What is $E[ \sum_{n = 1}^\infty X_n]$ when $E[X_n] = 0$, $E[X_n^2] = \frac{1}{n^2}$ and $\sum_{n = 1}^\infty X_n < \infty$ a.s?

Question

Suppose $(X_n)$ is a sequence of independent real valued random variables, such that $E[X_n] = 0$, $E[X_n^2] = \frac{1}{n^2}$ and $\sum_{n = 1}^\infty X_n < \infty$ a.s. I need to compute $E[ \sum_{n = 1}^\infty X_n]$, but I do not know how to start. Help is much appreciated :)

Answer 1

The series $\sum X_n$ converges in $L^{2}(\Omega, \mathcal F, P)$ because it is Cauchy: $ E(\sum\limits_{k=n}^{m} X_k)^{2}=\sum\limits_{k=n}^{m} \frac 1 {k^{2}} \to 0$ as $m>n \to \infty$. This implies convergence in $L^{1}$ also. Hence the mean of $\sum X_k$ is the limit of the means of the partial sums which is $0$.