The key observation is that even if the sum of expectations of individual $X_n$ diverges, if correlations among the $\{X_{k_1}, X_{k_2}\ldots X_{k_m}\}$ with $k_m = \sup (k_i)$ do not fall off sufficiently fast as $k_m \to \infty$, then it is possible to "save" the convergence, that is, to have a finite  probability of that the sum does not diverge.

With that concern in mind, here is 
a sufficient condition (though I would not say this is anything like a necessary condition) that $\sum_1^\infty X_n$ diverges a.s., expressed in terms of the moments (including of course the correlation expectations):

If for some $C \in \Bbb{R}, C > 0$ 
$$
\lim_{n\to \infty} n E(X_n) = C
$$
and for all partitions of integers $k = \sum_{i=1}^m k_i$ with $k > 1$, and $ k_i \geq 0$ for all $i$, 
 $$\lim_{n\to \infty} n^{k+1} \left[\prod_{i=1}^k E(X_i^{k_i})- E\left((\prod_{i=1}^k X_i^{k_i}\right)\right]=0
$$
then $\sum_{n=1}^\infty X_n$ diverges a.s.