How to find the sum of series $\sum_{i=1}^{\infty}\frac{i}{2^i}$? I am learning about series of numbers at the moment. In the book there is an exercise in which I need to find the sum of :
$$\sum_{i=1}^{\infty}\frac{i}{2^i}$$
I know it is equal to $2$. But how do I get to that result?
Are there any general ways of finding the sums of series? In the books I am using, there is a lot about series, their convergence etc. but almost no examples.

Let
$$S := \sum_{k=1}^\infty \frac{k}{2^k}.$$
Since all of its elements are positive and the series is absolutely convergent, we can do the following:
\begin{align*}
S &= \sum_{k=1}^\infty \frac{k}{2^k} = \sum_{k=1}^\infty \frac{k-1}{2^k} + \sum_{k=1}^\infty \frac{1}{2^k} = \frac{1}{2} \sum_{k=1}^\infty \frac{k-1}{2^{k-1}} + 1 = \frac{1}{2} \sum_{k=2}^\infty \frac{k-1}{2^{k-1}} + 1 = \frac{1}{2} \sum_{k=1}^\infty \frac{k}{2^k} + 1 \\
&= \frac{1}{2}S + 1,
\end{align*}
so $S = 2$.
