2 Taylor series

In HELM booklet  16 we encountered Maclaurin series and their generalisation, Taylor series. Taylor series are a useful way of approximating functions by polynomials. The Taylor series expansion of a function $f\left(x\right)$ about $x=a$ may be stated

$f\left(x\right)=f\left(a\right)+\left(x-a\right)f^{\prime }\left(a\right)+\frac{1}{2}{\left(x-a\right)}^2{f}^{″}\left(a\right)+\frac{1}{3!}{\left(x-a\right)}^3{f}^{\‴}\left(a\right)+\dots .$

(The special case called Maclaurin series arises when $a=0$ .)

The general idea when using this formula in practice is to consider only points $x$ which are near to $a$ . Given this it follows that $\left(x-a\right)$ will be small, ${\left(x-a\right)}^2$ will be even smaller, ${\left(x-a\right)}^3$ will be smaller still, and so on. This gives us confidence to simply neglect the terms beyond a certain power, or, to put it another way, to truncate the series.

Example 2

Find the Taylor polynomial of degree 2 about the point $x=1$ , for the function $f\left(x\right)=ln\left(x\right)$ .

Solution

In this case $a=1$ and we need to evaluate the following terms

$f\left(a\right)=ln\left(a\right)=ln\left(1\right)=0,f^{\prime }\left(a\right)=1∕a=1,f^{″}\left(a\right)=-1∕a^2=-1.$

Hence

$ln\left(x\right)\approx 0+\left(x-1\right)-\frac{1}{2}{\left(x-1\right)}^2=-\frac{3}{2}+2x-\frac{x^2}{2}$

which will be reasonably accurate for $x$ close to 1, as you can readily check on a calculator or computer. For example, for all $x$ between 0.9 and 1.1, the polynomial and logarithm agree to at least 3 decimal places.

One drawback with this approach is that we need to find (possibly many) derivatives of $f$ . Also, there can be some doubt over what is the best choice of $a$ . The statement of Taylor series is an extremely useful piece of theory, but it can sometimes have limited appeal as a means of approximating functions by polynomials.

Next we will consider two alternative approaches.