因为,a、b为有理数,且|a-1|+（ab-2）²=0
所以,a-1=0且ab-2=0,
解之得a=1,b=2,代入所求式得
所求式=1/(1x2)+1/(2x3)+1/(3x4)+.+1/(2010x2011)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+.+(1/2010-1/2011)
=1+(-1/2+1/2)+(-1/3+1/3)+(-1/4+1/4)+.+(-1/2010+1/2010)-1/2011
=1-1/2011
=2010/2011
a+1009应该是a+2009吧