1/1x2+1/2x3+...+1/2009x2010+1/2010x2011
=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...+(1/2009-1/2010)+(1/2010-1/2011)
=1+1/2011
=2012/2011