超难问题！！ 呵呵 来考你们哦～！

来源：百度知道 编辑：UC知道 时间：2024/07/01 02:14:28
what is the value of :
(x+1)(x+2006)[1/(x+1)(x+2)+...+1/(x+2005)(x+2006)]?

1/(x+1)(x+2)=1/(x+1)-1/(x+2)
1/(x+2)(x+3)=1/(x+2)-1/(x+3)
1/(x+1)(x+2)+...+1/(x+2005)(x+2006)
=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)....+1/(x+2005)-1/(x+2006)
=1/(x+1)-1/(x+2006)
(x+1)(x+2006)[1/(x+1)(x+2)+...+1/(x+2005)(x+2006)]
=x+2006-(x+1)
=2005

(x+1)(x+2006)[1/(x+1)(x+2)+...+1/(x+2005)(x+2006)]

==(x+1)(x+2006)[1/(x+1) - 1/(x+2) + …… + 1/(x+2005) -
1/(x+2006)]

==(x+1)(x+2006)[1/(x+1) - 1/(x+2006)]

==(x+1)(x+2006) * [（x+2006）-（x+1）]/(x+1)(x+2006)

==2005

这里用到了裂项求和

(x+1)(x+2006)[1/(x+1)(x+2)+...+1/(x+2005)(x+2006)]
=(x+1)(x+2006)[(1/(x+1)-1/(x+2006)]
=x+2006-(x+1)
=2005

这题我见过，你写错了吧

2005一眼就看出来了！

悲哀呀