泰勒公式的证明及应用

The Proof of Taylor Formula and Its Application

泰勒公式集中体现了微积分"逼近法"的精髓,在微积分学及相关领域的各个方面都有重要的应用。在现行教材对泰勒公式证明基础上,介绍泰勒公式的一种新的更为简单的证明方法,并归纳了其在求极限与导数、判定级数与广义积分敛散性、不等式证明、定积分证明,行列式计算与中值公式、导数的中值估计、界的估计等方面的应用。

Taylor formula is the embodiment of ＂approximatioss＂ of calculous and has important applications in various aspects of calculus.This paper introduces a simpler new method of proof of Taylor formula on the basis of theorems,which have been stated in current general textbooks,and its proofs.In view of extensive use of Taylor formula in analysis and study of problems in mathematical analysis and solutions of practical applications,the paper have generalized applications in nine aspects which include limit and differential coefficient calculation,judgement of convergence and divergence of progression and improper integral,proof of inequality,proof of definite integral,determinant calculation and mean-value formula,mean-value estimate of derivative,boundary estimate,and illustrated relevant techniques of applications.