1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+.1/(a+2009)(b+2009)
=1/(1×2）+1/(2×3）+.1/(2010×2011）
=1-1/2+1/2-1/3+1/3-1/4+.+1/2010-1/2011
=1-1/2011
=2010/2011