Taylor Series

Let's compute the derivatives at tex2html_wrap_inline161 :

tex2html_wrap_inline163 , so f(0)=1.

tex2html_wrap_inline167 , so f'(0)=0.

tex2html_wrap_inline171 , so tex2html_wrap_inline173 .

tex2html_wrap_inline175 , so f'''(0)=0.

tex2html_wrap_inline179 , so tex2html_wrap_inline181 .

Only the even order derivatives are non-zero (since tex2html_wrap_inline183 is an even function!). We can model this for the general term by using the counter 2n instead of n (use 2n+1, if you want to catch odd terms!).

Since the power of the term 2 equals the order of the derivative, the general term has the form


What about the alternating sign? If n=0, we want to create a positive term, so let's use tex2html_wrap_inline193 . For n=1, this produces the desired negative term (associated with the second derivative). Thus


This yields the Taylor series expansion


Use the ratio test to show that this series converges for all x.

Using substitutions

Computing a Taylor series by successively taking derivatives is often not very practical, and should really be considered a last resort. An elegant way to derive Taylor series for functions is to use substitution.

An example.

Consider for example the function tex2html_wrap_inline207 . It is not much fun to compute its derivatives. Instead we will use the formula for the geometric series:


We can rewrite


which suggests the substitution tex2html_wrap_inline209 . Substituting in the series yields


Since the formula for the geometric series holds for |q|<1, the new series will converge for


Another example.

Let's find the Taylor series for tex2html_wrap_inline219 with center tex2html_wrap_inline221 .

The tricky part is to rewrite this expression to exploit the geometric series once again. Note that we need the term tex2html_wrap_inline223 to show up in our substitution to end up with the right center! So, let's start by writing


In order to use the geometric series, we will have to "replace" the 3 in the denominator by 1. Here is how:


This suggests the substitution tex2html_wrap_inline225 .


The Taylor series will represent the function as long as


Thus the series will work for -1<x<5.

A last example.

This works for other series than the geometric series as well. If you know that


then you can easily compute the Taylor series for tex2html_wrap_inline233 . Set tex2html_wrap_inline235 , to obtain


Try it yourself!

Find the Taylor series with center tex2html_wrap_inline237 for tex2html_wrap_inline239 . Where does the Taylor series coincide with the function?

Click here for the answer, or to continue.

Helmut Knaust
Tue Jul 16 11:39:32 MDT 1996

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