ABSTRACT
The representation and approximation of a complex mathematical object by means of a series of simpler components is extremely common and useful. If the components of a series are sufficiently simple, then the representation may be more efficient with which to work, and may offer a deeper understanding of the properties of the object. In this chapter, we shall examine several classes of series representations and consider some of their applications. We first discuss the nature of the computational representation of numbers, and their interpretation as series. We then go on to discuss the symbolic representation of functions using Taylor series and Fourier series. Taylor series provide a useful way of approximating well known functions, but their main utility is in the analysis of error. Specifically, we shall use Taylor series to analyse the error of three quadrature algorithms introduced in the last chapter. Our discussion of Fourier series, among other things, will help us to characterise the elusive notion of the “ frequency” of a function alluded to in Chapter 3.
