Fun With Fourier Series

TL;DR

This paper explores experimental manipulations of Fourier series using computers, revealing series with invariant sums under specific modifications and providing a Mathematica package for Fourier series calculations.

Contribution

It introduces novel Fourier series constructions with interesting invariance properties and offers a Mathematica package for computing and visualizing Fourier series.

Findings

Series whose sums are unchanged when multiplied by sin(n)/n

Examples including a series for π/4 and sums involving sin(n)/n

Mathematica package simplifies Fourier series calculations

Abstract

By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the $n^{th}$ term is multiplied by $\sin(n)/n$. One example is this classic series for $\pi/4$: \[ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = 1 \cdot \frac{\sin(1)}{1} - \frac{1}{3} \cdot \frac{\sin(3)}{3} + \frac{1}{5} \cdot \frac{\sin(5)}{5} - \frac{1}{7} \cdot \frac{\sin(7)}{7} + \dots . \] Another example is \[ \sum_{n=1}^{\infty} \frac{\sin(n)}{n} = \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\right)^2 = \frac{\pi-1}{2}. \] This paper also discusses an included Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.