where the are independent, identically distributed random variables taking the values +1 and −1 with equal probability , is a well-known example in probability theory for a series of random variables that converges with probability 1. the fact of this convergence is an easy consequence of either the kolmogorov three-series theorem or of the closely related kolmogorov maximal inequality. byron schmuland of the university of alberta further examined the properties of the random harmonic series, and showed that the convergent series is a random variable with some interesting properties. in particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value ..., differing from by less than 10−42. schmuland's paper explains why this probability is so close to, but not exactly, . the exact value of this probability is given by the infinite cosine product integral divided by .