1/1x3+1/3x5+1/5x7..+1/2009x2011
=1/2x[(1-1/3)+(1/3-1/5)+(1/5-1/7)+...+(1/2009-1/2011)]
=1/2x(1-1/2011)
=1/2x2010/2011
=1005/2011