January 2014
Intermediate to advanced
144 pages
4h 2m
English
Content preview from Mathematical Formulas for Industrial and Mechanical Engineering
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5.32 Taylor and Maclaurin Series
If
f ðxÞ5
X
N
n50
c
n
ðx2aÞ
n
has a power series representation, then
c
n
5
f
n
ðxÞ
n!
and
f ðxÞ5
X
N
n50
f
n
ðxÞ
n!
ðx2aÞ
n
5 f ðaÞ1 f
0
ðaÞðx 2 aÞ1
f
00
ðaÞ
2 !
ðx2aÞ
2
1
f wða Þ
3!
ðx2aÞ
3
1 ? 1
f
n
ðaÞ
n!
ðx2aÞ
n
1 ?
which is called a Taylor series at x 5 a.
If a 5 0, then
f ðxÞ5
X
N
n50
f
n
ðxÞ
n!
x
n
5 f ð0Þ1 f
0
ð0Þx 1
f
00
ð0Þ
2!
x
2
1
f wð0 Þ
3!
x
3
1 ? 1
f
n
ð0Þ
n!
x
n
1 ...
is the Maclaurin series of f centered at x 5 0.
Expansions for some function:
1
x
5 1 2 ðx 2 1Þ1 ðx21Þ
2
2 ðx21Þ
3
1 ðx21Þ
4
2 ? 1 ð21Þ
n
ðx21Þ
n
1
1 2 x
5
X
N
n50
x
n
5 1 1 x 1 x
2
1 x
3
1 x
4
1 ?
e
x
5
X
N
n50
x
n
n!
5 1 1
x
1!
1
x
2
2!
1
x
3
3!
1
x
4
4!
1 ?
sin x 5
X
N
n50
ð21Þ
n
x
2n11
ð2n 1 1Þ!
5 x 2
x
3
3!
1
x
5
5!
2
x
7
7!
1 ?
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I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.