Question

用放缩法证明两个等式谢了

1.n(n+1)/2<根号1*2+根号2*3+……+根号n(n+1)<(n+1)^2/2
2.1+1/根号2+1/根号3+……+1/根号n<2*根号n
xiele

Answer 1

1.
根号1*2+根号2*3+……+根号n(n+1)
<根号2*2+根号3*3+……+根号(n+1)(n+1)
=2+3+……+(n+1)=(n+1)^2/2,
根号1*2+根号2*3+……+根号n(n+1)
>根号1*1+根号2*2+……+根号n*n
=1+2+……+n=n(n+1)/2, 得证。
2.
1/根号n=2/(2根号n)<2/(根号n+根号(n-1))=2(根号n-根号(n-1))
1+1/根号2+1/根号3+……1/根号n
<2(1-0)+2(根号2-1)+……+2(根号n-根号(n-1))
=2根号n