用数学归纳法证明(1+2+…+n)(1+1/2+…+1/n)>=n^2

Question

用数学归纳法证明(1+2+…+n)(1+1/2+…+1/n)>=n^2

Answer 1

(1).当N=3时,左边=(1+2+3)*(1+1/2+1/3)=11
右边=3^2+3-1=11
左边=右边,原式成立
(2)设当N=K时原式成立,有(1+2+3+……+K)(1+1/2+1/3+……+1/K)≥K^2+K-1
当=k+1时(1+2+3+...+k+k+1)(1+1/2+1/3+.+1/k+1/(k+1))=
(1+2+3+...+k)(1+1/2+1/3+...+1/k)+(k+1)(1+1/2+1/3+...)+1/(k+1)(1+2+3+...+k)+1>
k^2+k-1+(k+1)+(k+1)/2+k(k+1)/2(k+1)+1>
k^2+2k+1+k+1-1=(k+1)^2+(k+1)-1
即当n=k+1时,不等式成立
由（1）（2）得,当N为正整数数且大于2时,原式成立
(1+2+3+...+k)(1+1/2+1/3+...+1/k)+ (k+1)(1+1/2+1/3+...)+1/(k+1)(1+2+3+...+k) +1>
k^2+k-1+ (k+1)+(k+1)/2+ k(k+1)/2(k+1) +1>
k^2+2k+1+k+1-1=(k+1)^2+(k+1)-1