Let X be a random variable. Will its expectation E[X] expressed as a (possibly infinite) sum converge for arbitrary X?
My intuition would be yes. Since E[X] is a function of a r.v., E[X] is a r.v. itself, so by definition it maps to the reals. The infinity is not a real number, so there will be a limit to which the E[X] will converge.
Of course you can construe a r.v. such that weighted sum will diverge, but is then the case that E[X] = $\infty$ or is E[X] said to be undefined?