A bounded set of derivatives (the easy case)

Let's start out with an easy case, where we know that the derivatives are all bounded (and we know that bound):

c is the center around which the Taylor series is expanded, nDegree is the degree Taylor function we want to use, and yRange is just the range of y values we want to see when we graph these (not used for anything else).

We want to know how large the domain of this Taylor series can be if we use a degree Taylor approximation.

In order for the maximum error bound to be correct, you must enter a number or a function of n here:

This is actually the most difficult part of the problem in the general case.  For , of course, this is trivial (why?), but for a generic function, finding the maximum size of its derivative is very hard (especially if you don't know in advance how many terms you will need to use).

We need to define a few functions for working with errors.  The first one just computes the actual error between the original function and the degree Taylor Polynomial (as a function of the degree).

So, for example, in our case:

Now, we need to define a function to approximate our maximum possible error (, where ):

A more accurate formula for the error is given by , which is sometimes useful (but generally much harder to work with), so we will define a second errorBoundFunction as:

The following function is designed to display an animation of two graphs side-by-side (see below for explanation).  Most of you will want to just execute this definition and move on; it isn't necessary for you to understand the details of this particular command.

Now, let's look at some graphs.  These graphs show the original function (green) and the degree Taylor series (black) in the graph on the left and the actual error (, in red) and the maximum error bound (, where where is any value in the interval under consideration, in yellow).  Notice a few things:

1)  Observe how the Taylor polynomial gradually fits itself to more and more of the graph of as the degree of the Taylor polynomial is increased.

2)  Notice how the errors become small over larger and larger intervals as the degree of the Taylor polynomial is increased.

3)  Notice how the magnitude of the maximum error bound is indeed always at least as big as the actual error everywhere.  (Does it matter that sometimes the actual error and the maximum error bound have different signs?)  This means that if you can keep your maximum error bound small enough, you can be assured of the accuracy you want.

So, to find our domain, we need to see when the error is either or (above or below):

We can safely ignore the "nonreal" solutions, so the domain is , including or excluding the endpoints, depending on whether you want to allow the error to actually equal .  This gives us a radius of convergence of: