Are there certain conditions that $a_n$ must meet in order for this series to converge?

Solution 1:

I don't think you're going to find a characterization that's not just a disguised version of "the series converges if and only if the series converges"; convergence of this series is a ding an sich (sp?), seems to me.

But the series does converge for "most" choices! If $(\epsilon_n)$ is a sequence of plus or minus ones chosen "at random" then the series $\sum \epsilon_n/n$ converges almost surely.

(I was about to give a more precise statement of that, decided that would be silly; anyone with the background to understand the precise statement should be able to formulate it for himself. Regarding proving it, look in some probability book in the neighborhood of the Law of Large Numbers...)