Introduction

In this Section we examine, briefly, the convergence characteristics of a Fourier series. We have seen that a Fourier series can be found for functions which are not necessarily continuous (there may be jumps in the curve) $-$ the only requirement that we have made is that the function be periodic. We have seen that the more terms we take in the Fourier series the better is the approximation to the given signal. But an obvious question to ask is what happens at the points of discontinuity? What does the Fourier series converge to at these points? It must converge to something (finite) since a Fourier series is a sum of very smooth continuous functions. In this Section we give the answer to this question.

Prerequisites

• know how to obtain a Fourier series
• be familiar with the limit process as applied to functions

Learning Outcomes

• determine what a Fourier series converges to at each point, including at a point of
  
  discontinuity
• use the convergence property of Fourier
  
  Series to obtain series for the number $\pi$