## Taylor Series with Remainder

We repeat the derivation of the preceding section, but this time we treat the error term more carefully.Again we want to approximate with an th-order

*polynomial*:

*Padé approximation*. Padé approximation sets the error value and its first derivatives to zero at a single chosen point, which we take to be . Since all ``degrees of freedom'' in the polynomial coefficients are used to set derivatives to zero at one point, the approximation is termed

*maximally flat*at that point. In other words, as , the th order polynomial approximation approaches with an error that is proportional to . Padé approximation comes up elsewhere in signal processing. For example, it is the sense in which Butterworth filters are optimal [53]. (Their frequency responses are maximally flat in the center of the pass-band.) Also, Lagrange interpolation filters (which are nonrecursive, while Butterworth filters are recursive), can be shown to maximally flat at dc in the frequency domain [82,36]. Setting in the above polynomial approximation produces

*slope*of the error to be exactly zero at . In the same way, we find

*All degrees of freedom*in the polynomial coefficients were devoted to minimizing the approximation error and its derivatives at . As you might expect, the approximation error generally worsens as gets farther away from 0. To obtain a more

*uniform*approximation over some interval in , other kinds of error criteria may be employed. Classically, this topic has been called ``economization of series,'' or simply polynomial approximation under different error criteria. In

`Matlab`or

`Octave`, the function

`polyfit(x,y,n)`will find the coefficients of a polynomial of degree

`n`that fits the data

`y`over the points

`x`in a least-squares sense. That is, it minimizes

**Next Section:**

Formal Statement of Taylor's Theorem

**Previous Section:**

Informal Derivation of Taylor Series