Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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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.
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