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arXiv · 1209.5030

A simple computation of $\zeta(2k)$ by using Bernoulli polynomials and a telescoping series

Ó. Ciaurri·L. M. Navas·F. J. Ruiz·J. L. Varona
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Abstract

We present a new proof of Euler's formulas for $\zeta(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $\zeta(2k)$ by summing a telescoping series. Only basic techniques from Calculus are needed to carry out the computation. The method also applies to $\zeta(2k+1)$ and the harmonic numbers, yielding integral formulas for these.

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Ó. Ciaurri, L. M. Navas, F. J. Ruiz, J. L. Varona. 2012-09-23. A simple computation of $\zeta(2k)$ by using Bernoulli polynomials and a telescoping series. https://doi.org/10.4169/amer.math.monthly.122.5.444

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