证明k＞7且k∈N,存在1/n + 1/n+1 + 1/n+2 + ……… + 1/kn-1 ＞ 3/2

Question

证明k＞7且k∈N,存在1/n + 1/n+1 + 1/n+2 + ……… + 1/kn-1 ＞ 3/2

Answer 1

原不等式左边=
1/n +1/（n+1）+1/（n+2）+…… +1/2n+
1/(2n+1)+1/(2n+2）+…… +1/3n+
1/(3n+1)+1/(3n+2) +…… +1/4n+
1/(4n+1)+ …… +1/5n+
1/(5n+1)+ …… +1/6n+
1/(6n+1)+ …… +1/7n+
1/(7n+1)+ …… +1/(8n-1)+1/8n+
1/(8n+1)+ …… +1/(kn-1)
因为
1/（n+1）+1/（n+2）+…… +1/2n>
1/2n+1/2n+1/2n +……1/2n(总共n个）=
n*1/2n=1/2
即1/（n+1）+1/（n+2）+…… +1/2n>1/2
同理可得
1/(2n+1)+1/(2n+2）+…… +1/3n>1/3
1/(3n+1)+1/(3n+2) +…… +1/4n>1/4
1/(4n+1)+ …… +1/5n>1/5
1/(5n+1)+ …… +1/6n>1/6
1/(6n+1)+ …… +1/7n>1/7
1/(7n+1)+ …… +1/(8n-1)+1/8n>1/8
以上所得不等式总记为M
当K>7,即k>=8时,
1/n + 1/n+1 + 1/n+2 + ……… + 1/kn-1>=
1/n +1/（n+1）+1/（n+2）+…… +1/2n+
1/(2n+1)+1/(2n+2）+…… +1/3n+
1/(3n+1)+1/(3n+2) +…… +1/4n+
1/(4n+1)+ …… +1/5n+
1/(5n+1)+ …… +1/6n+
1/(6n+1)+ …… +1/7n+
1/(7n+1)+ …… +1/(8n-1)
(上式由k分别2,3,4,5,6,7,8所得,当k>=8显然成立）
由上面所推导出的M式得
1/n + 1/n+1 + 1/n+2 + ……… + 1/kn-1>
1/n+(1/2+1/3+1/4+1/5+1/6+1/7+1/8)-1/8n>
1/2+1/3+1/4+1/5+1/6+1/7+1/8>
1/2+(1/4+1/4)+(1/8+1/8+1/8+1/8)=
1/2+1/2+1/2=3/2
综上可知,当k＞7且k∈N,存在1/n + 1/n+1 + 1/n+2 + ……… + 1/kn-1 ＞ 3/2,得证.