Does infinite sum of almost surely 0 random variables converge to 0?

Question

Suppose we have an infinite sequence $X_i^{(n)}$ of random variables such that for each $i$, $X_i^{n} \overset{a.s.}{\to} 0$ as $n \to \infty$. With this in hand, which additional conditions need to hold (if any) to ensure that $$\sum_{i=1}^{\infty}X_i^{n} \overset{a.s.}{\to} 0$$ as $n \to \infty$?