1/1x3+1/2x4+1/3x5+...+1/2011x2013=1/2*(1-1/3+1/2-1/4+1/3-1/5+,+1/2010-1/2012+1/2011-1/2013)=1/2*(1+1/2-1/2012-1/2013)=1/2*(6071209/4050156)=6071209/8100312