已知:a=1 b=2 求1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)...+1/(a+2007)(b +2007)的值

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已知:a=1b=2求1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)...+1/(a+2007)(b+2007)的值已知:a=1b=2求1/ab+1/(a+1)(b+1)+1/(a+2)(

已知:a=1 b=2 求1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)...+1/(a+2007)(b +2007)的值
已知:a=1 b=2 求1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)...+1/(a+2007)(b +2007)的值

已知:a=1 b=2 求1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)...+1/(a+2007)(b +2007)的值
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 + …… + 1/2008*2009
=(1-1/2)+(1/2-1/3)+……+(1/2008-1/2009)
=1-1/2+1/2-1/3+……+1/2008-1/2009
=1-1/2009
=2008/2009

因为b-a=1, 所以1/(a+n)-1/(b+n)=1/(a+n)(b+n)
1/ab+1/(a+1)(b+1)+1/(a+2)(b+2)...+1/(a+2007)(b +2007)
=1/a-1/b+1/(a+1)-1/(b+1)+1/(a+2)-1/(b+2)...+1/(a+2007)-1/(b +2007)
把b=a+1代进去就会发现,中间的项都消掉了,只剩下
1/a-1/(a+2008)
=2008/2009