Question

用数学归纳法证明:1/n+1/(n+1)+1/(n+2)+…+1/n^2大于1（n大于1）

Answer 1

1/n+1/(n+1)+1/(n+2)+…+1/n^2>1(n>1且n是整数)
证明:
(1)当n=2,
1/2+1/3+1/4=13/12>1成立
(2)假设当n=k时,
1/n+1/(n+1)+...+1/n^2>1
所以:
1/n+1/(n+1)+...+1/k^2>1
所以当n=k+1时,有:
1/n+1/(n+1)+...+1/k^2+1/(k^2+1)+1/(k^2+2)+...+1/(k^2+2k+1)
>1+1/(k^2+1)+1/(k^2+2)+1/(k^2+2k+1)
因为:
1/(k^2+1)+1/(k^2+2)+...+1/(k^2+2k+1)>0
所以:
1/n+1/(n+1)+...+1/k^2+1/(k^2+1)+1/(k^2+2)+...+1/(k^2+2k+1)
>1+0
=1
所以当n=k+1原式也成立
综上,有:
1/n+1/(n+1)+1/(n+2)+…+1/n^2>1(n>1且n是整数)
没说是整数就不能用数学归纳法