about 1/2,000, right?

331
00:19:48,320 --> 00:19:50,530
The denominators keep
getting bigger and bigger,

332
00:19:50,530 --> 00:19:54,470
but the numerators stay
the same; they're always x.

333
00:19:54,470 --> 00:19:57,590
So when I take the product,
if I go far enough out,

334
00:19:57,590 --> 00:20:00,070
I'm going to be multiplying,
by very, very small numbers

335
00:20:00,070 --> 00:20:01,730
and more and more of them.

336
00:20:01,730 --> 00:20:05,980
And so no matter what
x is, these numbers

337
00:20:05,980 --> 00:20:07,220
will converge to 0.

338
00:20:07,220 --> 00:20:11,550
They'll get smaller and
smaller as x gets to be bigger.

339
00:20:11,550 --> 00:20:16,590
That's the sign that x is inside
of the radius of convergence.

340
00:20:16,590 --> 00:20:21,360
This is the sign for
you that this series

341
00:20:21,360 --> 00:20:23,780
converges for that value of x.

342
00:20:23,780 --> 00:20:33,600
And because I could do
this for any x, this works.

343
00:20:33,600 --> 00:20:40,260
This convergence to
0 for any fixed x.

344
00:20:40,260 --> 00:20:43,330
That's what tells
you that you can

345
00:20:43,330 --> 00:20:46,060
take-- that the radius of
convergence is infinity.

346
00:20:46,060 --> 00:20:49,575
Because in the
formula, in the fact,

347
00:20:49,575 --> 00:20:53,700
in this property that
the radius of convergence

348
00:20:53,700 --> 00:20:56,340
talks about, if R is
equal to infinity,

349
00:20:56,340 --> 00:20:58,160
this is no condition on x.

350
00:20:58,160 --> 00:21:02,250
Every number is less than
infinity in absolute value.

351
00:21:02,250 --> 00:21:05,930
So if this convergence
to 0 of the general term

352
00:21:05,930 --> 00:21:10,320
works for every x, then radius
of convergence is infinity.

353
00:21:10,320 --> 00:21:11,700
Well that was kind
of fast, but I

354
00:21:11,700 --> 00:21:13,790
think that you've heard
something about that

355
00:21:13,790 --> 00:21:16,020
earlier as well.

356
00:21:16,020 --> 00:21:19,140
Anyway, so we've got the
sine function, a new function

357
00:21:19,140 --> 00:21:20,880
with its own power series.

358
00:21:20,880 --> 00:21:23,730
It's a way of computing sin(x).

359
00:21:23,730 --> 00:21:26,660
If you take enough
terms you'll get

360
00:21:26,660 --> 00:21:28,560
a good evaluation of sin(x).

361
00:21:28,560 --> 00:21:30,000
for any x.

362
00:21:30,000 --> 00:21:32,200
This tells you a lot
about the function sin(x)

363
00:21:32,200 --> 00:21:33,750
but not everything at all.

364
00:21:33,750 --> 00:21:36,590
For example, from
this formula, it's

365
00:21:36,590 --> 00:21:39,745
very hard to see that the
sine of x is periodic.

366
00:21:39,745 --> 00:21:41,930
It's not obvious at all.

367
00:21:41,930 --> 00:21:44,090
Somewhere hidden away
in this expression

368
00:21:44,090 --> 00:21:47,400
is the number pi, the
half of the period.

369
00:21:47,400 --> 00:21:51,100
But that's not clear from
the power series at all.

370
00:21:51,100 --> 00:21:53,320
So the power series are
very good for some things,

371
00:21:53,320 --> 00:21:55,510
but they hide other
properties of functions.

372
00:21:58,150 --> 00:22:00,830
Well, so I want to spend
a few minutes telling you

373
00:22:00,830 --> 00:22:04,350
about what you can do
with a power series,

374
00:22:04,350 --> 00:22:07,620
once you have one, to get new
power series, so new power

375
00:22:07,620 --> 00:22:08,530
series from old.

376
00:22:18,300 --> 00:22:25,490
And this is also called
operations on power series.

377
00:22:25,490 --> 00:22:27,989
So what are the things that
we can do to a power series?

378
00:22:27,989 --> 00:22:29,905
Well one of the things
you can do is multiply.

379
00:22:33,990 --> 00:22:37,310
So, for example, what if
I want to compute a power

380
00:22:37,310 --> 00:22:40,970
series for x sin(x)?

381
00:22:40,970 --> 00:22:44,160
Well I have a power series
for sin(x), I just did it.

382
00:22:44,160 --> 00:22:45,920
How about a power series for x?

383
00:22:48,910 --> 00:22:51,480
Actually, I did that here too.

384
00:22:51,480 --> 00:22:55,120
The function x is a
very simple polynomial.

385
00:22:55,120 --> 00:22:58,330
It's a polynomial where
that's 0, a_1 is 1,

386
00:22:58,330 --> 00:23:00,930
and all the other
coefficients are 0.

387
00:23:00,930 --> 00:23:04,870
So x itself is a power
series, a very simple one.

388
00:23:04,870 --> 00:23:08,317
sin(x) is a powers series.

389
00:23:08,317 --> 00:23:09,900
And what I want to
encourage you to do

390
00:23:09,900 --> 00:23:12,720
is treat power series
just like polynomials

391
00:23:12,720 --> 00:23:14,330
and multiply them together.

392
00:23:14,330 --> 00:23:16,960
We'll see other operations too.

393
00:23:16,960 --> 00:23:20,860
So, to compute the power series
for x sin(x), of I just take

394
00:23:20,860 --> 00:23:24,400
this one and multiply it by x.

395
00:23:24,400 --> 00:23:26,650
So let's see if I
can do that right.

396
00:23:26,650 --> 00:23:30,050
It distributes through:
x^2 minus x^4 / 3!

397
00:23:33,000 --> 00:23:42,190
plus x^6 / 5!, and so on.

398
00:23:42,190 --> 00:23:44,180
And again, the
radius of convergence

399
00:23:44,180 --> 00:23:47,130
is going to be the smaller of
the two radii of convergence

400
00:23:47,130 --> 00:23:48,220
here.

401
00:23:48,220 --> 00:23:51,840
So it's R equals
infinity in this case.

402
00:23:51,840 --> 00:23:54,010
OK, you can multiply
power series together.

403
00:23:54,010 --> 00:23:56,910
It can be a pain if the
power series are very long,

404
00:23:56,910 --> 00:24:01,800
but if one of them is
x, it's pretty simple.

405
00:24:01,800 --> 00:24:06,040
OK, that's one thing I can do.

406
00:24:06,040 --> 00:24:08,634
Notice something by the way.

407
00:24:08,634 --> 00:24:10,175
You know that even
and odd functions?

408
00:24:13,180 --> 00:24:17,390
So, sine is an odd function,
x is an odd function,

409
00:24:17,390 --> 00:24:20,570
the product of two odd
functions is an even function.

410
00:24:20,570 --> 00:24:24,070
And that's reflected in the fact
that all the powers that occur

411
00:24:24,070 --> 00:24:26,790
in the power series are even.

412
00:24:26,790 --> 00:24:30,640
For an odd function, like the
sine, all the powers that occur

413
00:24:30,640 --> 00:24:32,589
are odd powers of x.

414
00:24:32,589 --> 00:24:33,380
That's always true.

415
00:24:37,510 --> 00:24:38,969
OK, we can multiply.

416
00:24:38,969 --> 00:24:40,010
I can also differentiate.

417
00:24:48,660 --> 00:24:56,950
So let's just do a
case of that, and use

418
00:24:56,950 --> 00:24:59,240
the process of
differentiation to find out

419
00:24:59,240 --> 00:25:03,580
what the power
series for cos(x) is

420
00:25:03,580 --> 00:25:06,560
by writing the cos(x) as
the derivative of the sine

421
00:25:06,560 --> 00:25:09,010
and differentiating
term by term.

422
00:25:09,010 --> 00:25:11,360
So, I'll take this
expression for the power

423
00:25:11,360 --> 00:25:14,200
series of the sine and
differentiate it term by term,

424
00:25:14,200 --> 00:25:18,210
and I'll get the power
series for cosine.

425
00:25:18,210 --> 00:25:19,030
So, let's see.

426
00:25:19,030 --> 00:25:22,510
The derivative of x is one.

427
00:25:22,510 --> 00:25:27,110
Now, the derivative of x^3 is
3x^2, and then there's a 3!

428
00:25:27,110 --> 00:25:28,910
in the denominator.

429
00:25:28,910 --> 00:25:34,440
And the derivative of x^5
5x^4, and there's a 5!

430
00:25:34,440 --> 00:25:38,680
in the denominator,
and so on and so on.

431
00:25:38,680 --> 00:25:40,950
And now some
cancellation happens.

432
00:25:40,950 --> 00:25:45,960
So this is 1 minus, well, the
3 cancels with the last factor

433
00:25:45,960 --> 00:25:48,730
in this 3 factorial
and leaves you with 2!.

434
00:25:52,460 --> 00:25:56,040
And the 5 cancels with the
last factor in the 5 factorial

435
00:25:56,040 --> 00:25:58,129
and leaves you with a 4!

436
00:25:58,129 --> 00:25:58,920
in the denominator.

437
00:26:01,570 --> 00:26:04,710
And so there you go, there's
the power series expansion

438
00:26:04,710 --> 00:26:05,980
for the cosine.

439
00:26:05,980 --> 00:26:07,970
It's got all even powers of x.

440
00:26:07,970 --> 00:26:12,720
They alternate, and you have
factorials in the denominator.

441
00:26:12,720 --> 00:26:15,510
And of course, you could
derive that expression

442
00:26:15,510 --> 00:26:19,130
by using Taylor's formula, by
the same kind of calculation

443
00:26:19,130 --> 00:26:22,290
you did here, taking higher
and higher derivatives

444
00:26:22,290 --> 00:26:22,970
of the cosine.

445
00:26:22,970 --> 00:26:25,720
You get the same
periodic pattern

446
00:26:25,720 --> 00:26:30,080
of derivatives and values
of derivatives at x = 0.

447
00:26:30,080 --> 00:26:33,200
But here's a cleaner way to
do it, simpler way to do it,

448
00:26:33,200 --> 00:26:36,830
because we already knew
the derivative of the sine.

449
00:26:36,830 --> 00:26:39,630
When you differentiate, you keep
the same radius of convergence.

450
00:26:44,420 --> 00:26:49,320
OK, so we can
multiply, I can add too

451
00:26:49,320 --> 00:26:52,400
and multiply by a
constant, things like that.

452
00:26:52,400 --> 00:26:54,280
How about integrating?

453
00:26:54,280 --> 00:26:56,580
That's what half of this
course was about isn't it?

454
00:26:56,580 --> 00:26:58,550
So, let's integrate something.

455
00:27:07,210 --> 00:27:15,960
So, the integration I'm
going to do is this one:

456
00:27:15,960 --> 00:27:20,160
the integral from 0
to x of dt / (1+x).

457
00:27:20,160 --> 00:27:21,980
What is that integral
as a function?

458
00:27:28,360 --> 00:27:32,060
So, when I find the
anti-derivative of this,

459
00:27:32,060 --> 00:27:39,660
I get ln(1+t), and then when
I evaluate that at t = x,

460
00:27:39,660 --> 00:27:42,600
I get ln(1+x).

461
00:27:42,600 --> 00:27:48,510
And when I evaluate the natural
log at 0, I get the ln 1,

462
00:27:48,510 --> 00:27:55,070
which is 0, so this
is what you get, OK?

463
00:27:55,070 --> 00:28:05,690
This is really valid, by the
way, for x bigger than -1.

464
00:28:05,690 --> 00:28:09,060
But you don't want to think
about this quite like this

465
00:28:09,060 --> 00:28:10,240
when x is smaller than that.

466
00:28:13,770 --> 00:28:18,900
Now, I'm going to try to apply
power series methods here

467
00:28:18,900 --> 00:28:22,660
and find-- use this integral
to find a power series

468
00:28:22,660 --> 00:28:28,230
for the natural log, and I'll
do it by plugging into this

469
00:28:28,230 --> 00:28:34,330
expression what the power
series for 1/(1+t) was.

470
00:28:34,330 --> 00:28:36,830
And I know what that is because
I wrote it down on the board

471
00:28:36,830 --> 00:28:38,330
up here.

472
00:28:38,330 --> 00:28:40,940
Change the variable
from x to t there,

473
00:28:40,940 --> 00:28:49,410
and so 1/(1+t) is 1 minus t
plus t^2 minus t^3, and so on.

474
00:28:52,620 --> 00:28:55,196
So that's the thing in the
inside of the integral,

475
00:28:55,196 --> 00:29:01,710
and now it's legal to
integrate that term by term,

476
00:29:01,710 --> 00:29:03,517
so let's do that.

477
00:29:03,517 --> 00:29:05,350
I'm going to get something
which I will then

478
00:29:05,350 --> 00:29:09,230
evaluate at x and at 0.

479
00:29:09,230 --> 00:29:14,093
So, when I integrate 1 I get
x, and when I integrate t,

480
00:29:14,093 --> 00:29:14,593
I get t.

481
00:29:14,593 --> 00:29:16,829
I'm sorry.

482
00:29:16,829 --> 00:29:29,100
When I integrate t, I get t^2
/ 2, and t^2 gives me t^3 / 3,

483
00:29:29,100 --> 00:29:29,970
and so on and so on.

484
00:29:32,520 --> 00:29:36,350
And then, when I
put in t = x, well,

485
00:29:36,350 --> 00:29:40,350
I just replace all the t's by
x's, and when I put in t = 0,

486
00:29:40,350 --> 00:29:41,780
I get 0.

487
00:29:41,780 --> 00:29:43,950
So this equals x.

488
00:29:43,950 --> 00:29:55,170
So, I've discovered that ln(1+x)
is x minus x^2 / 2 plus x^3 / 3

489
00:29:55,170 --> 00:30:02,020
minus x^4 / 4, and
so on and so on.

490
00:30:02,020 --> 00:30:04,720
There's the power series
expansion for ln(1+x).

491
00:30:07,800 --> 00:30:10,280
And because I began
with a power series

492
00:30:10,280 --> 00:30:13,030
whose radius of
convergence was just 1,

493
00:30:13,030 --> 00:30:15,930
I began with this power
series, the radius

494
00:30:15,930 --> 00:30:18,150
of convergence of this
is also going to be 1.

495
00:30:22,200 --> 00:30:25,670
Also, because this function,
as I just pointed out,

496
00:30:25,670 --> 00:30:29,080
this function goes bad when
x becomes less than -1,

497
00:30:29,080 --> 00:30:32,160
so some problem happens,
and that's reflected

498
00:30:32,160 --> 00:30:35,590
in the radius of convergence.

499
00:30:35,590 --> 00:30:36,750
Cool.

500
00:30:36,750 --> 00:30:41,110
So, you can integrate.

501
00:30:41,110 --> 00:30:45,570
That is the correct power series
expansion for the ln(1+x),

502
00:30:45,570 --> 00:30:49,340
and another victory of Euler's
was to use this kind of power

503
00:30:49,340 --> 00:30:52,270
series expansion to calculate
natural logarithms in a much

504
00:30:52,270 --> 00:30:54,350
more efficient way than
people had done before.

505
00:30:57,850 --> 00:31:08,380
OK, one more property, I think.

506
00:31:12,920 --> 00:31:17,080
What are we at here, 3?

507
00:31:17,080 --> 00:31:18,550
4.

508
00:31:18,550 --> 00:31:19,050
Substitute.

509
00:31:25,410 --> 00:31:28,669
Very appropriate for me
as a substitute teacher

510
00:31:28,669 --> 00:31:29,960
to tell you about substitution.

511
00:31:32,810 --> 00:31:35,480
So I'm going to try to find
the power series expansion

512
00:31:35,480 --> 00:31:36,385
of e^(-t^2).

513
00:31:36,385 --> 00:31:36,885
OK?

514
00:31:41,740 --> 00:31:45,190
And the way I'll do that is
by taking the power series

515
00:31:45,190 --> 00:31:50,050
expansion for e^x,
which we have up there,

516
00:31:50,050 --> 00:32:01,630
and make the substitution x =
-t^2 in the expansion for e^x.

517
00:32:01,630 --> 00:32:03,125
Did you have a question?

518
00:32:03,125 --> 00:32:04,625
AUDIENCE: Well,
it's just concerning

519
00:32:04,625 --> 00:32:07,624
the radius of convergence.

520
00:32:07,624 --> 00:32:11,660
You can't define x so that is
always positive, and if so,

521
00:32:11,660 --> 00:32:14,360
it wouldn't have a radius
of convergence, right?

522
00:32:14,360 --> 00:32:20,740
PROFESSOR: Like I say, again the
worry is this ln(1+x) function

523
00:32:20,740 --> 00:32:24,120
is perfectly well
behaved for large x.

524
00:32:24,120 --> 00:32:27,720
Why does the power series
fail to converge for large x?

525
00:32:27,720 --> 00:32:30,260
Well suppose that
x is bigger than 1,

526
00:32:30,260 --> 00:32:32,200
then here you get
bigger and bigger powers

527
00:32:32,200 --> 00:32:35,160
of x, which will
grow to infinity,

528
00:32:35,160 --> 00:32:40,330
and they grow large faster
than the numbers 2, 3, 4, 5, 6.

529
00:32:40,330 --> 00:32:45,850
They grow exponentially, and
these just grow linearly.

530
00:32:45,850 --> 00:32:49,430
So, again, the general term,
when x is bigger than one,

531
00:32:49,430 --> 00:32:51,600
the general term will
go off to infinity,

532
00:32:51,600 --> 00:32:53,840
even though the function
that you're talking about,

533
00:32:53,840 --> 00:32:56,750
log of net of 1 plus
x is perfectly good.

534
00:32:56,750 --> 00:33:01,615
So the power series is not
good outside of the radius

535
00:33:01,615 --> 00:33:02,240
of convergence.

536
00:33:02,240 --> 00:33:04,650
It's just a fact of life.

537
00:33:04,650 --> 00:33:05,150
Yes?

538
00:33:05,150 --> 00:33:06,025
AUDIENCE: [INAUDIBLE]

539
00:33:18,332 --> 00:33:20,290
PROFESSOR: I'd rather--
talk to me after class.

540
00:33:20,290 --> 00:33:22,620
The question is why is
it the smaller of the two

541
00:33:22,620 --> 00:33:24,110
radii of convergence?

542
00:33:24,110 --> 00:33:30,050
The basic answer
is, well, you can't

543
00:33:30,050 --> 00:33:33,370
expect it to be bigger than that
smaller one, because the power

544
00:33:33,370 --> 00:33:34,790
series only gives
you information

545
00:33:34,790 --> 00:33:37,020
inside of that range
about the function, so.

546
00:33:37,020 --> 00:33:37,895
AUDIENCE: [INAUDIBLE]

547
00:33:41,010 --> 00:33:43,390
PROFESSOR: Well, in this
case, both of the radii

548
00:33:43,390 --> 00:33:44,890
of convergence are infinity.

549
00:33:44,890 --> 00:33:48,110
x has radius of convergence
infinity for sure,

550
00:33:48,110 --> 00:33:49,260
and sin(x) does too.

551
00:33:49,260 --> 00:33:52,140
So you get infinity
in that case, OK?

552
00:33:54,850 --> 00:33:58,162
OK, let's just do
this, and then I'm

553
00:33:58,162 --> 00:33:59,620
going to integrate
this and that'll

554
00:33:59,620 --> 00:34:03,830
be the end of what I
have time for today.

555
00:34:03,830 --> 00:34:05,880
So what's the power
series expansion for this?

556
00:34:05,880 --> 00:34:07,490
The power series
expansion of this

557
00:34:07,490 --> 00:34:11,160
is going to be a
function of t, right,

558
00:34:11,160 --> 00:34:13,530
because the variable here is t.

559
00:34:13,530 --> 00:34:19,360
I get it by taking my expansion
for e^x and putting in what x

560
00:34:19,360 --> 00:34:20,460
is in terms of t.

561
00:34:30,210 --> 00:34:32,180
Whoops!

562
00:34:32,180 --> 00:34:36,700
And so on and so on.

563
00:34:36,700 --> 00:34:42,100
I just put in -t^2 in place of
x there in the series expansion

564
00:34:42,100 --> 00:34:44,120
for e^x.

565
00:34:44,120 --> 00:34:47,910
I can work this out
a little bit better.

566
00:34:47,910 --> 00:34:49,030
-t^2 is what it is.

567
00:34:49,030 --> 00:34:53,760
This is going to give me a t^4
and the minus squared is going

568
00:34:53,760 --> 00:34:55,790
to give me a plus,
so I get t^4 / 2!.

569
00:34:58,730 --> 00:35:05,750
Then I get (-t)^3, so there'll
be a minus sign and a t^6

570
00:35:05,750 --> 00:35:08,190
and the denominator 3!.

571
00:35:08,190 --> 00:35:10,300
So the signs are
going to alternate,

572
00:35:10,300 --> 00:35:13,430
the powers are all even,
and the denominators

573
00:35:13,430 --> 00:35:15,380
are these factorials.

574
00:35:20,160 --> 00:35:23,950
Several times as this
course has gone on,

575
00:35:23,950 --> 00:35:27,216
the error function has
made an appearance.

576
00:35:27,216 --> 00:35:31,790
The error function was, I guess
it gets normalized by putting

577
00:35:31,790 --> 00:35:41,550
a 2 over the square
root of pi in front,

578
00:35:41,550 --> 00:35:46,830
and it's the integral of
e^(-t^2) dt from 0 to x.

579
00:35:46,830 --> 00:35:54,700
And this normalization
is here because as x

580
00:35:54,700 --> 00:36:01,300
gets to be large
the value becomes 1.

581
00:36:01,300 --> 00:36:04,305
So this error function is
very important in the theory

582
00:36:04,305 --> 00:36:06,080
of probability.

583
00:36:06,080 --> 00:36:09,120
And I think you calculated
this fact at some point

584
00:36:09,120 --> 00:36:12,236
in the course.

585
00:36:12,236 --> 00:36:14,960
So the standard definition of
the error function, you put a 2

586
00:36:14,960 --> 00:36:16,460
over the square
root of pi in front.

587
00:36:16,460 --> 00:36:18,425
Let's calculate its
power series expansion.

588
00:36:21,320 --> 00:36:23,220
So there's a 2 over
the square root of pi

589
00:36:23,220 --> 00:36:27,280
that hurts nobody
here in the front.

590
00:36:27,280 --> 00:36:30,530
And now I want to
integrate e^(-t^2),

591
00:36:30,530 --> 00:36:34,010
and I'm going to use this
power series expansion for that

592
00:36:34,010 --> 00:36:36,350
to see what you get.

593
00:36:36,350 --> 00:36:38,800
So I'm just going to
write this out I think.

594
00:36:38,800 --> 00:36:41,340
I did it out carefully in
another example over there,

595
00:36:41,340 --> 00:36:43,100
so I'll do it a
little quicker now.

596
00:36:43,100 --> 00:36:45,606
Integrate this term
by term, you're

597
00:36:45,606 --> 00:36:47,730
just integrating powers of
t so it's pretty simple,

598
00:36:47,730 --> 00:36:51,830
so I get-- and then I'm
evaluating at x and then at 0.

599
00:36:51,830 --> 00:37:03,690
So I get x minus x^3 /
3, plus x^5 / (5*2!),

600
00:37:03,690 --> 00:37:07,790
5 from integrating
the t^4, and the 2!

601
00:37:07,790 --> 00:37:11,390
from this denominator
that we already had.

602
00:37:11,390 --> 00:37:18,040
And then there's a -x^7
/ (7*3!), and plus,

603
00:37:18,040 --> 00:37:21,620
and so on, and you can imagine
how they go on from there.

604
00:37:24,490 --> 00:37:27,500
I guess to get this
exactly in the form

605
00:37:27,500 --> 00:37:32,510
that we began talking about,
I should multiply through.

606
00:37:32,510 --> 00:37:35,810
So the coefficient of x is 2
over the square root of pi,

607
00:37:35,810 --> 00:37:39,465
and the coefficient of x^3 is
-2 over 3 times the square root

608
00:37:39,465 --> 00:37:41,012
of pi, and so on.

609
00:37:41,012 --> 00:37:43,470
But this is a perfectly good
way to write this power series

610
00:37:43,470 --> 00:37:45,630
expansion as well.

611
00:37:45,630 --> 00:37:49,020
And, this is a very good way to
compute the value of the error

612
00:37:49,020 --> 00:37:49,570
function.

613
00:37:49,570 --> 00:37:53,210
It's a new function
in our experience.

614
00:37:53,210 --> 00:37:55,470
Your calculator
probably calculates it,

615
00:37:55,470 --> 00:37:58,710
and your calculator probably
does it by this method.

616
00:38:01,270 --> 00:38:07,480
OK, so that's my sermon
on examples of things

617
00:38:07,480 --> 00:38:10,260
you can do with power series.

618
00:38:10,260 --> 00:38:13,870
So, we're going to do the
CEG thing in just a minute.

619
00:38:13,870 --> 00:38:17,740
Professor Jerison wanted
me to make an ad for 18.02.

620
00:38:17,740 --> 00:38:20,620
Just in case you were thinking
of not taking it next term,

621
00:38:20,620 --> 00:38:21,980
you really should take it.

622
00:38:21,980 --> 00:38:24,810
It will put a lot of
things in this course

623
00:38:24,810 --> 00:38:26,720
into context, for one thing.

624
00:38:26,720 --> 00:38:29,030
It's about vector
calculus and so on.

625
00:38:29,030 --> 00:38:32,190
So you'll learn about
vectors and things like that.

626
00:38:32,190 --> 00:38:34,701
But it comes back and
explains some things

627
00:38:34,701 --> 00:38:36,700
in this course that might
have been a little bit

628
00:38:36,700 --> 00:38:43,460
strange, like these strange
formulas for the product

629
00:38:43,460 --> 00:38:48,710
rule and the quotient rule and
the sort of random formulas.

630
00:38:48,710 --> 00:38:50,790
Well, one of the things
you learn in 18.02

631
00:38:50,790 --> 00:38:54,560
is that they're all special
cases of the chain rule.

632
00:38:54,560 --> 00:38:57,480
And just to drive
that point home,

633
00:38:57,480 --> 00:39:02,530
he wanted me to show you
this poem of his that

634
00:39:02,530 --> 00:39:06,000
really drives the points
home forcefully, I think.