Question

用数学归纳法证明不等式1/(n+1)+1/(n+2)+…+1/(n+n)>13/24

Answer 1

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
==========================================
附加告诉你两点：
第一：n=1时，不等式变为1/2+1/2>13/24,所以是成立的！
第二：这个不等式是泰勒级数，属于高等数学的东西。
有些加强命题，不妨试试看：
3/4>1/(n+1)+1/(n+2)+…+1/(n+n)>13/24
1/（n+1）
+
1/（n+2）+...+
1/2n≤7/10-1/(4n+1)