关于高二一道数列证明题令bn=lnn/n^2,前n项和为Tn,求证:Tn〈(2n^2-n-1)/(4n+4)

Question

关于高二一道数列证明题
令bn=lnn / n^2,前n项和为Tn,求证:
Tn〈 (2n^2-n-1) / (4n+4)

Answer 1

归纳法
设n成立,n+1时只要证明
(2(n+1)^2-n-2)/(4n+8)-(n^2-n-1)/(4n+4)>bn+1=ln(n+1)/(n+1)^2就行了
化简一下要证
(n^2+3n+1)/2(n+2)>ln(n+1)/(n+1)
因为2(n+2)ln(n+1)/(n+1)
(n^2+3n+1)/3>ln(n+1)
这是显然的,事实上n>ln(n+1)