一个重要极限limn→∞n^k/an=0,（a〉1,k∈N）的多种证法  

Many Testimonies of An Important Limitation of ＂limn→∞n^k/an=0,（a〉1,k∈N）＂

作　　者：胡其明 HU Qi-ming （Xingyi Normal University for Nationalities,Xingyi ,Guizhou 562400,China）

机构地区：兴义民族师范学院,贵州兴义562400

出　　处：《兴义民族师范学院学报》2010年第1X期102-103,107,共3页Journal of Minzu Normal University of Xingyi

摘　　要：在数列极限中，有一个使用频率较高的极限limn→∞n^k/an=0,（a〉1,k∈N），而在一般微积分的教材中又没有给出证明，仅有少数教材对该极限的特殊情形“limn→∞n^k/an=0,（a〉1,k∈N）”进行了证明。为了加深对该极限的理解和应用，这篇文章用七种不同的方法对该极限进行了详细证明，体现了证明数列极限的灵活性和多样性。通过对证明方法的归纳和总结，可以拓展证明数列极限的思路，起到举一反三的作用。In one of the number sequence limitations, limn→∞n^k/an=0,（a〉1,k∈N）is used with greater frequency and hasn＇t been testified to in general calculus course books, and its special case of ＂limn→∞n^k/an=0,（a〉1,k∈N）＂ has been testified by only a few course books. To furtherly comprehend and apply it, seven approaches will be shown to testify to this limitation in the essay, which reveals the flexibility and diversity of testifying number sequence limitations so as to expand the ways for the testimony of number sequence limitations and infer other things from one fact.

关 键 词：极限 “ε-N”语言 单调有界 海涅定理 洛必达法则 收敛级数 迫敛性 斯笃茨定理 

分 类 号：O171[理学—数学;理学—基础数学]