a=1,b=2.
则1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+...+1/(a+2009)(b+2009)
=1/1x2+1/2x3+1/3x4+…x1/2010x2011
=1-1/2+1/2-1/3+1/3-1/4+……+1/2010-1/2011
=1-1/2011
=2010/2011