1/(1*3)+1/(3*5)+.+1/(2007*2009)
=1/2*(1-1/3)+1/2*(1/3-1/5)+.+1/2*(1/2007-1/2009)
=1/2*(1-1/3+1/3-1/5+.+1/2007-1/2009)
=1/2*(1-1/2009)
=1/2*2008/2009
=1004/2009