TITLE: Expected value of infinite sum
QUESTION [16 upvotes]: $x_1, x_2, 
\dots, x_n, \dots$ - independent random variables.
Is it true that
$$
\sum_{i = 1}^{\infty}Ex_i = E(\sum_{i = 1}^{\infty} x_i)
$$
?

REPLY [10 votes]: 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.