解方程1/(x+1)(x+2)+1/(x+2)(x+3)+......+1/(x+2005)(x+2006)=1/2x+4012

1/(x+1)(x+2)+1/(x+2)(x+3)+......+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)
=2005/(x+1)(x+2006)
即原式化为：2005/(x+1)(x+2006)=1/2（x+2006）
解得x=4009

左=1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)...
=1/(x+1)-1/(x+2006)
=右=1/2(x+2006)
(你是不是打错了？=1/（2x+4012）括号没打？)
所以2x+4012=3x+3
x=4009

1/(x+1)(x+2)+1/(x+2)(x+3)+......+1/(x+2005)(x+2006)=1/(2x+4012)
1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+......+1/(x+2005)-1/(x+2006))=1/(2x+4012)
1/(x+1)-1/(x+2006)=1/(2x+4012)
2005/(x+1)(x+2006)=1/2（x+2006）
4010/(x+1)(x+2006)=1/（x+2006）
4010/(x+1)(x+2006)=x+1/(x+1)(x+2006)
所以4010=x+1
x=4009
注：1/(x+1)(x+2)+1/(x+2)(x+3)+......+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)
=2005/(x+1)(x+2006)
公式：1/a(a+1)=1/a-1/(a+1)