sequences and series - How to compute this infinite sum?

I'm trying to compute the infinite sum

$\sum_{n=1}^{\infty}n(\frac{1}{2})^n$

which I believe should represent the expected amount of coin flips needed to get a head. Can someone remind me how to do this?

Answer

The key is that the infinite sum $\sum x^n $ converges to $\frac 1{1 - x}$ under certain conditions on $x$, and differentiating the resulting inequality gives that $\sum nx^{n-1}$ is convergent to the derivative of $\frac 1{1 - x}$, under the same conditions. Multiplying this by $x$ gives the sum $\sum nx^{n}$, which is the result you are looking for with $x = \frac 12$, which does fall under the set for which the first equality holds.