a question about $E(\sum_{n=1}^{+\infty}X_{k}$)=$\sum_{n=1}^{+\infty}EX_{k}$ $X_{k}$ are random variables and they are not independent,
I wonder if $E(\sum_{n=1}^{+\infty}X_{k}$)=$\sum_{n=1}^{+\infty}EX_{k}$.
if the equation does not hold, what conditions are required in order to make it right. 

You can conclude this if $X_i$'s are non-negative. The proof follows immediately from monotone convergence theorem.
