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. 

If the $X_i$ are nonnegative, it is always true, by either monotone convergence or Tonelli's theorem (thinking of the sum as an integral over $\mathbb{N}$ with respect to counting measure).  If we have
$\sum_k E|X_k| < \infty$ or equivalently $E \sum_k |X_k| < \infty$, then it is also true, by either dominated convergence or Fubini's theorem. 
Otherwise it can fail: let $U \sim U(0,1)$, $Y_k = k 1_{\{U \le 1/k\}}$, so that $Y_k \to 0$ a.s. but $E Y_k = 1$ for each $k$.  Let $Y_0 = 0$ and $X_k = Y_k - Y_{k-1}$ for $k \ge 1$.  Then you can easily check that 
$$E \sum_{k=1}^\infty X_k = 0 \ne 1 = \sum_{k=1}^\infty E X_k.$$ 
