一元函数泰勒公式及泰勒级数的应用探究

Application of Taylor Formula and Taylor Series for Single Variable Function

摘要: 泰勒公式及泰勒级数是重要的数学工具，但通常求解微积分题目过程中不易想到。针对一些典型问题，用常规的手段解决比较困难时，可尝试用泰勒公式及泰勒级数求解。本文将带有Peano余项泰勒公式、带有Lagrange余项泰勒公式及泰勒级数应用到了以下几个方面：求极限、误差估计、不等式证明、级数敛散性判别、寻找非初等原函数、求定积分、证明无理数。对这些典型的微积分题目给出了一种新的解法。希望这对扩展初学者视野及启发应用有所帮助。

Abstract: Taylor formula and Taylor series are important mathematical tools, but it’s not always easy to think about when you are solving calculus problems. For some typical problems, when it is difficult to solve them by conventional means, we can try to solve them by Taylor formula and Taylor series. In this paper, Taylor’s formula with Peano remainder, Taylor’s formula with Lagrange remainder and Taylor series are applied to the following aspects: The limit, Estimation of error, Proof of inequality, Criterion on convergence and diverge of series, Looking for non-elementary primitive function, Evaluating of definite integral, Proof of irrational. A new method for solving these typical calculus problems is given. I hope this will be helpful for beginner to expand views and inspire application.