如果有理数a，b满足|ab-2|+（1-b）2=0，试求1ab+1(a+1)(b+1)+1(a+2)(b+2)+…+1(a+2007)(b+2007)的值．

Question

如果有理数a，b满足|ab-2|+（1-b）²=0，试求(1)/(ab)+(1)/((a+1)(b+1))+(1)/((a+2)(b+2))+…+(1)/((a+2007)(b+2007))的值．

Answer 1

∵|ab-2|≥0，（1-b）²≥0，且|ab-2|+（1-b）²=0，
∴ab-2=0，且1-b=0，解得ab=2，且b=1，
把b=1代入ab=2中，解得a=2，
则(1)/(ab)+(1)/((a+1)(b+1))+(1)/((a+2)(b+2))+…+(1)/((a+2007)(b+2007))
=(1)/(2)+(1)/(3×2)+(1)/(4×3)+…+(1)/(2009×2008)
=（1-(1)/(2)）+（(1)/(2)-(1)/(3)）+（(1)/(3)-(1)/(4)）+…+（(1)/(2008)-(1)/(2009)）
=1-(1)/(2)+(1)/(2)-(1)/(3)+(1)/(3)-(1)/(4)+…+(1)/(2008)-(1)/(2009)
=1-(1)/(2009)
=(2008)/(2009)．