题目

帮我用数学归纳法证明一道数学题
1/1*3+1/+……+1/〔2n-1〕*〔2n+1〕=n/2n+1
帮我看看,

提问时间：2021-04-02

答案

1/(1×3)+1/+……+1/[(2n-1)×(2n+1)]=n/(2n+1)
证：
当n=1时,1/(1×3)=1/(2×1-1),命题成立；
设当n=k时,命题成立,
即：1/(1×3)+1/+……+1/[(2k-1)×(2k+1)]=k/(2k+1)；
当n=k+1时,
1/(1×3)+1/+……+1/[(2k-1)×(2k+1)]+1/[(2k+1)(2k+3)]
=k/(2k+1)+1/[(2k+1)(2k+3)]
=k(2k+3)/[(2k+1)(2k+3)]+1/[(2k+1)(2k+3)]
=(2k^2+3k+1)/[(2k+1)(2k+3)]
=(2k+1)(k+1)/[(2k+1)(2k+3)]
=(k+1)/(2k+3)
=(k+1)/[2(k+1)+1]
即：
1/(1×3)+1/+……+1/[(2k-1)×(2k+1)]+1/[(2k+1)(2k+3)]=(k+1)/[2(k+1)+1]
命题成立.
所以：1/(1×3)+1/+……+1/[(2n-1)×(2n+1)]=n/(2n+1)
证毕.