I was going through the following 2-page handout:
https://stanford.edu/class/msande321/Handouts/Appendix%20B%20-%20Limits%20and%20Expectations.pdf
In Page 2, it is mentioned that if $X_n \geq 0$, by Monotone Convergence Theorem (MCT) on Page 1, we can conclude that
$\mathbb{E} \Big[ \sum_{n=1}^{\infty} X_n \Big] = \sum_{k=1}^{\infty} \mathbb{E}[X_{n}]$.
My question is, is there an implicit assumption on the existence of
$\lim_{n \rightarrow \infty} X_{n} = X_{\infty}$?
Because MCT on page 1 proves this.
The answer is no.
When you are adding non-negative terms then you can only increase the result: thus you can swap $\lim$ with $\lim\sup$ and the latter always exists!