TITLE: What is the probability that this harmonic series with randomly chosen signs will converge?
QUESTION [9 upvotes]: Suppose we fix $p$ between $0$ and $1$ (without loss of generalization, we can assume $p \leq 1/2$). Then suppose we form the series $\sum_n a_n / n$ where the $a_n$ are independent random variables and each $a_n$ equals $1$ with probability $p$ and equals $-1$ with probability $1-p$. (Hence why we can assume $p \leq 1/2$.) What is the probability that this series converges, as a function of $p$? Clearly if the series diverges with probability 1 for $p = 1/2$ then it always diverges with probability $1$ for any $p$, so perhaps the case $p=1/2$ is the most interesting. (However, if other values of $p$ give non-zero probability of convergence that would also obviously be interesting.)

REPLY [9 votes]: Random harmonic series - A little self promotion.