Is there any sufficient condition in terms of moments under which $$ \sum_{n=1}^{\infty} X_n$$ diverges a.s.?Here $X_n$ are not independent

I am given that $\sum_n E[X_n]$ diverges. Actually, I am interested in the following special case:

Let $\{F_n\}$ be a filtration, and $\{A_n\}$ be a sequence of events such that $A_n \in F_n$. Suppose $P(A_n) > 0 \forall n$. Then what is the weakest additional condition that will ensure the following

$$ \sum_{n=1}^{\infty} P(A_n | F_{n-1}) = \infty \mbox{ a.s.}$$

One condition is $P(A_n i.o) =1$, but I want something verifiable.