用数学归纳法证明1/(1x3)+1/(3x5)+1/(5x7)…1/(2n-1)(2n+1)=n/(2n+1)

问题描述:

用数学归纳法证明1/(1x3)+1/(3x5)+1/(5x7)…1/(2n-1)(2n+1)=n/(2n+1)
我证明完n=k+1后与结论不符,不知哪错了
当n=k时成立
即1/(1x3)+1/(3x5)+1/(5x7)…1/(2k-1)(2k+1)=k/(2k+1)
则n=k+1时
1/(1x3)+1/(3x5)+1/(5x7)…1/(2k-1)(2k+1)+1/(2k+1)(2k+3)
=k/(2k+1)+1/(2k+1)(2k+3)
=2k^2+3k+1/(2k+1)(2k+3)
=(k+1/2)(k+1)/(2k+1)(2k+3)
=(k+1)/2(2k+3)
而原式应为(k+1)/(2k+3)

=k/(2k+1)+1/(2k+1)(2k+3)=(2k+1)(k+1/(2k+3))=(2k+1)((2k方+3k+1)/(2k+3))=1/(2k+1)*((2k+1)(k+1)/(2k+3))=(k+1)/(2k+3)成立你算的=(2k^2+3k+1)/(2k+1)(2k+3)=(2k+1)(k+1)/(2k+1)(2k+...