The Taylor series is a generalisation of the Maclaurin series being a power series developed in powers of rather than in powers of . Thus
If the function can be differentiated as often as required at then:
The reader will see that the Maclaurin expansion is the Taylor expansion obtained if is chosen to be zero.
Obtain the Taylor series expansion of about . (That is, find a power series in powers of .)
First, obtain the first three derivatives and the derivative of :
Now evaluate these derivatives at :
Hence, write down the Taylor expansion of about :
- Show that the series obtained in the last Task is convergent if
- Sketch the linear, quadratic and cubic approximations to obtained from the series in the last task and compare to .
In the following diagrams some of the terms from the Taylor series are plotted to compare with