I am stuck working an exercise. The setting is given as follows:
Let $(X_n)_{n\geq 1}$ be a sequence of independent random variables defined on some probability space where \begin{equation} P\left(X_n = \frac{1}{n}\right) = P\left(X_n = -\frac{1}{n}\right) = \frac{1}{2} \end{equation} for every $n$. Let $Y = \sum_{n\geq 1} X_n$.
The first question I got was to show that $|Y|<\infty$ almost surely. Which I managed to do after some time.
The second question is to compute the expectation of $Y$ (using the previous result). I'm not entirely sure how I should do this. From the previous result I found that $E(Y)<\pi/\sqrt{6}$ and from a symmetry argument it would seem that the expectation should be equal to $0$. Yet I have no idea how I could show this. Is it allowed to interchange limits and expectation in this case?
There is no need for interchange of sum and expectation .
Each $X_n$ has a symmetric distribution. Together with independence this implies that $Y$ also has a symmetric distribution. If you have already proved that $Y$ has finite expectation it follows that $EY=E(-Y)$ so $EY=0$.
Note: The fact that $Y$ has symmetric distribution is consequence of the fact that the sequences $(X_n)$ and $(-X_n)$ induce the same distribution on $\mathbb R^{\infty}$.