1+2+...+n=n*(n+1)/2
1/(1+2+..+n)=2/[n*(n+1)]=2*[1/n-1/(n+1)]
原式
=2*(1/2-1/3)+2*(1/3-1/4)+...+2*(1/2000-1/2001)
=2*(1/2-1/3+1/3-1/4+...+1/2000-1/2001)
=2*(1/2-1/2001)
=1-2/2001
=1999/2001