级数求和问题的研究.pdf

级数求和问题的研究 摘 要 收敛级数的求和问题是一个复杂多变的问题。本文通过利用数学分析、复变函 数、线性代数、组合数学等许多数学方面的知识，归纳总结了收敛级数（包括数项 级数和函数项级数）求和的若干的不同方法和技巧，包括利用傅立叶级数展开式求 和法、利用幂级数展开式求和法、利用欧拉公式法、利用交换求和顺序求和法、利 用列项消去法求和法、利用收敛级数定义求和法、利用部分和子列求和法、Abel 求 和法、利用逐项微分和逐项积分求和法、关于一类三角级数求和法、利用两个级数 的Cauchy 乘积求和法、利用微分方程求和法以及 P 级变号级数求和法等许多方法， 并通过列举实例来说明这些求和方法和技巧的应用。 本文还通过 matlab 数学软件工具给出了几个关于级数敛散性判别方法的程序 实现以及 matlab 在级数求和方面的相关内置函数应用的一些介绍。我们相信这项工 作在理论上对提高有关级数理论方面的学习效果和相关数学竞赛是有许多帮助的。 关键词：级数，求和，收敛，matlab 级数求和问题的研究 Abstract It is not easy to get the sum of the convergent series. Based on Mathematical Analysis, Complex Vaniable Function, Linear Algebra and Combinatorial Mathematics and much mathematical knowledge, several different techniques and methods ( including the usage of the expansion of the Fourier series, the usage of the expansion of the power series, the usage of the Eulers formula, the usage of the exchanges of the order, the elimination, the usage of the definition of the series , the Abel method, the usage of the differential and the integral summation for every items, the summation about one type of triangular series, the Cauchy product for two series, the usage of the differential equation, the usage of the p series and so on ) to the sum of the different series ( numeral series and series of function ) are given, and the usage of the summations is illustrated further with some related examples. This article also gives some computer programs about some parts of the methods in the convergence of series with the mathematical soft tools –matlab and the introduction of the usage of some built-in functions ( symsum ) about the sum of the convergent series. We think this work is theoretically very helpful for the learning efficiency on the series theory