Another elementary proof of ${\sum }_{n\ge 1}1/n^2={\pi }^2/6$ and a recurrence formula for $\zeta (2k)$

Abstract

In this shortnote, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values $\zeta (2k+1)$, $\zeta (s)$ being the Riemann zeta function and $k$ a positive integer, is modified in a manner to furnish the even zeta values $\zeta (2k)$. As a result, I find an elementary proof of ${\sum }_{n=1}^{\infty }1/n^2={\pi }^2/6$, as well as a recurrence formula for $\zeta (2k)$ from which it follows that the ratio $\zeta (2k)/{\pi }^{2k}$ is a rational number, without making use of Euler's formula and Bernoulli numbers.

arXiv comment5 pages, no figures. Small corrections, some examples added at the end of the paper. Submitted to J. Comb. Number Theory (11/16/2012)

Cited by in corpus (2)

• Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2022) and another new proof
• A One-Sentence and Truly Elementary Proof of the Basel Problem