两个极限相等的有趣数列

证明了两个有趣数列{n~(1/2)∫_0~(π/2)sin~nxdx},{(n+1)~(1/2)∫_0~(π/2)sin~nxdx}的极限均为(π/2)~(1/2),且(π/2(n+1))~(1/2)∫_0~(π/2)sin~nxdx(π/2n)~(1/2).     

一个有趣数列的单调性

利用不等式(π4(n-1)/2n 4(n+))^(1/2)1<∫_0^(π/2)sin^nxdx<(π(4n+5)/2 (n+1)(4n+3))^(1/2),对有趣数列{(n+c)^(1/2)∫_0^(π/2)sin^nxdx}的单调性再次进行了分析论证,并对已有结论进行了改进.     

一个与Wallis不等式有关的单调减少数列

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~nxdxπ(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).     

推广的Wallis数列的单调性

讨论了推广的Wallis数列{(n+c)(1/2)∫π/20sin~nxdx}(n≥1,c为非负常数)的单调性.黄永忠等(2016)证明了当0≤c≤1/2该数列严格递增;当1/2c≤1该数列对于充分大的n严格递减.本文给出了此结论的一个新的简洁证明,并对相关问题做了讨论.进一步,证明了当且仅当c2π~2-16/16-π~2=0.609945…,推广的Wallis数列为严格递减数列.     

关于积的三个三角恒等式的推广

一易证下列三个恒等式成立: (1)sinθsin(θ+π/ 3)sin(θ+2π/ 3) =sin3θ/4; (2)cosθcos(θ+π/3)cos(θ+2π/3) =-1/4cos3θ; (3)tgθtg(θ+π/3)tg(θ+2π/3) =-tg3θ。本文把上述三个恒等式予以推广,其一般形式为: (Ⅰ) multiply form j=1 to n sin(θ+(j-1)/nπ)=sinnθ/2~(n-1); (Ⅱ) multiply form j=1 to n cos(θ+(j-1)/nπ) =(-1)~(n-2) sinnθ/2~(n/1) (n为偶数), (-1)~(n-1)~2 cosnθ/2~(n-1)(n为奇数);     

一个有极限的单调增加数列

证明了{n(16n^2+4n+3)/16n^2-4~n+3^(1/2) integral from 0 to π/2 sin^nxdx}为严格单调增加数列,且极限为π/2^(1/2),因而得π(16n^2+36n+23)/2(n+1)(16n^2+28n+15)^(1/2)相似文献   

RIGIDITY THEOREMS OF CLIFFORD TORUS

In this article, we prove that the Clifford torus S1(1-r2~(1/2)) × Sn-1(r) is the only closed hypersurface in the unit sphere Sn+1(1) with infinite fundamental group, which satisfy r2 ≥ (n-1)/n, RicM ≤ C-(H), and S ≤ S+(H). Moreover, we give a characterization of Clifford torus S1(1-r2~(1/2)) × Sn-1(r) with r2 = 2(n-1)+nH2±|H| n2H2+4(n-1)(1/2)/2n(1+H2) .     

求等差数前列n项和的一个公式

设a_0,a_2,…,a_n,a_(n+1),…为等差数列,其公差为d,则有公式 (?)a_i~3=(a_n·a_(n+1))~2+(a_1a_0)~2/4d 下面给出证明。给定n个等式。 (a_n~2+da_n)~2-(a_n~2-da_n)~3=4da_n~3; (a_(n-1)~2+da_(n-1))-(a_(n-1)~2-da_(n-1))~2=4da_(n-1)~3; (a_(n-2)~2+da_(n-2))~2-(a_(n-2)~3 2-da_(n-2))~2=4da_(n-2)~3,…, (a_3~2+da_3)~2-(a_3~2-da_3)~2=4da_3~3,     

2011年清华金秋营数学试题及解答

1.求sinπnsin2πn…sin(n-1)πn的值.解设ε=cosπn+isinπn(i为虚数单位),则1,ε,ε2,…,ε2(n-1)为x2n-1=0的根,且sinkπn=εk-ε-k2i=ε2k-12iεk,所以sinπnsin2πn…sin(n-1)πn=(ε2-1)(ε4-1)…[ε2(n-1)-1]2n-1in-1ε12n(n-1)()n-1(2)(4)…[2(n-1)]     

满足AB=αBA的方阵A,B和AB的特征值的性质及其联系

设m阶方阵A,B满足AB=αBA,其中α=e~(2kπi/n),k,n为互素整数且n≥2.证明了σ(AB)■{α~(j-((n-1)/2))λ_AλB|λA∈σ(A),λB∈σ(B),j=0,1,…,n-1}及其它相关的结果,其中σ(A)表示方阵A的所有特征值的集合.