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!


1 Answer 1



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.


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