不等式的证明方法论文

不等式的证明方法

摘 要

不等式的形式与结构多种多样，其证明方法繁多，技巧性强，也没有通法，所以研究范围极广，难度极大．目前国内外研究者已给出很多不等式的证明方法，已有文献分别就不等式的性质、各种证明方法及应用作了论述．论文以现有研究成果为基础，整理和归纳了常用的不等式证明方法，包括构造几何图形、构造复数、构造定比分点、构造主元、构造概率模型、构造方差模型、构造数列、构造向量、构造函数、代数换元、三角换元、放缩法、数学归纳法，让每一种方法兼具理论与实践性．旨在使学生对不等式证明问题有一个较为深入的了解，进而在解决相关不等式证明问题时能融会贯通、举一反三，达到事半功倍的效果，同时为从事教育的工作者提供参考．

关键词：不等式；证明；方法

Methods for Proving Inequality

Abstract： The form of structure of inequality is diversity, and the proving methods of it are various which requires lots of skills, and there is no common way, so it is a extremely difficult study. Researchers have been given a lot of inequality proof methods at home and abroad, the existing literature, respectively, the nature of inequality, certificate of various methods and application are discussed. The paper on the basis of existing research results and summarizes the commonly used methods of inequality proof, including structural geometry, structure complex, the score point, tectonic principal component, structure, tectonic sequence probability model, structure of variance model, vector construction, constructor, algebra in yuan, triangle in yuan, zoom method, mathematical induction, making every kind of method with both theory and practice. The aim is to make the student have a more thorough understanding on the inequality problems , and in solving the problem of relative inequality proof can digest the lines, to achieve twice the result with half the effort, at the same time provide a reference for engaged in education workers.

Key words: inequality; proof; method

目 录

1 引言 ................................................................. 1 2 文献综述 ............................................................. 1 2．1 国内外研究状况 ....................................................... 1 2．2 国内外研究评价 ....................................................... 2 2．3 提出问题 ............................................................. 2 3 构造法 ............................................................... 2 3．1 构造几何图形 ......................................................... 2 3．2 构造复数 ............................................................. 3 3．3 构造定比分点 ......................................................... 4 3．4 构造主元，局部固定 ................................................... 5 3．5 构造概率模型 ......................................................... 5 3．6 构造方差模型 ......................................................... 6 3．7 构造数列 ............................................................. 7 3．8 构造向量 ............................................................. 8 3．9 构造函数 ............................................................. 8 4 换元法 .............................................................. 10 4．1 代数换元 ............................................................ 10 4．2 三角换元 ............................................................ 11 5 放缩法 .............................................................. 11 5．1 添加或舍弃一些正项（或负项） ........................................ 12 5．2 先放缩再求和（或先求和再放缩） ...................................... 12 5．3 先放缩，后裂项（或先裂项再放缩） .................................... 13 5．4 放大或缩小因式 ...................................................... 13 5．5 固定一部分项，放缩另外的项 .......................................... 14 5．6利用基本不等式放缩 .................................................. 14

6 数学归纳法 .......................................................... 15 7 结论 ................................................................ 16 7．1主要发现 ............................................................ 16 7．2启示 ................................................................ 16 7．3 局限性 .............................................................. 16 7．4 努力方向 ............................................................ 17 参考文献 ............................................................ 18