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# Revision of Taylor Series

Many of the numerical methods used in this course make use of a Taylor series either for deriving the formula to use or for estimating the error of the numerical solution. The function can be expanded in a power series in power of as
 (1.1)

where the error is given by
 (1.2)

Note that lies between the values of and . Here we have assumed that . In (1.1) we are defining the derivative, evaluated at the point by

Finally, we have assumed that the function is at least times differentiable. In this course we will assume that all our function will possess suitable number of derivatives. We will not be concerned with functions like which does not have a derivative at .

As an example, if , then

and so

Suppose , then

If we set

then an equivalent expression to (1.1) is given by
 (1.3)

where the error term is
 (1.4)

You need to learn this formula and be able to use it confidently. It is useful when you need to approximate the function in the neighbourhood of a particular point.

Example 1. .1Expand about so that . Remember to use radians when caculus is involved.

Therefore, from (1.1), the Taylor series for about is

As an example of the use of a Taylor series, we can use this to estimate the first positive zero of . Obviously the answer is but for other functions this will not be known and the Taylor series can sometimes provide a useful first estimate.

If we only take the first two non-zero terms then we approximate by

Hence, the first positive zero is approximately

If we now take the first three non-zero terms then we have

Now we can solve the quadratic in

Why did we take the negative square root? Obviously we are looking for the first positive zero and the positive square root will give an estimate for the second root at .

Can we do any better than this? Since is an estimate for the first zero, we could try and form the Taylor series about 1.59 instead of about 0. I could have used 1.5924 but 1.59 will do just as well. We now form the table as shown in Table 1.2.

The Taylor series is

If , then setting the left hand side equal to 0 and rearranging gives

Hence,

Not a bad estimate. In fact, we have actually undertaken one step of the Newton-Raphson method and that will be described in detail later.