Question

若n为大于1的自然数,求证：1/n+1+1/n+2+…+1/2n＞13/24

Answer 1

n为大于1的自然数
可以用数学归纳法来证：
（1）当n=2时
1/(2+1)+1/(2+2)=1/3+1/4=7/12=14/24>13/24成立
（2）假设当n=k时成立
即：1/(k+1)+1/(k+2)+1/(k+1)+---+1/(k+k)>13/24
那么当n=k+1时
1/(k+2)+1/(k+1)+---+1/(k+k)+1/(2k+1)+1/(2k+2)
=1/(k+1)+1/(k+2)+1/(k+1)+---+1/(k+k)+1/(2k+1)+1/(2k+2)-1/(k+1)
>13/24+1/(2k+1)+1/(2k+2)-1/(k+1)
>13/24+1/(2k+2)+1/(2k+2)-2/(2k+2)=13/24
说明当n=k+1时也成立
由（1）（2）可知不等式对于大于1的自然数都成立