Abstract
The general form of Taylor's theorem gives the formula, f = Pn + Rn, where Pn is the Newton's interpolating polynomial, computed with respect to a confluent vector of nodes, and Rn is the remainder. When f' /= 0, for each m = 2; : : : ; n + 1, we describe a "determinantal interpolation formula", f = Pm,n + Rm,n, where Pm,n is a rational function in x and f itself. These formulas play a dual role in the approximation of f or its inverse. For m = 2, the formula is Taylor's and for m = 3 it gives Halley's iteration function, as well as a Padé approximant. By applying the formulas to Pn, for each m >= 2, Pm,m-1,. . . , Pm,m+n-2, is a set of n rational approximations that includes Pn, and may provide a better approximation to f , than Pn. Thus each Taylor polynomial unfolds into an infinite spectrum of rational approximations. The formulas also give an infinite spectrum of rational inverse approximations, as well as a family of iteration functions for real or complex root finding, more fundamental than the Euler-Schröder family, or any other family. Given m >= 2, for each k <= m, we obtain a k-point iteration function, defined as the ratio of two determinants that depend on the first m - k derivatives, and Toeplitz for k = 1. The order of convergence ranges from m to the limiting ratio of the generalized Fibonacci numbers of order m. By applying these formulas, Hadamard's inequality, Gerschgorin's theorem, and a new lower bound on determinants, we express roots of numbers, e, and π, as the limiting ratio of Toeplitz determinants.