1/2*1+1/3*2+1/4*3+.+1/2006*2005
=(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