New series representations for any positive power of $\pi$ from a relation involving trigonometric functions

TL;DR

This paper introduces a general trigonometric series relation that enables the derivation of series representations for any positive power of pi, expanding on previous specific cases involving pi cubed and pi to the fifth.

Contribution

It presents a new general relation involving trigonometric functions and infinite series, allowing the derivation of series for any positive power of pi, including previous special cases.

Findings

Derived a general relation involving trigonometric functions and series

Provided a method to generate series for any positive power of pi

Extended previous specific series representations to a broad class

Abstract

In previous works, we presented series representations for $\pi^3$ and $\pi^5$, in which the prefactor depends only on the golden ratio appears. In this article, we derive a general relation involving trigonometric functions and an infinite series. Such an identity is likely to provide many series representations for any positive power of $\pi$, among them the above mentioned representations for $\pi^3$ and $\pi^5$.