Perhaps the most important application of the higher derivatives introduced in Section 3.4 is that, provided they exist, they enable functions in the neighbourhood of a given point to be approximated by polynomials in such a way that the accuracy of the approximation increases as the order of the polynomials increases. Such so-called *Taylor expansions* are useful because the resulting polynomials are often much easier to study and evaluate than the original functions themselves, and they have many applications, as we will see. Firstly, however, we must introduce some basic ideas about series and expansions in general.

## 5.1 Series

A series is the sum *u*_{0} + *u*_{1} + *u*_{2} + ⋅⋅⋅ of an ordered sequence {*u*_{n}} of elements *u*_{n}(*n* = 1, 2, … ). The elements may be numbers, for example , obtained from

or functions, such as *u*_{n} = 1, *x*, *x*^{2}, … , obtained from

and the sequence may contain a finite number of (*N* + 1) terms,

or an infinite number of terms

where ...