SERIES AND LIMITS
4.6 Taylor seriesTaylor’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 theoremSuppose 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+haf′(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+haf′(x)dx. (4.15)Afirst approximationforf(a+h) may be obtained by substitutingf′(a)forf′(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+ha[f′(a)+(x−a)f′′(a)]dx≈f(a)+hf′(a)+h^2
2f′′(a).We may repeat this procedure as often as we like (so long as the derivativesoff(x) exist) to obtain higher-order approximations tof(a+h); we find the