1/a（a+1）+1/(a+1)(a+2)+...+1/(a+2007)(a+2008)这题怎么做?

1/a（a+1）+1/(a+1)(a+2)+...+1/(a+2007)(a+2008)这题怎么做?
1/a（a+1）+1/(a+1)(a+2)+...+1/(a+2007)(a+2008)这题怎么做?

1/a（a+1）+1/(a+1)(a+2)+...+1/(a+2007)(a+2008)这题怎么做?
1/a（a+1）=1/a-1/(a+1)
所以1/a（a+1）+1/(a+1)(a+2)+...+1/(a+2007)(a+2008)=1/a-1/(a+1)+1/(a+1)-1/(a+2)+...+1/(a+2007)-1/(a+2008)中间项消掉了
=1/a-1/(a+2008)=2008/a(a+2008)

1/a（a+1）=1/a-1/（a+1）
1/(a+1)(a+2)=1/(a+1)-1/(a+2)
................
1/(a+2007)(a+2008)=1/(a+2007)-1/(a+2008)
裂项后
1/a－1/(a+2008)＝2008／a(a+2008)
注释：1/n(n+1)= 1/n - 1/(n+1)

1/a（a+1）+1/(a+1)(a+2)+...+1/(a+2007)(a+2008)
=1/a-1/(a+1) + 1/(a+1)-1/(a+2)+....+...1/(a+2007)-1/(a+2008)
=1/a+(-1/(a+1)+ 1/(a+1))+(-1/(a+2)+....+...1/(a+2007))-1/(a+2008)
=1/a -1/(a+2008)
原理 ：1/n(n+1)= 1/n - 1/(n+1) 对每个因子 这样展开，那么就会出现相互抵消的情况了