[TV] Taylor expansion is approximating a function F (normally from |R ---> |R) by a 'Taylor' polynomial T of degree N which is based on the first N derivatives at a point a of the domain of F.

The idea is to make a polynomial by looking at the function value, and make it follow the first N derivatives at a point a as an approximation of F around F(a).

This works so well that a lot of relevant theory is based on the main theorem:
                                           2          (n)         n
   t (x) = f(a) + f'(a)(x-a) + f''(a) (x-a)  + ... + f   (a) (x-a)
    n                          ------                -------
                                 2!                     n!


It can be proven that the absolute error in the approximation is upper bounded by the next term of the expansion.

In various theoretical mathematical proofs, the theorem has a prominent place (also in engineering because of its practical value) mainly because of its general applicability: any continuous, bounded function can be fully represented, without error by an infinite order (or as many orders as the function has) Taylor expansion in one point only.

It would be interesting to have a math lib which can differentiate on lists in Tcl, but at this point, I didn't write one, or know of one. Wolfram licences are expensive (at least they were 10 years ago).