Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITS4.6 Taylor seriesTaylor’s theorem provides a way of expressing a function as a power series in x,known as a Taylor series, but it can be applied only to those functions that arecontinuous and differentiable within the x-range of interest.4.6.1 Taylor’s theoremSuppose that we have a function f(x) that we wish to express as a power seriesin x − a about the point x = a. We shall assume that, in a given x-range, f(x)is a continuous, single-valued function of x having continuous derivatives withrespect to x, denoted by f ′ (x), f ′′ (x) and so on, up to and including f (n−1) (x). Weshall also assume that f (n) (x) exists in this range.From the equation following (2.31) we may write∫ a+haf ′ (x) dx = f(a + h) − f(a),where a, a + h are neighbouring values of x. Rearranging this equation, we mayexpress the value of the function at x = a + h in terms of its value at a byf(a + h) =f(a)+∫ a+haf ′ (x) dx. (4.15)A first approximation for f(a + h) may be obtained by substituting f ′ (a) forf ′ (x) in (4.15), to obtainf(a + h) ≈ f(a)+hf ′ (a).This approximation is shown graphically in figure 4.1. We may write this firstapproximation in terms of x and a asand, in a similar way,f(x) ≈ f(a)+(x − a)f ′ (a),f ′ (x) ≈ f ′ (a)+(x − a)f ′′ (a),f ′′ (x) ≈ f ′′ (a)+(x − a)f ′′′ (a),and so on. Substituting for f ′ (x) in (4.15), we obtain the second approximation:f(a + h) ≈ f(a)+∫ a+ha≈ f(a)+hf ′ (a)+ h22 f′′ (a).[f ′ (a)+(x − a)f ′′ (a)] dxWe may repeat this procedure as often as we like (so long as the derivativesof f(x) exist) to obtain higher-order approximations to f(a + h); we find the136