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.
Share and Cite:
Weiss, J. (2014) The Summation of One Class of Infinite Series.
Applied Mathematics,
5, 2815-2822. doi:
10.4236/am.2014.517269.
Conflicts of Interest
The authors declare no conflicts of interest.
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