1\1*5+1\5*9+1\9*3+...+1\2009*2013
=1/4*(1-1/5)+1/4*(1/5-1/9)+1/4*(1/9-1/13)+.1/4*(1/2009-1/2013)
=1/4*(1-1/5+1/5-1/9+1/9-1/13+.+1/2009-1/2013)
=1/4*(1-1/2013)
=1/4*2012/2013
=503/2013