What is the series $\sum_{i=n}^{\infty}\frac{i}{2^i}$ When computing the expected value for a random variable I reached the following series:
$$\sum_{i=n}^{\infty}\frac{i}{2^i}$$
I am confident it is convergent, but have no idea how to compute it.

$$ \sum_{i = n}^{\infty} \frac{i}{2^i}$$
$$\frac{i}{2^i} < 1, \space \forall i \ge 0$$
$$ \lim_{i\to\infty}{\frac{i}{2^i}} \to0$$
And apply ratio test of Convergence 
$$ \lim_{n\to\infty}\frac{U_{n+1}}{U_n} = \frac{n\times{2^{n+1}}}{(n+1)\times{2^n}} = \frac{n}{(n+1)}\times 2 = 2 < M$$
Here $M$ is finite positive no. Hence the given series will converge.
