浅谈幂级数的应用摘要 幂级数是一类最简单的函数项级数，它可以看作是多项式函数的延伸，幂级数在理论和实际上都有很多应用，特别在应用它表示函数方面，利用其性质往往能使一些函数的形式进行转化，进而在题目计算过程中起到化繁为简的作用.本文首先介绍了幂级数，包括幂级数的定义、敛散性、分析性质、函数的幂级数展式等，然后整理分析了近 15 年考研数学真题，并从中挑选了典型的与幂级数的应用有关的题目，通过举例、归纳的方法，讨论了幂级数在多种计算等方面中的应用，包括在近似计算、求定积分、求函数极限、证明不等式、求高阶导数、求函数项级数的和函数、求微分方程的解等方面的应用，结合实例探讨并阐述了利用幂级数求解相关问题的方法与技巧 .很多与函数相关的疑难问题，将原函数展开为幂级数后，再利用幂级数的性质进行运算和变形，往往就迎刃而解了.关键词 幂级数 应用 不等式 极限 微分方程 On the Application of Power SeriesAbstract Power series is a type of function term series, and its analytical nature can often transform the form of some functions, and then play a role in simplifying the problem calculation process. This paper first introduces the basic knowledge of power series, including the definition of power series, convergence and divergence, analytical properties, power series expansion of functions, and important theorems. The problems related to the application of series are analyzed by examples and inductive methods, and the application of power series in various calculations is analyzed, including in approximate calculation, definite integration, function limit, proof inequality, and higher order derivatives , The application of series sum function, etc., combined with examples to discuss and expound the method and skill of using power series to solve related problems.Key words Power Series application inequality limit Differential equation目 录1 前言……………………………………………………………………………………………………… 11.1 背景和意义…...
