Monotone convergence implies $\mathbb{E}\sum X_n = \sum \mathbb{E}X_n$?

Question

My professor stated the following implication made from the Monotone Convergence Theorem:

\begin{align}\mathbb{E}[\sum X_n] = \sum \mathbb{E}[X_n] \end{align}

Up till now I have been assuming the following operation holds by the linearity of the expectation: \begin{align} \mathbb{E}[X_1+X_2 + \cdots + X_n] = \mathbb{E}[X_1] + \cdots + \mathbb{E}[X_n]\end{align}

Aren't these two equalities identical? However, I am confused why I was allowed to assume the second equality, thus far, by the linearity of the expectations when in fact it is given by the Monotone Convergence Theorem...

Basically, is there a difference?

Thank you.

Answer 1

$ \mathbb{E}[X_1+X_2 + \cdots + X_n] = \mathbb{E}[X_1] + \cdots + \mathbb{E}[X_n]$ holds for any finite $n$ where $\mathbb{E}[\sum X_n] = \sum \mathbb{E}[X_n]$ is an infinite summation.

In order to exchange the expectation and sum, you need some form of monotone convergence, dominated convergence, Fubini's, etc.