Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.6 TAYLOR SERIES

f(a)

f(x)

a a+h

x

P

Q

R

θ

hf′(a)

h

Figure 4.1 The first-order Taylor series approximation to a functionf(x).

The slope of the function atP,i.e.tanθ,equalsf′(a). Thus the value of the

function atQ,f(a+h), is approximated by the ordinate ofR,f(a)+hf′(a).

(n−1)th-order approximation§to be

f(a+h)≈f(a)+hf′(a)+

h^2

2!

f′′(a)+···+

hn−^1

(n−1)!

f(n−1)(a). (4.16)

As might have been anticipated, the error associated with approximatingf(a+h)

by this (n−1)th-order power series is of the order of the next term in the series.

This error orremaindercan be shown to be given by

Rn(h)=

hn

n!

f(n)(ξ),

for someξthat lies in the range [a, a+h]. Taylor’s theorem then states that we

may write theequality

f(a+h)=f(a)+hf′(a)+

h^2

2!

f′′(a)+···+

h(n−1)

(n−1)!

f(n−1)(a)+Rn(h).

(4.17)

The theorem may also be written in a form suitable for findingf(x) given

the value of the function and its relevant derivatives atx=a, by substituting

§The order of the approximation is simply the highest power ofhin the series. Note, though, that

the (n−1)th-order approximation containsnterms.