The properties of the series are studied for both bounded and unbounded random variables. The technique is used to find efficient series for computation of the distributions of sums of uniform random variables and sums of Rayleigh random variables. A useful closed-form expression for the characteristic function of a Rayleigh random variable is presented, and an efficient method for computing a confluent hypergeometric function is given. An infinite series for the probability density function of a sum of independent random variables is also derived. The inversion of characteristic functions, a trapezoidal rule for numerical integration, and the sampling theorem in the frequency domain are related to, and interpreted in terms of, the results.< >