<s>[INST] Using power series, compute: $\sum\limits_{n=1}^{\infty}\frac{n}{2^n}$ 
Possible Duplicate:
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? 

I need to compute the sum of $$\sum_{n=1}^{\infty}\frac{n}{2^n}$$
using power series. 
Any hints of how should I do that?
 [/INST] So you want to compute $$S = \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \cdots$$ 
Now consider $$ f(x)= \frac{x}{2} + \frac{x^2}{2^2} + \frac{x^3}{2^3} + \cdots = \displaystyle\frac{\frac{x}{2}}{1 - \frac{x}{2}}$$ From here evaluate the value of $f'(1)$.
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