6分之1+12分之1+20分之1

1/n(n+1)=1/n-1/(n+1)
1/6+1/12+1/20+1/30+……+1/2005*2006
=1/2*3+1/3*4+1/4*5+……1/2005*2006
=1/2-1/3+1/3-1/4+1/4-1/5+……+1/2005-1/2006
=1/2-1/2006
=1002/2006
=501/1003