eΠ等无理数的有理逼近问题 数学和应用数学专业论文

摘 要ⅇ和π是人们不断研究探讨的两个特殊的无理数，本文将这两个数结合在一起对其的有关性质进行了分析讨论，讨论了ⅇ和π的无理性。根据实数的完备性可以知道无理数是可以用有理数的极限逼近表示的。进而用傅里叶级数展开和夹逼定理等方法，研究一些有理数列的极限，探讨有关ⅇ和π用有理数列极限逼近的问题。ⅇ、π相关的逼近问题研究结果可以推广到其他一些无理数用有理数列极限逼近问题，加深ⅇ和π此类特殊无理数有深刻的理解和熟练的应用，也同时可以帮助在一些有理数列极限的证明方面获得更多思路。关键字：ⅇ和π 无理数 有理逼近 极限Abstractⅇand π are two special irrational numbers that have been studied and discussed continuously for many years. This paper analyzes and discusses the properties ofⅇ and π including their irrationality. According to the completeness of the real numbers, we know that irrational numbers can be represented by the limit approximation of rational numbers. We can use the Fourier series expansion, Squeeze theorem and other methods to study the limit of some rational number sequences and discuss the problem of using the limit of rational number sequences to approximate ⅇ and π. The results of the approximation problem research on ⅇand πcan be extended to some other problems of using the limit of rational number sequences to approximate irrational number, which can deepen our understanding and application of special irrational numbers such as ⅇ andπ also help ourselves generate more ideas on proving the limit of rational number sequences at the same time. Key words: ⅇ、π ,irrational, rational approximation, limit目 录摘 要..............................................................................................2Abstract............................................................................................21引言.....................................................................................