若a，b为实数，且|a-1|+ab−2=0，求1ab+1(a+1)(b+1)+1(a+2)(b+2)+…+1(a+2009)(b+2009)的值．

Question

若a，b为实数，且|a-1|+√(ab−2)=0，求(1)/(ab)+(1)/((a+1)(b+1))+(1)/((a+2)(b+2))+…+(1)/((a+2009)(b+2009))的值．

Answer 1

∵|a-1|+√(ab−2)=0，
∴a-1=0，ab-2=0，
解得a=1，b=2，
(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)/(2010)-(1)/(2011)
=1-(1)/(2011)
=(2010)/(2011)．