(12.9) Power Series for Functions
Recall the formula for geometric series:
∞
±
n
=0
ar
n
=
a
1

r
provided

r

<
1. (We really
mean,
∞
±
n
=0
ar
n
converges to
a
1

r
provided

r

<
1.) So if we let
a
= 1 and
r
=
x
, we have
∞
±
n
=0
x
n
=1+
x
+
x
2
+
x
3
+
···
=
1
1

x
provided

x

<
1, or

1
<x<
1. So we think
of
f
(
x
)=
∞
±
n
=0
x
n
as a function of
x
. We therefore know is that it is really the function
f
(
x
)=
1
1

x
, but only for

1
<x<
1 (for
x
outside that interval, the function
f
(
x
)i
s
unde±ned).
We can use this series to ±nd power series for other functions.
Example
Find a power series for the function
f
(
x
)=
2
x
x

3
.
We start with
1
1

x
=1+
x
+
x
2
+
x
3
+
···
. Trick: substitute
x
3
for
x
. We have
1
1

x
3
=1+
x
3
+
²
x
3
³
2
+
²
x
3
³
3
+
···
or
3
3

x
=1+
x
3
+
x
2
9
+
x
3
27
+
···
. We adjust this a little (by dividing by 3 then multiplying
by

1):
1
x

3
=

1
3

x
9

x
2
27

x
3
81
···
.
This is
1
x

3
=
∞
±
n
=0

x
n
3
n
+1
. Now multiply by 2
x
to get the desired result:
2
x
x

3
=

2
x
3

2
x
2
9

2
x
3
27

2
x
4
81
···
=
∞
±
n
=0

2
x
n
+1
3
n
+1
=
∞
±
n
=0

2
x
n
3
n
.
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 Fall '08
 VALDIMARSSON
 Geometric Series, Power Series, Mathematical Series, John Machin

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