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

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!

Answer

Hint

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 $\frac {1}{2}$.

I am sure that you can take from here.