Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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SERIES AND LIMITS


4.6 Taylor series

Taylor’s theorem provides a way of expressing a function as a power series inx,


known as aTaylor series, but it can be applied only to those functions that are


continuous and differentiable within thex-range of interest.


4.6.1 Taylor’s theorem

Suppose that we have a functionf(x) that we wish to express as a power series


inx−aabout the pointx=a. We shall assume that, in a givenx-range,f(x)


is a continuous, single-valued function ofxhaving continuous derivatives with


respect tox, denoted byf′(x),f′′(x) and so on, up to and includingf(n−1)(x). We


shall also assume thatf(n)(x) exists in this range.


From the equation following (2.31) we may write
∫a+h

a

f′(x)dx=f(a+h)−f(a),

wherea,a+hare neighbouring values ofx. Rearranging this equation, we may


express the value of the function atx=a+hin terms of its value ataby


f(a+h)=f(a)+

∫a+h

a

f′(x)dx. (4.15)

Afirst approximationforf(a+h) may be obtained by substitutingf′(a)for

f′(x) in (4.15), to obtain


f(a+h)≈f(a)+hf′(a).

This approximation is shown graphically in figure 4.1. We may write this first


approximation in terms ofxandaas


f(x)≈f(a)+(x−a)f′(a),

and, in a similar way,


f′(x)≈f′(a)+(x−a)f′′(a),

f′′(x)≈f′′(a)+(x−a)f′′′(a),

and so on. Substituting forf′(x) in (4.15), we obtain thesecond approximation:


f(a+h)≈f(a)+

∫a+h

a

[f′(a)+(x−a)f′′(a)]dx

≈f(a)+hf′(a)+

h^2
2

f′′(a).

We may repeat this procedure as often as we like (so long as the derivatives

off(x) exist) to obtain higher-order approximations tof(a+h); we find the

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