1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+……+1/(x+2007)(x+2008)求当X=1时,该式的值请给出准确步骤

1/x(x+1)=1/x-1/(x+1)
1/(x+1)(x+2)=1/(x+1)-1/(x+2)
所以原式=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+2)+1/(x+3)....-1/(x+2007)+1/(x+2007)-1/(x+2008)
=1/x-1/(x+2008)
当x=1时，原式=1-1/2009=2008/2009

1/x*(x+1)=1/x-1/(x+1)
1/(x+1)*(x+2)=1/(x+1)-1/(x+2)
依次展开
1/(x+2007)*(x+2008)=1/(x+2007)-1/(x+2008)
原式=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)....-1/(x+2007)+1/(x+2007)-1/(x+2008)
=1/x-1/(x+2008)
当x=1 时
原式=2008/2009

根据裂项可知 如下：
1/X(X+1)=1/X-1/(X+1)
由此可知
原式=1/X-1/(X+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)+……+1/(x+2007)-1/(x+2008)=1/X-1/(X+2008)
把X=1代入
得：原式=2008/2009

裂项 课本上应该有

1/x(x+1)=1/x-1/(x+1)1/(x+1)(x+2)=1/(x+1)-1/(x+2)..1/(x+2007)(x+2008)=1/(x+2007)-1/(x+2008)所有式子相加得,右边两两相消1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+……+1/(x+2007)(x+2008)=1/x-1/(x+1)...

1/x(x+1)+1/(x+1)(x+2)+1/(x+2)(x+3)+1/(x+3)(x+4)+……+1/(x+2007)(x+2008)
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+....+1/(x+2007)-1/(x+2008)
=1/x-1/(x+2008)
=(x+2008-x)/[x(x+2008)]
=2008/[x(x+2008)]
当X=1时，该式的值:2008/(1+2008)=2008/2009

1/x*(x+1)=1/x-1/(x+1)
1/(x+1)*(x+2)=1/(x+1)-1/(x+2)
....
1/(x+2007)*(x+2008)=1/(x+2007)-1/(x+2008)
原式=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)....-1/(x+2007)+1/(x+2007)-1/(x+2008)
=1/x-1/(x+2008)
当x=1 原式=1-1/2009
=2008/2009