1/1*2+1/2*3+1/3*4+.+1/998*999+1/999*1000

1/1*2+1/2*3+1/3*4+.+1/998*999+1/999*1000
1/1*2+1/2*3+1/3*4+.+1/998*999+1/999*1000

1/1*2+1/2*3+1/3*4+.+1/998*999+1/999*1000
1/1*2+1/2*3+1/3*4+.+1/998*999+1/999*1000
=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+.+(1/998-1/999)+(1/999-1/1000)
中间正负抵消
=1-1/1000
=999/1000

没分不答

裂项法
1/1*2+1/2*3+1/3*4+...........+1/998*999+1/999*1000
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+....+(1/998-1/999)+(1/999-1/1000)中间约掉
=1-1/1000
=999/1000

1/1*2=1/1-1/2
1/2*3=1/2-1/3
1/3*4=1/3-1/4
……
……
原式＝1/1-1/2+1/2-1/3+1/3-1/4+……+1/998-1/999+1/999-1/1000
=1-1/1000
=999/1000