证:n=k时不等式成立,即:1/2²+1/3²+1/(k+1)²>1/2-1/(k+2)
那么n=k+1时,1/2²+1/3²+1/(k+1)²+1/(k+1+1)²>1/2-1/(k+2)+1/(k+1+1)²
=1/2-(k+2)/(k+2)²+1/(k+2)²
=1/2-(k+1)/(k+2)²
∵(k+1)(k+3)<(k+2)²
∴ (k+1)/(k+1)(k+3)>(k+1)/(k+2)²即:1/(k+3)>(k+1)/(k+2)²
1/2-1/(k+3)<1/2-(k+1)/(k+2)²
∴n=k+1时,1/2²+1/3²+1/(k+1)²+1/(k+1+1)²>1/2-(k+1)/(k+2)²>1/2-1/(k+3)
命题得证