无穷级数收敛方法综述

重庆三峡学院毕业设计（论文）题目：数项级数收敛方法综述

专业：数学与应用数学

年级：2006级

学号：200606030161

作者：王超

指导老师：杜祥林（教授）

完成时间：2010年5月

目录

摘要 .................................................................... I Abstract ............................................... 错误！未定义书签。1引言. (1)

2 数项级数收敛的定义 (2)

2．1 级数的定义 (2)

2．2 数项级数收敛的定义 (2)

3 数项级数收敛的性质 (3)

4 数项级数的收敛方法 (4)

4．1 数项级数的常用收敛方法 (4)

4．2 正项数项级数的收敛方法 (10)

4．2．1 同号级数的定义 (10)

4．2．2 正项级数的收敛方法 (10)

4．3 交错级数的收敛方法 (28)

4．3．1 变号级数与交错级数的定义 (28)

4．3．2 交错级数的收敛方法 (28)

4．4 一般项级数的收敛方法 (30)

4．4．1 绝对收敛与条件收敛 (30)

4．4．2 一般收敛级数判别法 (30)

5 数项级数的收敛方法的优缺点比较 (34)

5．1 数项级数的收敛方法概述 (34)

5．2 各种收敛方法优缺点比较 (35)

5．2．1 对于级数收敛的判别方法优缺点比较 (35)

5．2．2 对于正项级数收敛的判别方法优缺点比较 (35)

5．2．3 对于一般项级数收敛的判别方法优缺点比较 (37)

致谢 (38)

参考文献 (38)

数项级数收敛方法综述

王超

（重庆三峡学院数学与计算机科学学院数学与应用数学专业2006级重庆万州 404000）

摘要：研究无穷级数的一个主要目的，就是判断一个级数是否收敛。因此，判断一个级数收敛的方法就显得格外重要。由于篇幅有限，本文只着重考察数项级数的收敛方法。

本文采用总——分——总结构，来综述数项级数的收敛方法。首先，引言部分描述无穷级数的由来、重要性，然后进一步说明研究无穷级数的收敛方法的重要性；其次从特殊的级数——数项级数出发，综述数项级数的收敛方法；同时，勾勒出本文的写作思路、创作方法。使笔者与读者产生思维碰撞，起到了总揽全局的作用。其次，从数项级数的收敛的定义、收敛的性质出发，挖掘出较为常规的数项级数收敛方法；再将数项级数分为两类：正项级数和一般项级数（包括交错级数），分别对两类级数的收敛方法进行综述，并加以论证和综合举例，浅谈各类方法的特点，从而，让读者徜徉其中，达到各个击破的目的。最后，呼应引言，概述所有的收敛方法，再进一步透彻分析各种方法，分析它们的解题思想，比较它们的优缺点，推导出它们的适用范围，使读者进一步加深认识，从而理论得到升华，鞭辟入里。进一步，提高了读者的逻辑思维能力。

关键词：数项级数；总——分——总；收敛方法；正项级数；交错级数；一般项级数；逻辑思维

Review the Method of Judging the Several Series of Convergence

Wang Chao

(Grade 2006, Mathematics and Applied Mathematics, College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou, Chongqing 404000 )

Abstract:A main purpose of the infinite series studying is judging a Infinite series of convergence. Therefore, the method of judging converge is especially important.Due to the limited space, this article only focuses on several series convergence method.

This article is based on the structure of total--points--total to review several series convergence method.Firstly, the introduction section describes the origin of the infinite series, further study, and the importance of the infinite series convergence method of importance, then base on the special series of several series, the paper presents several series convergence method. At the same time, to outline the thoughts and methods of this paper. And to make the author and the readers sparkle collision of thinking overarching role. Secondly, from the convergence of several series of definitions, the convergence of the nature, excavate relatively conventional several method converges, Then several series is divided into two categories: positive series and general series (including staggered series), two kinds of methods were reviewed, and demonstrated, for example, and discuss the method of characteristics, thus, let the reader roam among them and have a intensive understanding. Finally, summarizes all the convergence of citations, further thorough analysis of various methods, analyzes their thoughts, problem solving, and their advantages and disadvantages are deduced, the scope for further understanding of the readers, thus theory and exact. To improve the reader’s logical thinking ability.

Keywords: Several Series；Total—points—total；The method of judging convergence；Positive series.