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Finding the Sum of a Power Series

by student45
Tags: power, series
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student45
#1
Nov27-06, 10:57 PM
P: n/a
I'm trying to find the sum of this:

[tex]
\[
\sum\limits_{n = 0}^\infty {( - 1)^n nx^n }
\]
[/tex]

This is what I have so far:

[tex]
\[
\begin{array}{l}
\frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {x^n } \\
\frac{1}{{(1 - x)^2 }} = \sum\limits_{n = 0}^\infty {nx^{n - 1} } = \sum\limits_{n = 1}^\infty {nx^{n - 1} } \\
\frac{x}{{(1 - x)^2 }} = \sum\limits_{n = 1}^\infty {nx^n } \\
\end{array}
\]
[/tex]

So how do I get the (-1)^n part in there? Any suggestions would be really helpful. Thanks.
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courtrigrad
#2
Nov27-06, 11:14 PM
P: 1,237
[tex] \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^{n}x^{n} [/tex]
student45
#3
Nov27-06, 11:28 PM
P: n/a
Oh, I see.. Where exactly does that come from?

courtrigrad
#4
Nov27-06, 11:29 PM
P: 1,237
Finding the Sum of a Power Series

[tex] \frac{1}{1+x} = \frac{1}{1-(-x)} = \sum_{n=0}^{\infty} (-x)^{n} = (-1)^{n}x^{n} [/tex]
student45
#5
Nov27-06, 11:44 PM
P: n/a
Ah! Of course. Okay. Thanks.


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