解题思路：1+2=2*3/2
1/(1+2)=2/(2*3)=2*(1/2-1/3)
1+2+3=3*4/2
1/(1+2+3)=2/(3*4)=2*(1/3-1/4)
………………
1+2+3+……+2000=2001*2000/2
1/(1+2+3+……+2000)=2*(1/2000-1/2001)
1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+...+1/(1+2+3+...+2000)
=2/(2*3)+2/(3*4)+2/(4*5)+……+2/(2000*2001)
=2[1/2-1/3+1/3-1/4+1/4-1/5+……+1/1998-1/1999+1/1999-1/2000+1/2000-1/2001)
=2(1/2-1/2001)
=1-2/2001)
=1999/2001