数列收敛判别法

学士学位毕业论文设计

数列收敛的判别法

所在系别：数学与应用数学系

专业：数学与应用数学

目录

中文摘要--------------------------------------------------------------------I 英文摘要-------------------------------------------------------------------II 前言------------------------------------------------------------------III 第一章数列极限的概念--------------------------------------------------------1

1.1 数列极限的定义-------------------------------------------------------1

1.2 收敛数列的定义-------------------------------------------------------2第二章判别数列收敛的方法----------------------------------------------------3

2.1 定义法---------------------------------------------------------------3 2.2 单调有界定理---------------------------------------------------------6 2.3 迫敛性定理-----------------------------------------------------------8 2.4 柯西收敛准则---------------------------------------------------------9 2.5 关于子列的重要定理--------------------------------------------------12参考文献-------------------------------------------------------------------14致谢-----------------------------------------------------------------------15

数列收敛的判别法

摘要：数列收敛是极限方法的基本情况，而极限方法是微积分学的基本方法，是初等

数学所没有的一套崭新的方法，它解决了“直与曲”、“均匀变化与非均匀变化”、“近似于精确”的矛盾，是客观世界中由量变到质变的一种反应。数列收敛恰是这些的基础，它的概念、性质、定理、推论为研究其它极限等数学理论研究起到铺垫作用。

本篇文章重点讨论的是判别数列收敛的一些方法，对于判断一个数列是否收敛有些茫然的人，本文会有针对性的对以上问题做细致的讲解和归纳。开篇第一章的内容是对一些基础概念做了叙述，以便于对后面的定理有更好的理解。在第二章重点介绍了判别数列收敛的方法，数列收敛的判别法有很多，对于简单的数列，通过定义其极限的存在常常可以通过观察直接看出，或通过极限的四则运算得出，研究数列收敛的判别法可以判断一些较复杂的极限，例如应用柯西收敛准则和迫敛性定理，它们是利用极限来研究微分学的许多理论问题时的有力工具，在近代分析中有极其重要的理论意义。

关键词：数列收敛、数列极限、判别法

Series convergence criterion

Abstract：Series is the ultimate way to convergence of the basic situation, and the limit is the basic calculus method is not elementary mathematics a new way, it has resolved the "straight and curly," "uniform change and nonuniform change," "close to accurate," the contradiction is the objective world from quantitative to qualitative changes in a response. Convergent series is just the foundation of the concept, nature, theorem, inference to study the other limit, such as paving the way mathematics has played the role of theoretical research.

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Key word:Sequence restraining, Sequence limit, Sequence restraining distinction way