# Finding the Sum of a Power Series

by student45
Tags: power, series
 P: n/a I'm trying to find the sum of this: $$$\sum\limits_{n = 0}^\infty {( - 1)^n nx^n }$$$ This is what I have so far: $$$\begin{array}{l} \frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {x^n } \\ \frac{1}{{(1 - x)^2 }} = \sum\limits_{n = 0}^\infty {nx^{n - 1} } = \sum\limits_{n = 1}^\infty {nx^{n - 1} } \\ \frac{x}{{(1 - x)^2 }} = \sum\limits_{n = 1}^\infty {nx^n } \\ \end{array}$$$ So how do I get the (-1)^n part in there? Any suggestions would be really helpful. Thanks.
 P: 1,236 $$\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^{n}x^{n}$$
 P: 1,236 Finding the Sum of a Power Series $$\frac{1}{1+x} = \frac{1}{1-(-x)} = \sum_{n=0}^{\infty} (-x)^{n} = (-1)^{n}x^{n}$$