论文标题
通过固定点迭代从微积分的基本定理证明泰勒的定理
Proving Taylor's Theorem from the Fundamental Theorem of Calculus by Fixed-point Iteration
论文作者
论文摘要
泰勒的定理（及其变体）广泛用于数学分析的几个领域，包括数值分析，功能分析和部分微分方程。本文解释了如何简单地证明泰勒的定理是微积分（FTOC）的基本定理的直接结果。证明显示了泰勒膨胀与定点迭代之间的深厚联系，这是数值和功能分析中的基础概念。证明的一种优雅变体还证明了在数学分析中的证明中使用组合和对称性。由于证明强调了在当前科学和行业中广泛使用的概念和技术，因此它可能是本科数学课程的宝贵补充。
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.
