Proving Taylor's Theorem from the Fundamental Theorem of Calculus by Fixed-point Iteration

TL;DR

This paper presents a simple proof of Taylor's theorem derived from the Fundamental Theorem of Calculus, highlighting its connection to fixed-point iteration and its educational value.

Contribution

It introduces a novel proof of Taylor's theorem based on the FTOC and fixed-point iteration, emphasizing conceptual clarity and pedagogical usefulness.

Findings

Proof demonstrates the link between Taylor's theorem and fixed-point iteration

Highlights the role of combinatorics and symmetry in analysis proofs

Provides an accessible approach suitable for undergraduate education

Abstract

Taylor's theorem (and its variants) is widely used in several areas of mathematical analysis, including numerical analysis, functional analysis, and partial differential equations. This article explains how Taylor's theorem in its most general form can be proved simply as an immediate consequence of the Fundamental Theorem of Calculus (FTOC). The proof shows the deep connection between the Taylor expansion and fixed-point iteration, which is a foundational concept in numerical and functional analysis. One elegant variant of the proof also demonstrates the use of combinatorics and symmetry in proofs in mathematical analysis. Since the proof emphasizes concepts and techniques that are widely used in current science and industry, it can be a valuable addition to the undergraduate mathematics curriculum.