证明不等式1/(n+1)+1/(n+2)+…+1/(n+n)>13/24

Question

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)=1/213/24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