Main

Taylor Series Redux

The last module took a sequence and turned it into a power series . This module turns this around and asks can we go from a function to a sequence ? The answer is yes, and this is the familiar process of computing the Taylor series for the function. Recalling the Taylor series for :

we see that is the sequence corresponding to . We could also center the Taylor series at a different point, and get a different sequence, but for now let's keep things centered at 0.

Now that we have turned our function into its Taylor series, we come back to the questions deferred from earlier in the course:

  1. For what values of does a function's Taylor series converge?
  2. Does the Taylor series converge to the function?

These questions are the topics of this module.

Taylor series convergence

We now have the tools to see when a power series converges, so the answer to the first question is that the series

converges absolutely for , where

Within the interval of convergence, differentiation and integration of a power series are nice, in that they can be done term by term:

Why is this useful? Being able to differentiate and integrate term by term allows us to compute the Taylor series for various functions by differentiating or integrating the Taylor series for other functions.

Example

Compute the Taylor series for by noting that .

We have that

for . Thus, for , we can safely integrate both sides of this equation to find

By checking , we find that the integration constant . Thus for we have

Example

Show that the power series can be written as for . Hint: try integrating both sides and see what familiar function you get.

Integrating both sides gives

by recognizing that this is the geometric series. Now, differentiating both sides gives that

as desired. Note that this only holds within the interval of convergence for the geometric series, .

Example

The Fresnel Integral is defined by

There is no elementary expression for this integral, but it can be expressed as a series by expanding the series for and then integrating term by term:

Taylor series convergence to a function

Now, we consider the second question above: when a Taylor series converges, does it always converge to the function? Unfortunately, not always. Even with a smooth , and within the interval of convergence, it is possible that the Taylor series does not converge to . The following definition is used for functions whose Taylor series do converge to the functions themselves:

Definition: Real-analytic function

A function is real-analytic at if for sufficiently close to ,

That is, a function is real-analytic at if the Taylor series for converges to near .

Almost all the functions we have encountered in this course are real-analytic. However, there are examples of smooth functions which are not real-analytic, as the next example shows.

Example

Consider the function

To show that this function is smooth, we must show that its derivative exists at (everywhere else it is a composition of nice functions, so we need not worry). We use the definition of the derivative:

This can either be thought of as a case for l'Hospital's rule, or we can do a change of variables , which gives the limit

since the exponential beats a polynomial asymptotically. So . It turns out that all of the higher derivatives of at 0 are 0 as well. So if we tried to expand this as a Taylor series, we would have

So the Taylor series for converges to 0, despite the fact that is non-zero for .