本 科 生 毕 业 论 文
论 文 题 目： 反常积分与无穷级数收敛关系的讨论
I
Huanggang Normal University
Thesis Graduates
Topic ： Discuss Improper Integrals and Infinite Series Converges Relations Author College
： CHEN Gan ： College of Mathematics and Physics Specialty ： Mathematics and Applied Mathematics Class Tutor
May 17th, 2015
： 201104 ： HE Chunling 郑重声明
本人所呈交的毕业论文(设计)是本人在指导教师 何春玲 的指导下独立研究并完成的. 除了文中特别加以标注引用的内容外,没有剽窃、抄袭、造假等违反学术道德、学术规范和侵权行为,本人完全意识到本声明的法律后果由本人承担.
特此郑重声明！
指导老师(手写签名)： 论文作者(手写签名)：
年 月 日
摘要
数学分析是一个研究变量的学科,既有连续变量,又有离散变量.级数和积分是数学分析中的两个重要概念,它们之间有着密切的联系,体现了离散与连续这一基本矛盾的对立与统一.因此深入研究两者关系,有助于我们理解数学分析原理,解决相关问题.二者似乎相距甚远,实则同出一源.它们本质上都是求和运算,只不过是对两种不同的变量求和,同时都是一个极限过程,因此“连续化”问题的积分理论（反常积分）和“离散化”问题的级数理论（数项级数）有很多性质、定理都是相互对应的,二者在研究问题与论证方法上极为相似.本文从判别法等方面对二者加以比较,列出了很多平行的结论,以及一些区别,指出它们之间的相互转化关系,并应用这种关系,通过某类问题的求解探究另一类问题的解法,从而使读者体会离散与连续的相互转化思想,学会数学知识的迁移. 关键词：反常积分；无穷级数；对比研究；审敛法
Abstract
Mathematical analysis is a subject mainly studying on variables, including the continuous and discrete ones. Series and integrals are two important concepts of it, there is a close relationship between them. They embodies the opposite and uniformity of basic contradiction of continuity and discreteness. So doing further research on the relationship between the two terms -based legacy with source. They are peace operations, is merely to two different variables summation, at the same time is a limit process, so \questions of integral theory (generalized integrals, with respect to the integral, etc.) and \ research on problems with similar reasoning methods. By comparing the concepts, convergence, nature and discriminant method of both aspects, this article lists many paralleled conclusions and some differences, as well as the translation between them. And solve some problems of one kind by applying the solutions of the other kind, thus between discrete and continuous thoughts, and be able to learn mathematics knowledge migration.
Keywords Improper integral; Infinite series; Comparative study; The Inspection
Technique