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

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.