## 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*:

Our problem is to find
so as to minimize
over some interval containing . There are many
``optimality criteria'' we could choose. The one that falls out
naturally here is called *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

Differentiating the polynomial approximation and setting gives

*slope*of the error to be exactly zero at .

In the same way, we find

From this derivation, it is clear that the approximation error (remainder
term) is smallest in the vicinity of . *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

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Formal Statement of Taylor's Theorem

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Informal Derivation of Taylor Series