Restzeros

Restzeros is a hypothetical concept used to illustrate the zeros of the Taylor remainder in finite expansions. For a smooth function f defined near a point a, the Taylor series gives f(x) = P_n(x) + R_n(x), where P_n is the degree-n Taylor polynomial about a and R_n is the remainder term. The restzeros of f at order n are defined as the set of points x in the domain for which R_n(x) = 0.

The notion highlights how, for a given n and center a, the truncated series may coincide with the function at certain points beyond the expansion point. If f is a polynomial of degree at most n, then R_n is identically zero, making every x a restzero in that degenerate sense. For analytic functions with nontrivial higher derivatives, R_n can have a nonempty zero set that depends on n and on the behavior of f, but the zeros are not guaranteed and may be limited to a small region. In many practical settings, R_n is analytic on the domain of interest, so restzeros form a discrete set within a compact region, besides the trivial zero at the expansion point a.

Applications of the concept are primarily pedagogical and methodological: understanding how the choice of n affects approximation accuracy, guiding adaptive selection of polynomial degree, and illustrating the relationship between remainder behavior and error bounds. For example, with f(x) = e^x about a = 0, the remainder R_n(x) = f^{(n+1)}(ξ)/(n+1)! x^{n+1} for some ξ between 0 and x vanishes only at x = 0, demonstrating how restzeros can be sparse. The term is not standard in mainstream literature but serves as a useful exploratory tool. See also Taylor series, remainder, numerical analysis.

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