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

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.

arXiv comment6 pages

Cited by in corpus (4)

• Revisiting The Riemann Zeta Function at Positive Even Integers
• Parseval's Identity and Values of Zeta Function at Even Integers
• Fourier Analysis and the closed form for the Zeta Function at even positive integers
• The computation of $\zeta (2k)$, $\beta (2k+1)$ and beyond by using telescoping series