Question

用数学归纳法证明1／n+1／n+1+1／n+2+...+1／n^2＞1（n＞1且为正整数

如题

Answer 1

令N=2，
原式=1/2+1/3+1/4=13/12>1成立
设N=K时成立.
即1/K+1/(K+1)+...+1/K^2>1
当N=K+1时
原式=1/(K+1)+1/(K+2)+...+1/K^2+1/(K^2+1)+...+1/(K^2+2K+1)

1/(K+1)+1/(K+2)+...+1/K^2+1/(K^2+1)+...+1/(K^2+2K+1)-(1/K+...+1/K^2)
=1/(K^2+1)+...+1/(K^2+2K+1)-1/K
>(2K+1)/(K^2+2K+1)-1/K>0
所以原式成立