概率方法證明不等式

摘要
在分析數(shù)學(xué)中有些不等式的證明往往比較復(fù)雜，若運(yùn)用代數(shù)方法較難得到解決，而且具體的直觀含義也比較抽象。因此，為了簡(jiǎn)化這些不等式的證明過(guò)程，通過(guò)閱讀大量的相關(guān)資料，本文從數(shù)學(xué)的基本概念入手，運(yùn)用了1種巧妙的方法——概率方法，即根據(jù)不等式的主要特征，結(jié)合概率論的1些基本概念和公式，通過(guò)建立1個(gè)適當(dāng)?shù)母怕誓Ｐ停�賦以1些隨機(jī)事件或隨機(jī)變量的具體含義，再利用概率論的理論加以證明，討論了柯西（Cauchy）不等式，級(jí)數(shù)不等式，詹森(Jensen)不等式和幾個(gè)1般不等式的證明。本文通過(guò)對(duì)利用概率方法證明不等式進(jìn)行了分析，從而得出：概率方法可以為抽象的數(shù)學(xué)問(wèn)題提供具體的概率背景，同時(shí)還溝通了各數(shù)學(xué)分支之間的聯(lián)系；顯示了概率應(yīng)用的巧妙性和優(yōu)越性，大大簡(jiǎn)化了不等式的證明過(guò)程。

關(guān)鍵詞：隨機(jī)變量；概率方法；數(shù)學(xué)不等式 

Abstract

In analysis mathematics，some identifications of inequalities are often more complicated，It will be difficult to prove that if using algebra method，and concrete ocular meaning is more abstract．Therefore，in order to simplify the proving process of these inequalities．Though reading a lot of relevant resource，we begin with the basic concept of math，and use an ingenious way――probabilistic method, which means that according to the main features of inequality theory，combining the basic concepts and formulas of probability，through creating one suitable probability model，giving some concrete meanings of random events or random variables，proving through probability theory，we discuss the Cauchy inequality，Class inequality，Jensen inequality，and several common inequality’s proofs．We analyze the inequalities which is proved by probabilistic method，and at last we obtain that probability method can provide a concrete probability background for abstract mathematics branch，and greatly simplify the proving process of inequalities．

Keywords: random variable; probability methob; mathematical inequalities 

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