x(x+1)分之一加(x+1)(x+2)分之一…一直加到(x+2005)(x+2006)分之一 等于多少？

来源：百度知道 编辑：UC知道 时间：2024/09/22 11:34:39
第一个给出正确答案的在追加100分 ，快，我要的速度

1/(x+n)(x+n+1)=1/(x+n)-1/(x+n+1)
裂项相消
原式=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-...-1/(x+2005)+1/(x+2005)
-1/(x+2006)
=1/x-1/(x+2006)
=2006/x(x+2006)

1/x(x+1)+1/(x+1)(x+2)+…+1/(x+2005)(x+2006)
=[1/x-1/(x+1)]+[1/(x+1)-1/(x+2)]+……+[1/(x+2005)-1/(x+2006)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+……+1/(x+2005)-1/(x+2006)
=1/x-1/(x+2006)
=2006/x(x+2006)

答案是1/X - 1/(X+2006)

现在我来给你写做题的步骤
因为1/X(X+1)=1/X-1/(X+1)
1/(X+1)*(X+2)=1/(X+1)-1/(X+2)
1/(X+2)*(X+3)=1/(X+2)-1/(X+3)
.....
.......
........
1/(X+2005)*(X+2006)=1/(X+2005)-1/(X+2006)
所以原式=1/X - 1/(X+2006)

1/[x(x+1)]+1/[(x+1)(x+2)]+......+1/[(x+2005)(x+2006)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+......+1/(x+2005)-1/(x+2006)
=1/x-1/(x+2006)

x(x+1)分之一加(x+1)(x+2)分之一…一直加到(x+2005)(x+2006)分之一
=[1/x-1/(x+1)]+[1/(x+1)-1/(x+2)]+[1/(x+2)-1/(x+3)]+……+[1/(x+2005)-1/(x+2006)]
=1/x-1/(x+2006)。