Question

证明不等式1/(n+1)+1/(n+2)+…+1/(n+n)>13/24

证明不等式1/(n+1)+1/(n+2)+…+1/(n+n)>13/24

Answer 1

确实有点问题，只有在n≥2的情况下才可能

数学归纳法:
①n=1时，不知道是不是这样：
左边=1/(1+1)=1/2<13/24=右边， 命题不成立
n=2时，1/(2+1)+1/(2+2)=14/24)>13/24命题成立
n=3时，1/(3+1)+1/(3+2)+1/(3+3)=74/120>13/24命题成立
②假设对n=k-1命题成立(k≥3),
即1/(k+1)+1/(k+2)+…+1/(k+k)>13/24，则
1/(k+1)+1/(k+2)+…+1/(k+k)
=1/k+1/(k+1)+1/(k+2)+…+1/(k+k-2) +1/(k+k)+1/(k+k-1)-1/k
>1/k+1/(k+1)+1/(k+2)+…+1/(k+k-2)+[1/(2k)+1/(2k)-1/k]
=1/k+1/(k+1)+1/(k+2)+…+1/(k+k-2)
>13/24

Answer 2

数学归纳法证明:
①n=1是显然成立。
②假设对n=k-1成立,即1/(k+1)+1/(k+2)+…+1/(k+k)>13/24，则
1/(k+1)+1/(k+2)+…+1/(k+k)
=1/k+1/(k+1)+1/(k+2)+…+1/(k+k-2) +1/(k+k)+1/(k+k-1)-1/k
>1/k+1/(k+1)+1/(k+2)+…+1/(k+k-2)+[1/(2k)+1/(2k)-1/k]
=1/k+1/(k+1)+1/(k+2)+…+1/(k+k-2)
>13/24

综合①②得1/(n+1)+1/(n+2)+…+1/(n+n)>13/24

Answer 3

该式不成立：

取n=1
原式左边=1/2<13/24