### 1 Introduction

You have met in earlier Mathematics courses the concept of representing a function by an infinite series of simpler functions such as polynomials. For example, the Maclaurin series representing ${e}^{x}$ has the form

$\phantom{\rule{2em}{0ex}}{e}^{x}=1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\dots $

or, in the more concise sigma notation,

$\phantom{\rule{2em}{0ex}}{e}^{x}={\sum}_{n=0}^{\infty}\frac{{x}^{n}}{n!}$

(remembering that 0! is defined as 1).

The basic idea is that for those values of $x$ for which the series converges we may approximate the function by using only the first few terms of the infinite series.

Fourier series are also usually infinite series but involve sine and cosine functions (or their complex exponential equivalents) rather than polynomials. They are widely used for approximating
**
periodic functions
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. Such approximations are of considerable use in science and engineering. For example, elementary a.c. theory provides techniques for analyzing electrical circuits when the currents and voltages present are assumed to be sinusoidal. Fourier series enable us to extend such techniques to the situation where the functions (or signals) involved are periodic but not actually sinusoidal. You may also see in
**
HELM booklet
**
25 that Fourier series sometimes have to be used when solving partial differential equations.