TITLE:
The Summation of One Class of Infinite Series
AUTHORS:
Jonathan D. Weiss
KEYWORDS:
Infinite Series, Hyperbolic Functions, Elliptical Nome
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.17,
October
28,
2014
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.