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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

abstract and applied analysis

Rapidly Converging Series for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>ζ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>from Fourier Series

Rapidly Converging Series forζ(2n+1)from Fourier Series