On the Summation of Infinite Series in Closed Form

Abstract

A direct approach to the problem of summing infinite series in closed form is described. This method is based on the parametric representation of the general term of a series, so as to produce either the geometric or exponential series inside one or more integral signs. Each of these may be summed in closed form, thereby permitting one to exhibit a given series as a combination of definite integrals. When the original series is summable, these integrations can be performed analytically and the sum presented in closed form. Such a procedure enjoys the evident advantage of exploiting the relative abundance of integral tables, while circumventing the need for extensive preparation in the field of infinite series. The relations of this development to other methods are indicated and a number of illustrative examples presented.