1/1*2+1/1*3+1*3*4+……+1/2000*1/2001

1/1*2+1/2*3+1/3*4+……+1/2000*1/2001
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+……+(1/2000-1/2001)
=1-1/2001
=2000/2001
本题利用裂项公式:1/n(n+1)=1/n-1/(n+1)