TaylorRestglieder

TaylorRestglieder is a term used to describe the remainder terms that appear after approximating a function by its Taylor polynomial around a chosen base point. If f is expanded about x0, its Taylor polynomial of degree n is Pn(x) = sum_{k=0}^n f^{(k)}(x0)/k! (x − x0)^k, and the TaylorRestglieder refers to the difference Rn(x) = f(x) − Pn(x). This remainder encapsulates the error of the approximation.

Variants of the TaylorRestglieder are common in analysis, depending on how the remainder is expressed. The Lagrange form gives Rn(x) = f^{(n+1)}(ξ)/(n+1)! (x − x0)^{n+1} for some ξ between x0 and x, providing a concrete error bound when derivatives are bounded. The Cauchy form and the integral form offer alternative representations that can be advantageous in different contexts. Collectively, these forms constitute the different ways to bound or estimate the TaylorRestglieder.

Etymology and usage: the term combines Taylor, after Brook Taylor, with Restglied, the German word for remainder term. In mathematical texts and numerical analysis, the concept of TaylorRestglieder is used to structure error analysis, convergence studies, and the design of approximation schemes. They are central to understanding how closely a polynomial approximation tracks the original function over a given interval.

Properties: the behavior of the TaylorRestglieder depends on the smoothness of f. If f is analytic near x0, the remainder tends to zero as n increases. Effective bounds require information about higher-order derivatives or integral representations. See also Taylor series, remainder term, and numerical analysis.