1/2+1/2*3+1/3*4+1/4*5+...+1/2000*2001
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/5-1/6)+……+(1/2000-1/2001)
=1-1/2+1/2-1/3+1/3-1/4+1/5-1/6+……+1/2000-1/2001
中间互相抵消
=1-1/2001
=2000/2001