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

题目

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

答案

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