归纳法证明1/2²+1/3²+1/(n+1)²>1/2-1/(n+2),n=k时不等式成立,n=k+1时应推得目标不等式为

假设n=k时不等式1/2²+1/3²+1/(k+1)²>1/2-1/(k+2)成立
n=k+1时应推得目标不等式为
1/2²+1/3²+1/(k+1)²+1/(k+2)² > 1/2 - 1/(k+3)
而1/2²+1/3²+1/(k+1)²+1/(k+2)²
> 1/2-1/(k+2) + 1/(k+2)²
> 1/2 - (k+1)/(k+2)²
所以要证明1/2²+1/3²+1/(k+1)²+1/(k+2)² > 1/2 - 1/(k+3)
证明(k+1)/(k+2)² 即证明(k+1)(k+3)展开,消去, 即证明0显然成立
所以归纳法第2步完成.

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+1）/(k+1)(k+3)>(k+1)/(k+2)² 即：1/(k+3)>(k+1)/(k+2)²
1/2-1/(k+3)∴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