Taylor Series for a Function
Up until now we have studied power series as defining a function on the interval of
convergence. One may turn the issue around though and ask the following question:
Given a function, is there a power series that represents the function on its interval of
convergence?
Theorem
: Let
f
be a given function. If there is a power series
(
29
0
n
n
n
c
x
a
∞
=

∑
for which
(
29
(
29
0
n
n
n
f x
c
x
a
∞
=
=

∑
on some interval, then
(
29
(
29
!
n
n
f
a
n c
=
, or equivalently,
(
29
(
29
!
n
n
f
a
c
n
=
for all
0
n
≥
. (Here we define
(
29
(
29
(
29
0
f
a
f a
≡
and0! 1
≡
.)
There are some observations about the above theorem that one needs to keep in mind.
The equality
(
29
(
29
0
n
n
n
f x
c
x
a
∞
=
=

∑
holds only on the IOC for
(
29
0
n
n
n
c
x
a
∞
=

∑
.
The coefficients
n
c
are related to the derivatives of the function
f
. So for the
function f to have a power series representation,
f
has to have infinitely many
derivatives at the center
a
. For example,
(
29
f x
x
=
will not be represented by
such a series about
0
a
=
because it is not differentiable at
0
a
=
. However
(
29
f x
x
=
does have infinitely many derivatives at
0
a
≠
. So it may be
represented by a series
(
29
0
n
n
n
c
x
a
∞
=

∑
for
0
a
≠
.
Since
f
is the sum of the power series
(
29
0
n
n
n
c
x
a
∞
=

∑
, one can write
(
29
(
29
(
29
(
29
0
!
n
n
n
f
a
f x
x
a
n
∞
=
=

∑
.
Definition
: Suppose
f
is a given function. The power series
(
29
(
29
(
29
0
!
n
n
n
f
a
x
a
n
∞
=

∑
is called
the Taylor Series for
f
about the point
a
. When
0
a
=
, the power series
(
29
(
29
(
29
0
0
!
n
n
n
f
f x
x
n
∞
=
=
∑
is sometimes called the McClaurin Series for
f
.
Examples
: Below are several examples how to find the Taylor Series for several
functions.
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View Full Document1)
Recall the Geometric Series, namely
(
29
0
1
1
n
n
f x
x
x
∞
=
=
=

∑
. Then
0
n
n
x
∞
=
∑
is the
McClaurin Series for
(
29
1
1
f x
x
=

. Since the coefficients are
1
n
c
=
for all
0
n
≥
, it
follows that
(
29
(
29
0
!
!
n
n
f
n c
n
=
=
for all
0
n
≥
.
2)
Find the McClaurin series for
(
29
x
f x
e
=
.
Note that
(
29
(
29
(
29
,
,
,
x
x
x
f
x
e
f
x
e
f
x
e
′
′′
′′′
=
=
=
K
and in general
(
29
(
29
n
x
f
x
e
=
for all
0
n
≥
. Since the problem asks for the McClaurin series
0
a
=
. Then
(
29
(
29
0
0
1
n
f
e
=
=
. So the coefficients of the series are
(
29
(
29
0
1
!
!
n
n
f
c
n
n
=
=
. It follows
that
0
!
n
x
n
x
e
n
∞
=
=
∑
on the IOC.
To find the IOC, apply the Ratio Test:
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 Spring '11
 BUEKMAN
 Power Series, Taylor Series, Mathematical Series

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