I think this is an old problem, almost certainly solved, but I don't think I've ever seen the solution: 

Given the harmonic series but with an independently chosen random sign in front of each term:

±1 ± 1/2 ± 1/3 ± ...,

then

a. what is the probability of convergence?

b. when it converges, what is the probability distribution of the sum?

Similarly, one could ask the same questions a. and b. about the closely related series:

u_1 + u_2 / 2 + u_3 / 3 +...

where each u_n is a complex number independently chosen at random from the unit circle.

Anyone know the answers, or a reference?

Remarks:
1. It's a bit tricky to even define rigorously what the probability in question a. means.

2. Intutitively it may seem obvious that the series, at least in the first case of
±1 ± 1/2 ± 1/3 ± ..., converges with probability 1.  But it's not quite that obvious, since the "arcsine law" says that if S_k is the sum of k independent choices of ±1 at random, then as k -> oo, the most probable fraction(s) of the time that S_k is positive are 0 and 1 -- rather than the 1/2 that most people would guess. (Ditto, of course, for the most probable fraction(s) of the time S_k is negative.) So it's not clear that the random harmonic series isn't closer to the actual harmonic series than to the series with alternating signs.

3. These can be considered the 1- and 2-dimensional cases of a question that can be easily stated n dimensions:  Let {v_k} be vectors chosen independently at random from the unit sphere in R^n, and consider the series

v_1 + v_2 / 2 + v_3 / 3 + ....

--Dan