Series representations of positive integral powers of pi

TL;DR

This paper derives new series representations for positive integer powers of pi using Fourier series techniques and verifies these formulas with computational tools.

Contribution

It introduces novel series formulas for powers of pi based on a pointwise Fejér's theorem approach, validated through software applications.

Findings

New series formulas for powers of pi derived

Formulas verified using R software

Provides a mathematical foundation for pi power series

Abstract

Using a pointwise version of Fej\'{e}r's theorem about Fourier series, we obtain two formulae related to the series representations of positive integral powers of $\pi$. We also check the correctness of our formulae by the applications of the R software.