11.9 Representations of Functions as Power Series
To represent a function as a sum of a power series we manipulate a geometric series or we
differentiate or integrate it.
Remember from our study of geometric series that
0


1
n
n
a
ar
r
r
∞
=
=
<
−
1
∑
.
Similarly, this tells us from the perspective of a power series that
0
1
,
1
1
1
n
n
x
r
x a
when
x
x
∞
=
=
=
=
−
<
−
∑
1
<
.
So, the function
1
( )
1
f x
x
=
−
can be represented as a power series for part of its domain.
Look at the graphs below.
The graph of the series is on the left and the function graph is on
the right. Notice that the graphs are the same.
100
0
1
n
n
x
for
x
=
−
<
<
∑
1
1
( )
1
f x
x
=
−
1
0.5
0.5
1
1
2
3
4
5
1
0.5
0.5
1
1
2
3
4
5
In similar ways other functions can be represented by power series.
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 Fall '08
 Estrada
 Calculus, Geometric Series, Power Series, Graph of a function

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