幂级数是数学分析中的重要概念，被作为基础内容应用到了实变函数、复变函数等众多领域当中。并且在考研数学里，幂级数也是级数这一章所需要掌握的重点。因此，总结幂级数的收敛半径，收敛域，以及和函数求法在现阶段备考数学中是非常重要的。首先，本文对幂级数收敛与和函数的概念与性质进行了简要概括。然后，对幂级数收敛与和函数在考研数学中的应用进行了总结归纳。其中，在求幂级数收敛域的问题中，本文分别对具体型与抽象型问题进行了讨论；在求和函数的问题中，本文对不同情况下和函数的求解思路进行总结，针对具体的例题给出相应的解题过程，并对幂级数和函数在不定积分、导数、数列极限、不等式等相关数学分支中的应用进行系统地论述。
Power series is an important concept in mathematical analysis, and is used as a basic content in many fields such as real variable functions and complex variable functions. And in the postgraduate mathematics, the power series is also the focus of the chapter of the series. Therefore, it is very important to summarize the convergence radius of the power series, the convergence domain, and the method of finding the sum function in the mathematics at this stage. First, this article briefly summarizes the concepts and properties of power series convergence and sum functions. Then, it summarizes the application of power series convergence and sum function in postgraduate mathematics. Among them, in the problem of finding the convergence series of power series, this article discusses the concrete and abstract problems respectively; in the problem of the sum function, this article summarizes the solution of the sum function in different situations, and gives the corresponding problem-solving process for specific examples, and it also systematically discusses the application of power series and functions in related mathematical branches such as indefinite integrals, derivatives, sequence limits, and inequalities.
