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The Summation of One Class of Infinite Series

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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: 2l∏λ and 2l∏λ, 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 x = ∏/4, ∏/2, ∏, and 2∏. 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.
Cite this paper
Weiss, J. (2014) The Summation of One Class of Infinite Series. Applied Mathematics, 5, 2815-2822. doi: 10.4236/am.2014.517269. 
 

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