一个与Wallis不等式有关的单调减少数列  

One Monotonic Decreasing Sequence Associated With The Wallis Inequality

作　　者：杨天虎[1,2] 李玉宏[2] YANG Tian-hu;LI Yu-hong(New Energy Engineering Department,Jiuquan Vocational and Technical College,Jiuquan Gansu 735000,China;Jiuquan Vocational and Technical College,Key Laboratory of the Solar Power System Engineering,Gansu Province,Jiuquan Gansu 735000,China)

机构地区：[1]酒泉职业技术学院新能源工程系,甘肃酒泉735000 [2]酒泉职业技术学院甘肃省太阳能发电系统工程重点实验室,甘肃酒泉735000

出　　处：《大学数学》2018年第5期48-53,共6页College Mathematics

基　　金：甘肃省科技创新平台专项资助(1505JTCF039);甘肃省科技计划项目(1309RTSF043)

摘　　要：证明了{n (64 n^3+16 n^2+72n+15)/64 n^3-16 n^2+72n-15^(1/2) integral from 0 to π/2 sin^nxdx}为严格单调减少数列,且极限为π/2^(1/2),因而得π(64 n^3-16 n^2+72n-15)/2n 64 n^3+16 n^2(+72n+15)^(1/2)<integral from 0 to π/2 sin^nxdx<π(64 n^3+208 n^2+296n+167)/2 n(+1)(64 n^3+176 n^2+232n+105)^(1/2),将Wallis不等式改进为512 n^3-64 n^2+144n-15/πn (512 n^3+64 n^2+144n+15)^(1/2)<2(n-1)!!/2(n)!!<512 n^3+832 n^2+592n+167/(πn+0.5)(512 n^3+704 n^2+464n+105)^(1/2).It was proved in this paper that n 64 n 3+16 n 2+72n+15 64 n 3-16 n 2+72n-15∫π2 0 sin n x d x is a strictly monotonic decreasing sequence,and the limit of valueπ2,thus we have the resultπ64 n 3-16 n 2+72n-15 2n 64 n 3+16 n 2+72n+15<∫π2 0 sin n x d x<π64 n 3+208 n 2+296n+167 2 n+1 64 n 3+176 n 2+232n+105,and the Wallis inequality is improved to 512 n 3-64 n 2+144n-15πn 512 n 3+64 n 2+144n+15<2n-1!!2n!!<512 n 3+832 n 2+592n+167πn+0.5 512 n 3+704 n 2+464n+105.

关 键 词：Wallis不等式 单调数列 数列极限 

分 类 号：O178[理学—数学]