令bn=(2^(n-1)+1)/((3n-2)an)数列{bn^2}的前n项和为tn,证明对于任意的n∈N+,都有tn
来源:学生作业帮 编辑:拍题作业网作业帮 分类:数学作业 时间:2024/05/16 20:14:34
令bn=(2^(n-1)+1)/((3n-2)an)数列{bn^2}的前n项和为tn,证明对于任意的n∈N+,都有tn
(1) = 2/4 = 1/2,
t(1) = 1/4 = 3/12 < 5/12.
n>=2时,b(n)= [2^(n-1)+1]/[(3n-2)a(n)] = 1/(3n-2).
[b(n)]^2 = 1/(3n-2)^2,
t(2) = [b(1)]^2 + [b(2)]^2 = 1/4 + 1/16 = 5/16 < 5/12.
n>=3时,3(n-1)-2 >= 4.
[b(n)]^2 = 1/(3n-2)^2 < 1/[(3n-2)(3n-5)] =(1/3) [1/(3n-5) - 1/(3n-2)]
t(n) = [b(1)]^2 + [b(2)]^2 + [b(3)]^2 + ...+ [b(n-1)]^2 + [b(n)]^2
= 1/4 + 1/16 + (1/3)[1/4 - 1/7 + 1/7 - 1/10 + ...+ 1/(3n-8)-1/(3n-5) + 1/(3n-5) - 1/(3n-2)]
= 5/16 + (1/3)[1/4 - 1/(3n-2)]
< 5/16 + 1/12
= 15/48 + 4/48
= 19/48
< 20/48
= 5/12.
综合,有,t(n) < 5/12
t(1) = 1/4 = 3/12 < 5/12.
n>=2时,b(n)= [2^(n-1)+1]/[(3n-2)a(n)] = 1/(3n-2).
[b(n)]^2 = 1/(3n-2)^2,
t(2) = [b(1)]^2 + [b(2)]^2 = 1/4 + 1/16 = 5/16 < 5/12.
n>=3时,3(n-1)-2 >= 4.
[b(n)]^2 = 1/(3n-2)^2 < 1/[(3n-2)(3n-5)] =(1/3) [1/(3n-5) - 1/(3n-2)]
t(n) = [b(1)]^2 + [b(2)]^2 + [b(3)]^2 + ...+ [b(n-1)]^2 + [b(n)]^2
= 1/4 + 1/16 + (1/3)[1/4 - 1/7 + 1/7 - 1/10 + ...+ 1/(3n-8)-1/(3n-5) + 1/(3n-5) - 1/(3n-2)]
= 5/16 + (1/3)[1/4 - 1/(3n-2)]
< 5/16 + 1/12
= 15/48 + 4/48
= 19/48
< 20/48
= 5/12.
综合,有,t(n) < 5/12
数列bn的前n项和为Tn,6Tn=(3n+1)bn+2,求bn
数列bn=1/(n^2)+1 前n项和为Tn,求证:对于任意正整数n 都有 Tn
an=3*2^(n-1),设bn=n/an求数列bn的前n项和Tn
已知数列{an}前n项和Sn=n^2+n,令bn=1/anan+1,求数列{bn}的前n项和Tn
数列{an}的前n项和为Sn=n平方+n,(1)求an,(2)令bn=2的an次方,证明bn为等比数列,并求前n项和Tn
设bn=(an+1/an)^2求数列bn的前n项和Tn
Sn=n^2,令bn=1/anan+1,Tn是数列bn的前n项和,试证明Tn
已知数列an的前n项和Sn=n^2,设bn=an/3n,记数列bn的前n项和为Tn,求证Tn=1-(n+1)/3^n
已知数列an的前n项和为sn=2n^2+5n+1,数列bn的前n项和tn满足Tn=(3/2)bn-3/2 求数列an的通
已知数列bn满足bn=b^2n,其前n项和为Tn,求(1-bn)/Tn
数学题:等差数列an,bn的前n项和分别为Sn,Tn,对于任意自然数n都有Sn/Tn=2n-3/4n-3,则a6/b6=
等差数列{An},{Bn}的前n项和为Sn与Tn,若Sn/Tn=2n/3n+1,则An/Bn的值是?