Proving Identity for Generalized Sum S(x)

[kreil]

Homework Statement

In order to solve the problem I am working on, I have to prove the following generalized problem,

$$S(x)=\sum_{n=0}^{\infty} n x^n =\frac{x}{(x-1)^2}$$ for |x|< 1

I evaluated this sum using Wolfram Alpha. Clearly it looks related to the geometric series solution, but I am unsure how to prove this identity. Any ideas to get me started?

 

[Office_Shredder]

Differentiation of the geometric series

 

[kreil]

Of course, I see it now. Thanks!