用数学归纳法证明：1n+1+1n+2+1n+3+…+12n＞1324（n≥2，n∈N*）

Question

用数学归纳法证明：(1)/(n+1)+(1)/(n+2)+(1)/(n+3)+…+(1)/(2n)＞(13)/(24)（n≥2，n∈N^*）

Answer 1

证明：（1）当n=2时，(1)/(2+1)+(1)/(2+2)＝(14)/(24)，(14)/(24)＞(13)/(24)命题成立．
（2）假设当n=k时，(1)/(k+1)+(1)/(k+2)+(1)/(k+3)+…+(1)/(2k)＞(13)/(24)成立
当n=k+1时，(1)/(k+2)+(1)/(k+3)+…+(1)/(2k)+(1)/(2k+1)+(1)/(2k+2)=(1)/(k+1)+(1)/(k+2)+(1)/(k+3)+…+(1)/(2k)+(1)/(2k+1)+(1)/(2k+2)−(1)/(k+1)＞(13)/(24)+(1)/(2k+1)+(1)/(2k+2)−(1)/(k+1)，
∵(1)/(2k+1)+(1)/(2k+2)−(1)/(k+1)＝(1)/(2(2k+1)(k+1))＞0，
∴(1)/((k+1)+1)+(1)/((k+1)+2)+(1)/((k+1)+3)+…+(1)/(2(k+1))＞(13)/(24)，
当n=k+1时命题成立．
所以对于任意n≥2，n∈N^*都成立．