Series Converging Almost Surely But Diverging in Mean

Question

I am looking for an example of independent, non-negative random variables $X_1, X_2, \dots$ such that

$$ \sum_{n=1}^{\infty} X_n \, \lt \, \infty $$

almost surely but

$$ \sum_{n=1}^{\infty} \mathbb{E}(X_n) \, = \, \infty $$

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I can find examples of sequences which converge almost surely but diverge in mean, but can’t seem to be able to cook up an example with a series.

Answer 1

For example, take $X_n$ s.t. $P(X_n = 2^n) = \frac{1}{2^n}$, $P(X_n = 0) = 1 - \frac{1}{2^n}$. Then a.s. all but finitely many $X_n$ are zeroes, so $\sum X_n$ converges a.s., but $\mathbb E (X_n) = 1$, so series of mean diverges.