均值不等式证明方法与应用.doc

第 PAGE \* MERGEFORMAT 3页 共20页 均值不等式的证明方法及应用 摘要 均值不等式在不等式理论中处于核心地位，是现代分析数学中应用最广泛的不等式之一。应用均值不等式，可以使一些较难的问题得到简化处理。本文首先系统全面地总结了均值不等式的十种证明方法,其中包括柯西法、数学归纳法、詹森不等式法、不等式法、几何法、排序法、均值变量替换法、构造概率模型法、逐次调整法、泰勒公式法；其次, 结合相关例题给出均值不等式在证明不等式、比较大小、求最值、证明极限的存在性、判断级数敛散性、证明积分不等式方面的应用。 关键词:均值不等式；数学归纳法；最值；极限；积分不等式 PROOFS AND APPLICATIONS ON AVERAGE VALUE INEQUALITY ABSTRACT Average value inequality occupies a core position in inequality theory and is one of the most widely used inequalities in modern mathematics. Using average inequality can make some difficult problems simple. In this paper, ten proof methods of average value inequality are first systematically summarized, including Cauchy method, mathematical induction, Jensen inequality, inequality method, geometry method, sorting method, variable substitution method of average value, constructing probability model method, successive adjustment method, Taylor formula method, respectively. Secondly, we give applications of average value inequality combining the corresponding examples on comparing the size, solving maximum and minimum, proving the existence of the limit, judging convergence of series and proving integral inequality. Key words: average value inequality; mathematical induction; maximum and minimum; limit; integral inequality 目 录 TOC \o 1-3 \h \z \u HYPERLINK \l _Toc358209346 前言 PAGEREF _Toc358209346 \h 4 HYPERLINK \l _Toc358209347 1 均值不等式的证明方法 PAGEREF _Toc358209347 \h 5 HYPERLINK \l _Toc358209348 1.1 柯西法 PAGEREF _Toc358209348 \h 5 HYPERLINK \l _Toc358209349 1.2 数学归纳法 PAGEREF _Toc358209349 \h 6 HYPERLINK \l _Toc358209350 1.3 詹森不等式法 PAGEREF _Toc358209350 \h 7 HYPERLINK \l _Toc358209351 1.4 不等式法 PAGEREF _Toc358209351 \h 7 HYPERLINK \l _Toc358209352 1.5 几何法 PAGEREF _Toc358209352 \h 8 HYPERLINK \l _Toc358209353 1.6 排序法 PAGEREF _Toc358209353 \h 9 HYPERLINK \l _Toc358209354 1.7 均值变量替换法 PAGEREF _Toc358209354 \h 9 HYPERLINK \l _Toc358209355 1.8 构造概率模型法 PAGEREF _Toc358209355 \h 9 HYPERLINK \l _Toc3582093