The Summation of One Class of Infinite Series

Abstract

This paper presents closed-form expressions for the series, , where the sum is from n = 1 to n = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of x for which these expressions can be obtained are of the form: and , where l is an integer between – ∞ and ∞. The values of λ include 1, , and 3. Examples of closed-form expressions obtained in this manner are first presented for , , , and . Additional examples are then presented for , , , and . This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for x = in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.