1×1/2=1/2=1-1/2
1/2×1/3=1/6=1/2-1/3
1/3×1/4=1/12=1/3-1/4
.
1/2005×1/2006=1/2005-1/2006
所以1×1/2+1/2×1/3+1/3×1/4……+1/2005×1/2006
=1-1/2+1/2-1/3+1/3-1/4+.+1/2005-1/2006
=1-1/2006
=2005/2006