Expectations of sums and products

In Theorem 8.1 (expectation of a sum) and Theorem 9.2 (expectation of a product)  we proved results for a finite number of r.vs, $X_1,\dotsc,X_n$. In general, the corresponding results for a countably infinite number of r.vs are not true. You might like to think about why the proofs do not work if $n$ is replaced by $\infty$. Can you find $X_1,X_2,\dotsc$ such that the following is not true?

$E\sum_{n=1}^\infty X_n = \sum_{n=1}^\infty EX_n$

Here is a hint. There do exist special cases for which the above is true, such as when $\sum_{n=1}^\infty E|X_n|$ converges.