Start with some function of t :

It looks like this:

To fourth order in t, we have, analytically,

We'll actually use 20th order for numerical calculations.

Create a function that numerically evaluates the Taylor series representation.

For example,

Compare the Taylor series with the original function:

The Taylor series is good to about .

Check the conversion routines:

Take a look at the first few terms of the Legendre polynomial representation:

Convert the Taylor series to different representations, for example Legendre and Chebyshev:

Numerically,