数项级数求和的若干方法

┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ 装 ┊ ┊ ┊ ┊ ┊ 订 ┊ ┊ ┊ ┊ ┊ 线 ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊

安徽工业大学 信息与计算科学系 毕业论文

题 目: 姓 名: 学 院: 专 业: 班 级: 学 号: 指导教师:

中文：级数求和的若干方法 English:Summation of several methods 徐科 数理学院 信息与计算科学 2009级1班 099084166 张 敬 和

2013年6月8日

┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ 装 ┊ ┊ ┊ ┊ ┊ 订 ┊ ┊ ┊ ┊ ┊ 线 ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊

安徽工业大学 信息与计算科学系 毕业论文

安徽工业大学 毕业设计(论文)任务书

课题名称 学 院 专业班级 姓 名 学 号

毕业设计(论文)的主要内容及要求：

1、了解正项级数，任意项级数，函数项级数，幂级数的相关概念。 2、熟悉各种级数收敛的的理论，理解部分定理的证明过程。 3、尽可能的对某些定理之间的区别以及联系稍加分析。 4、借助幂级数，数列等知识给出数项级数求和的若干方法。

5、参阅数学分析，常微分方程等与级数相关的教材或者文献，充分利用图书馆以及电子阅览室。提高自己查阅资料的能力。

6、论文必须符合科技论文的要求，格式严格按照本科毕业论文的规范来撰写。 7、查阅相关文献资料，至少10篇，其中英文文献不少于2篇。

8、翻译一篇跟本设计有关的外文文献，要求翻译无错误，可以通顺阅读。 9、熟悉微软Word或者金山的WPS的使用方法和技巧，以期提高使用计算机的能力。

级数求和的若干方法

数理学院

信息与计算科学 091班

徐科 099084166

指导教师签字：

┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ 装 ┊ ┊ ┊ ┊ ┊ 订 ┊ ┊ ┊ ┊ ┊ 线 ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊

安徽工业大学 信息与计算科学系 毕业论文

级数求和的若干方法

安徽工业大学 数理学院 信息与计算科学系

091班 徐科 学号：099084166

摘 要

级数，重要的数学工具。无论是对数学学科本身，还是在其他学科及技术的研究与发展方面，都发挥着特别重要的作用和影响，且其与我们的日常生活息息相关。需要我们去掌握并利用，我们也应该去发掘出它更为广泛的应用领域，为我们的研究与学习奠定基础。

级数求和，作为级数理论及应用的主要板块之一。它有着比较繁多的方法和很强的技巧性，而目前国内大多数数学教材及其他相关书籍中没有专门针对级数求和的常用方法设立板块，若要理解并掌握它的方法和技巧，则需要借鉴一些国内外涉及此内容的数学书籍，进行总结和提炼。

本文对级数的有关概念，收敛的定义以及部分定理给与了证明，介绍了运用裂项相消, 错位相减, 逐项微分, 逐项积分, 运用特殊级数求和等等这几种方法求数项级数的和, 并通过实例说明了这些方法的应用.

关键词：级数，收敛，数项级数求和，幂级数。

I

┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ 装 ┊ ┊ ┊ ┊ ┊ 订 ┊ ┊ ┊ ┊ ┊ 线 ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊

安徽工业大学 信息与计算科学系 毕业论文

Summation of several methods

ANHUI UNIVERSITY OF TECHNOLOGY

The Mathematical Institute Information and computing science department

Class 091 Xu Ke Student ID: 099084166

Abstract

Progression , important mathematical tools ! Both for mathematics itself , or in other disciplines and technology research and development, has played a particularly important role and influence , and its our daily lives. We need to grasp and use , we should go to discover its broader application areas for our research and learning foundation . Summation as a series theory and application of the main plate. It has a relatively strong variety of methods and techniques , while most domestic mathematics textbooks and other books not specifically for the establishment of a common method Summation sector , to understand and grasp its methods and techniques , then needs to learn some of this content and abroad involved in the mathematical books were summarized and refined .

In this paper, the concept series , convergence theorems give some definitions and proved , introduces the use of destructive Splitting , dislocation subtract itemized differential , itemized points , the use of these types of special summation etc. method for solving a number of series and , and through examples illustrate the application of these methods .

Keywords : series , convergence, Summation , power series .

II

┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ 装 ┊ ┊ ┊ ┊ ┊ 订 ┊ ┊ ┊ ┊ ┊ 线 ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊ ┊

安徽工业大学 信息与计算科学系 毕业论文

目 录

摘要 ···························································································································· Ⅰ Abstract ················································································································· Ⅱ 一、综述 ······················································································································ 1

1.1 级数的背景知识 ··························································································· 1 1.2 研究现状 ······································································································· 2 1.3 研究意义 ······································································································· 2 二、基础知识 ·············································································································· 3 2.1 引言 ··············································································································· 3

2.2 级数的分类及定义 ······················································································· 3 2.2.1 数项级数 ···························································································· 3 2.2.2 函数项级数 ························································································ 3 2.2.3 三个重要级数 ···················································································· 4 2.3 级数收敛的定义 ··························································································· 4

2.4 级数收敛的判断 ··························································································· 4

2.4.1 正项级数收敛的判断 ········································································ 5 2.4.1.0 级数收敛的必要条件 ····························································· 5 2.4.1.1 定理02 ···················································································· 5 2.4.1.2 正项级数的收敛原理 ····························································· 5 2.4.1.3 常用级数 ················································································· 5 2.4.1.4 正项级数的各种判别法 ························································· 7 2.4.1.5 引理 ······················································································· 11 2.4.2 任意项级数收敛的判断 ·································································· 14 2.4.3 函数项级数收敛的判断 ·································································· 16 2.4.4 幂级数 ······························································································ 17 2.4.4.1 幂级数的基本概念和定理 ··················································· 18 2.4.4.2 函数的幂级数的展开 ··························································· 21 三、级数求和 ············································································································ 26

<一> 简单易用的求和方法 ·············································································· 26 3.1 根据定义求级数的和 ········································································· 26 3.2 首尾相加法 ························································································· 26 3.3 错位相减法 ························································································· 27 3.4 分组求和法 ························································································· 28

3.5 微分方程法 ························································································· 28

III

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说综合文库数项级数求和的若干方法在线全文阅读。