题目

如果有理数a,b满足 |ab-2|+(1-b)^2=0,
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+3)(b+3)+.+1/（a+2007)(b+2007)的值.此题有规律可循吗?

提问时间：2020-12-06

答案

|ab-2|+（1-b）^2=0∴ab-2=01-b=0∴a=2b=1∴1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)+...+1/(a+2007)(b+2007）=1/1×2+1/2×3+1/3×4+1/4×5+……+1/2008×2009=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+……+1/2008-1/2009=1-1/2009=...