1/1X2+1/2X3+1/3X4+1/4X5+...+1/1999X2000=?

1/1×2+1/2×3+1/3×4+1/4×5+...+1/1999×2000
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+…+1/1999-1/2000
=1-1/2000
=1999/2000
其规律是：
1/1×2=1/2,而1-1/2=1/2,即1/1×2=1-1/2；
1/2×3=1/6,而1/2-1/3=1/6,即1/2×3=1/2-1/3；
1/3×4=1/12,而1/3-1/4=1/12,即1/3×4=1/3-1/4；
……
故1/n×（n+1）=1/n - 1/n+1

1/1X2=1/2-1/3,1/2X3=1/2-1/3……………………1/1999X2000=1/1999-1/2000

1/1x2+1/2x3........+1/1999x2000
=1-1/2+1/2-2/3+2/3........-1/1999+1/1999-1/2000
=1-1/2000
=1999/2000
在1-1/2+1/2-2/3+2/3................-1/1999+1/1999这就全部抵消了

1/n*(n+1)