### Introduction

In this Section we examine how functions may be expressed in terms of power series. This is an extremely useful way of expressing a function since (as we shall see) we can then replace ‘complicated’ functions in terms of ‘simple’ polynomials. The only requirement (of any significance) is that the ‘complicated’ function should be
**
smooth
**
; this means that at a point of interest, it must be possible to differentiate the function as often as we please.

#### Prerequisites

- have knowledge of power series and of the ratio test
- be able to differentiate simple functions
- be familiar with the rules for combining power series

#### Learning Outcomes

- find the Maclaurin and Taylor series expansions of given functions
- find Maclaurin expansions of functions by combining known power series together
- find Maclaurin expansions by using differentiation and integration

#### Contents

1 Maclaurin and Taylor series2 The Maclaurin series

3 Differentiation of Maclaurin series

4 The Taylor series