∵ n=2时S(2)=1/3+1/4=14/24>13/24,猜想m=13,即当n≥2时,S(n)≥13/24.下面用数学归纳法证明之:
(i) n=2时,S(2)=14/24>13/24, 猜想成立.
(ii) 假设n=k时,S(k)=1/(k+1)+1/(k+2)+…+1/(2k)>13/24,则当n=k+1时,S(k+1)=1/(k+2)+1/(k+3)+…+1/(2k)+1/(2k+1)+1/[2(k+1)]
=1/(k+1)+1/(k+2)+1/(k+3)+…+1/(2k)+1/(2k+1)+1/[2(k+1)]-1/(k+1)
=S(k)+1/(2k+1)+1/[2(k+1)]-1/(k+1)>13/24+1/(2k+1)+1/[2(k+1)]-1/(k+1)
∵ 1/(2k+1)+1/[2(k+1)]-1/(k+1)=1/(2k+1)-1/(2k+2)>0,
∴ S (k+1)>13/24,　即n=k+1时,猜想成立.
∴ 对一切n∈N+,猜想成立. ∴ m=13