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)]
问题描述：
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)]
答
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/x-1/(x+3)/3+[1/(x+3)-1/(x+6)]/3+...+[1/(x+27)-1/(x+30)]/3
=[1/x-1/(x+30)]/3
=[30/x(x+30)]/3
=10/x(x+30)
答
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+20...