Proving zeta(2) through an evolution of the Mengoli's series to the set of rational numbers

TL;DR

This paper develops a new method to evaluate the Riemann zeta function at 2 using an evolved form of Mengoli's series, providing a novel proof of its value through rational summations and limits.

Contribution

It introduces a new function that computes sums of a specific rational form and demonstrates its application to prove zeta(2) using limits, extending Mengoli's series.

Findings

Derived a function for summing rational series of a specific form.

Proved the value of zeta(2) through limits of the new series.

Extended Mengoli's series to rational numbers for zeta function evaluation.

Abstract

The present paper is an evolution of the Mengoli's series to the set of rational numbers, which eventually will allow developing the summation, by limits, obtaining the value of zeta(2); problem which Mengoli himself was the first to postulate. More specifically in this paper is postulated and demonstrated the function which resolves every summation since n=1 to infinite of the type 1/((n+a/w)(n+b/w)...(n+z/w)) for all a,b,...,z,w belonging to the set of integers, where a>b>...>z are different; a/w,b/w,...,z/w not equal to -1,-2,-3,... Finally limits to the past summation is applied to demonstrate zeta(2).