Approximations for Apery's constant $\zeta(3)$ and rational series representations involving $\zeta(2n)$

TL;DR

This paper introduces new series representations for Apery's constant and related functions involving zeta values and Euler numbers, recovering known series for pi and providing novel insights into these constants.

Contribution

It derives new series representations for (3) and (2n) involving Euler numbers, expanding the analytical tools for studying these constants.

Findings

New series representations for (3) using Clausen function

Representation of (2n) involving Euler numbers

Recovery of known series for  and (2n) in special cases

Abstract

In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $\zeta(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new representations of Apery's constant $\zeta(3)$. In particular cases, we recover some well-known series representations of $\pi$.