Compute infinite sum of a arithmetico-geometric series $\sum_{i=0}^{\infty} \frac{i}{2^i}$ [duplicate]

I am trying to compute the sum

$\sum_{i=0}^{\infty} \frac{i}{2^i}$

which I know should be equal to $2$, but I cannot prove it.

If I am not mistaken, it should be a arithmetico-geometric series (Wikipedia), hence the title.

Any help greatly appreciated!

marked as duplicate by Martin Sleziak, egreg, user127.0.0.1, Davide Giraudo, Hans LundmarkApr 18 '14 at 10:49

Consider the series $$\sum_{i=0}^{\infty} {i}{x^i}=x \sum_{i=0}^{\infty} {i}{x^{i-1}}=x \frac {d}{dx}\sum_{i=0}^{\infty} {x^{i}}$$ Now you have a geometric series. Compute its sum, take its derivative, multiply by $x$ and replace $x$ by $\frac {1}{2}$.