Originally Posted by

**CaptainBlack** Is that what you really want to ask? What you are asking appears to be:

does (or rather what is the probability that):

$\displaystyle \displaystyle \sum_{n=1}^{\infty}\dfrac{z_n}{n}$

converges where the $\displaystyle $$z_n$'s are sampled uniformly on the top half of the unit circle in the complex plane.

If that is what you mean, then the answer is probably (I will need to work out how to prove it but this is what I would place my money on): the real part converges with probability 1 and the imaginary part converges with probability 0.

The heuristic argument behind this is the the real part behaves on average like the alternating harmonic series, while the imaginary part behaves on average like the harmonic series.

CB