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
来源:学生作业帮助网 编辑:作业帮 时间:2024/08/28 01:02:25
![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](/uploads/image/z/2586669-69-9.jpg?t=%E5%BD%92%E7%BA%B3%E6%B3%95%E8%AF%81%E6%98%8E1%2F2%26sup2%3B%2B1%2F3%26sup2%3B%2B1%2F%28n%2B1%29%26sup2%3B%3E1%2F2-1%2F%28n%2B2%29%2Cn%3Dk%E6%97%B6%E4%B8%8D%E7%AD%89%E5%BC%8F%E6%88%90%E7%AB%8B%2Cn%3Dk%2B1%E6%97%B6%E5%BA%94%E6%8E%A8%E5%BE%97%E7%9B%AE%E6%A0%87%E4%B8%8D%E7%AD%89%E5%BC%8F%E4%B8%BA)
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
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
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
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
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 <...
全部展开
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 <1/(k+3) 就行了.
即证明(k+1)(k+3)<(k+2)²
展开,消去, 即证明0<1
显然成立
所以归纳法第2步完成.
收起