1\2*5+1\5*8+8*11.+1\2003*2006+1\2006*2009
=1/3*(1/2-1/5)+1/3*(1/5-1/8)+1/3*(1/8-1/11)+.+1/3*(1/2003-1/2006)+1/3*(1/2006-1/2009)
=1/3*(1/2-1/5+1/5-1/8+1/8-1/11+.+1/2003-1/2006+1/2006-1/2009)
=1/3*(1/2-1/2009)
=1/3*2007/4018
=669/4018