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

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 $\pi$.