Expected value of infinite sum
Here it says:
It can be shown that linearity of expectation also holds for countably infinite summations in certain cases. For example, it holds that:
$E\left[\sum\limits_{i=1}^{\infty}X_i\right]=\sum\limits_{i=1}^{\infty}E[X_i]$
if $\sum\limits_{i=1}^{\infty}E[|X_i|]$ converges.
Also This Book says the same thing.