Formulas for odd zeta values and powers of $\pi$

TL;DR

This paper uncovers a general pattern for formulas expressing odd zeta values and powers of pi, confirming conjectures about their rapid convergence and providing proofs for these formulas.

Contribution

It generalizes Plouffe's conjectured formulas for all nonnegative integers, establishing a unified pattern and offering rigorous proofs.

Findings

Derived explicit formulas for all odd zeta values and powers of pi.

Confirmed rapid convergence of the series formulas.

Provided proofs for the conjectured formulas.

Abstract

Plouffe conjectured rapidly converging series formulas for $\pi^{2n+1}$ and $\zeta (2n+1)$ for small values of $n$. We find the general pattern for all nonnegative integer values of $n$ and offer a proof.