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Taylor's theorem (and its variants) is widely used in several areas ofmathematical analysis, including numerical analysis, functional analysis, andpartial differential equations. This article explains how Taylor's theorem inits most general form can be proved simply as an immediate consequence of theFundamental Theorem of Calculus (FTOC). The proof shows the deep connectionbetween the Taylor expansion and fixed-point iteration, which is a foundationalconcept in numerical and functional analysis. One elegant variant of the proofalso demonstrates the use of combinatorics and symmetry in proofs inmathematical analysis. Since the proof emphasizes concepts and techniques thatare widely used in current science and industry, it can be a valuable additionto the undergraduate mathematics curriculum.

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