答：
1*3分之1+3*5分之1...+2009*2011分之1+2011*2013分之1
=1/(1*3)+1/(3*5)+1/(5*7)+.+1/(2009*2011)+1/(2011*2013)
=(1/2)*(1-1/3+1/3-1/5+1/5-1/7+.+1/2009-1/2011+1/2011-1/2013)
=(1/2)*(1-1/2013)
=(1/2)*(2012/2013)
=1006/2013