The expectation of an infinite series of random variables Let $(X_n)_{n = 1}^\infty$ be a sequence of random variables taking values in the extended real line, and suppose that $\sum_{n = 1}^\infty |E(X_n)| < \infty$.
(a) Is it possible that $\sum_{n = 1}^\infty X_n$ is not defined?
(b) If the series $\sum_{n = 1}^\infty X_n$ is defined, is it possible that its expectation is not defined?
(c) If its expectation is defined, is it possible that $E(\sum_{n = 1}^\infty X_n) \neq \sum_{n = 1}^\infty E(X_n)$?
 A: (a) Yes, it is possible. Example: Let $Y \sim -1 + 2 \cdot \operatorname{Ber}\left(\frac{1}{2}\right)$. For every $m \in \mathbb{N}_1$ define $X_{2m - 1} := Y$, $X_{2m} := -Y$.
(b) Yes, it is possible. Example: Let $Y$ be a random variable for which the following holds. For every $n \in \mathbb{N}_1$,
$$
P(Y = n) = P(Y = -n) = \frac{3}{\pi^2 n^2}
$$
$Y$ has no expectation (cf. counterexample 3.2 of Romano & Siegel, pp. 40-41).
Now, for every $m \in \mathbb{N}_1$, define
$$
X_m := Y \mathbb{1}_{\left\{Y \in \{\pm n\}\right\}}
$$
(c) Yes, it is possible. Consider the canonical probability space (i.e. $\Omega = (0,1)$, $\mathcal{F} =$ the collection of all Borel subsets of $\Omega$, $P = $ the Lebesgue measure). For every $n \in \mathbb{N}_1$, define $Y_n := n\mathbb{1}_{(0,1/n)}$. For every $m \in \mathbb{N}_1$ define $X_m := Y_{m + 1} - Y_m$. (cf. counterexample 5.32 of Romano & Siegel, p. 106).

Works Cited
Romano, Joseph P. and Siegel, Andrew F., "Counterexamples in Probability and Statistics", Wadsworth & Brooks/Cole Advanced Books & Software, 1985
