数学分析中的反例问题

摘要

数学分析是一门非常重要的基础课程,反例对理解数学分析有关定义和定理的内涵和外延有着不可替代的作用,反例的地位在数学的学习中占有很重要的地位,对培养我们的逆向思维至关重要,恰当的运用反例对我们数学能力的提高起着事半功倍的效果,我们希望定理中的条件是最简的,在我们一步步削弱条件的时候,反例的作用就越来越明显,一个特列不能说明一个命题是对的,但一个反例完全可以证明一个命题是错的.反例的作用和构造也越来越受到重视.本文介绍了数列,函数,导数,积分,无穷积分,级数等中的一些典型问题的反例,对一些逆命题的成立与否通过反例做了简单的论证,通过反例把一些看似相关性很大的定义和定理的区别又做了进一步的比较和分析,对一些反例的构造过程和思路做了详细介绍,回答了为什么这样构造的问题,可以让读者在错综复杂的关系里得到清晰的逻辑和思路.

关键词：命题;反例;构造;数学分析;体现

ABSTRACT

Mathematical analysis is a very important basic course, counterexample has an irreplaceable role in understanding mathematical analysis about definition and theorem of connotation and denotation , counter example role has a extremely important position in learning mathematics occupies,it is very important to educate our reverse thinking, appropriate mathematical ability for us to use counterexample improve play a extremely important position, we hope that the conditions of the theorem is one of the most simple, when we weaken conditions step by step, the counter example of the role is more and more obvious, a special example does not justify a question is right, but a counter example can prove that a theorem is wrong. counterexample and structure is becoming more and more important. According to the general mathematical analysis teaching material order, this paper introduces the sequence, function, derivative,and series of a reverse case of some typical problems, such as, for some of the establishment of the converse proposition, seemingly through counterexamples correlation definition theorem of great difference and do a further comparison and analysis of the construction process of some counter example ,it also made a detailed introduction, why and how structure counterexample get a answer in this paper, reader can get a clear logic in this paper.

Key words:proposition; counter example;structure;mathematical analysis;reflect

目 录

1.引言 .................................................................. 1

2.反例在加深理解定义及相关概念中的体现................................... 1 2.1周期函数 ............................................................ 1 2.2复合函数 ............................................................ 1 2.3极值 ................................................................ 2 2.4一致连续 ............................................................ 2 2.5导数 ................................................................ 3

3.反例在掌握定理的内涵与外延中的体现 .................................... 3 3.1柯西收敛准则 ........................................................ 3 3.2 STOLZ 公式 ............................................................ 4 3.3 比式判别法 .......................................................... 5 3.4 比较原则 ............................................................ 5 3.5 阿贝尔判别法 ........................................................ 6 3.6 莱布尼茨判别法 ...................................................... 7

4.反例在辨析重要结论的逆命题中的体现 .................................... 7

5.反例在论证辩证关系中的体现 ........................................... 10 5.1

lim ()x f x →+∞和'

lim ()

x f x →+∞

的关系 (10)

5.2 原函数与可积函数之间的关系 ......................................... 10 5.3 ()a

f x d x

+∞

⎰

收敛与lim ()x f x →+∞

=0

的关系 (11)

5.4 可积和绝对可积以及平方可积之间的关系 ............................... 12

6.结论 ................................................................. 13 参 考 文 献 ............................................................ 14 致 谢 ................................................. 错误！未定义书签。