Rapidly Converging Series for ζ(2n + 1) from Fourier Series

Junesang Choi

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

Abstract: Ever since Euler first evaluated ζ(2) and ζ(2m), numerous interesting solutions of the problem of evaluating the ζ(2m) (m ∈ ℕ) have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of ζ(2m) (m ∈ ℕ) is known for ζ(2m + 1) (m ∈ ℕ) or even for any special case such as ζ(3). Instead, various rapidly converging series for ζ(2m + 1) have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for ζ(2m + 1). In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of β(2m + 1) and ζ(2m) (m ∈ ℕ), which have been treated in earlier works.

Date: 2014
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https://doi.org/10.1155/2014/457620

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:457620

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