Random Harmonic Series

Here the positive and negative terms partly cancel, allowing the series to converge. To a probabilist, this alternating series suggests choosing plus and minus signs at random, by tossing a fair coin. Formally, let (ej) ∞ j=1 be independent random variables with common distribution P (ej = 1) = P (ej = −1) = 1/2. Then, Kolmogorov’s three series theorem [1, Theorem 22.8] or the martingale convergence theorem [1, Theorem 35.4] shows that the sequence ∑n j=1 ej/j converges almost surely. In this note, we investigate the distribution of the sum X := ∑∞ j=1 ej/j.