Mathematical Methods for Physics and Engineering - Matematica.NET

4.6 TAYLOR SERIESf(x)QRf(a)Pθhhf ′ (a)aa + hxFigure 4.1 The first-order Taylor series approximation to a function f(x).The slope of the function at P ,i.e.tanθ, equalsf ′ (a). Thus the value of thefunction at Q, f(a + h), is approximated by the ordinate of R, f(a)+hf ′ (a).(n − 1)th-order approximation § to bef(a + h) ≈ f(a)+hf ′ (a)+ h22! f′′ (a)+···+ hn−1(n − 1)! f(n−1) (a). (4.16)As might have been anticipated, the error associated with approximating f(a+h)by this (n − 1)th-order power series is of the order of the next term in the series.This error or remainder can be shown to be given byR n (h) = hnn! f(n) (ξ),for some ξ that lies in the range [a, a + h]. Taylor’s theorem then states that wemay write the equalityf(a + h) =f(a)+hf ′ (a)+ h22! f′′ (a)+···+ h(n−1)(n − 1)! f(n−1) (a)+R n (h).(4.17)The theorem may also be written in a form suitable for finding f(x) giventhe value of the function and its relevant derivatives at x = a, by substituting§ The order of the approximation is simply the highest power of h in the series. Note, though, thatthe (n − 1)th-order approximation contains n terms.137