请根据等式(b-a)/(ab)=b/(ab)-a/(ab)=1/a-1/b将分式1/[(x+1)]化为两个分式之差的形式

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/12 06:24:36

请根据等式(b-a)/(ab)=b/(ab)-a/(ab)=1/a-1/b将分式1/[(x+1)]化为两个分式之差的形式.
用上述规律,计算：
1/[x(x+1)]1/[(x+1)(x+2)]+……+1/[(x+2007)(x+2008)]
和：
1/[x(x+3)]1/[(x+3)(x+6)]+……+1/[(x+27)(x+30)]

1/[x(x+1)]
=[(x+1)-x]/[x(x+1)]
=(x+1)/[x(x+1)]-x/[x(x+1)]
=1/x-1/(x+1)
1/[x(x+1)]1/[(x+1)(x+2)]+……+1/[(x+2007)(x+2008)]
=1/x-1/(x+1)-1/(x+1)-1/(x-2)+……+1/(x+2007)-1/(x+2008)
=1/x-1/(x+2008)
=2008/[x(x+2008)]
1/[x(x+3)]+1/[(x+3)(x+6)]+……+1/[(x+27)(x+30)]
=(1/3){3/[x(x+3)]+3/[(x+3)(x+6)]+……+3/[(x+27)(x+30)]}
=(1/3)[1/x-1/(x+3)+1/(x+3)-1/(x+6)+……+1/(x+27)-1/(x+30)]
=(1/3)[1/x-1/(x+30)]
=10/[x(x+30)]

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