9.3 Finite-difference derivatives  9. Numerical Methods  9.1 Introduction

9.2 The Taylor series

Many numerical methods are based on the Taylor expansion, so a brief review of the Taylor series is included here. The Taylor series expansion of a function f(x) is

$$f(x)=f(x_0)+\sum_{k=1}^\infty {{(x-x_0)^k}\over{k!}}f^{(k)}(x_0)$$ (9.1)

 

where f^(k) is the kth derivative of the function. Differing approximations to the function are obtained from this series by truncation. If the series is truncated at the nth term, the maximum error in the approximation is

$${{\vert x-x_0\vert^{n+1}}\over{n!}}\left\vert f^{(n+1)}\right\vert _{max}$$ (9.2)

 

where the "max" subscript indicates the maximum value of the derivative in the interval from x to x₀. The error is said to be of order (x-x₀)ⁿ⁺¹.

The Taylor expansion must be used with some caution because the series does not converge for all values and sometimes converges very slowly.