Enhancing the accuracy of the Taylor polynomial by determining the remainder term

TL;DR

This paper introduces a method to improve Taylor polynomial accuracy by explicitly calculating the remainder term using a differential equation approach and spline approximation, significantly enhancing approximation quality especially outside the convergence radius.

Contribution

It presents a novel approach to determine the Taylor remainder term via solving an initial-value problem and spline approximation, leading to substantial accuracy improvements.

Findings

Accuracy improved by orders of magnitude

Effective beyond the radius of convergence

Method applicable to various functions

Abstract

We determine the Lagrange function in Taylor polynomial approximation by solving an appropriate initial-value problem. Hence, we determine the remainder term which we then approximate by means of a natural cubic spline. This results in a significant improvement in the quality of the Taylor approximation. We observe improvements in the accuracy of the approximation of many orders of magnitude, including a case when the independent variable x lies beyond the relevant radius of convergence.