Taylor Series

Let's compute the derivatives at :

, so f(0)=1.

, so f'(0)=0.

, so .

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

, so .

Only the even order derivatives are non-zero (since 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 . 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 . 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 . 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 with center .

The tricky part is to rewrite this expression to exploit the geometric series once again. Note that we need the term 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 .

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 . Set , to obtain

Try it yourself!

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