(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+（2000*2000+2001*2001)/(2000*2001)

可以观察其中任一项n，
(n*n+(n+1)*(n+1))/n(n+1)=n/(n+1)+(n+1)/n=1-1/(n+1)+1+1/n=2+1/n-1/(n+1)
所以(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+（2000*2000+2001*2001)/(2000*2001)
=2+1-1/2+2+1/2-1/3+......+2+1/2000-1/2001
=2*2000+1-1/2001
=4001-1/2001
=800600/2001
(不懂请追问^_^o~ 努力！)

(1*1+2*2)/(1*2)+(2*2+3*3)/(2*3)+……+（2000*2000+2001*2001)/(2000*2001)
=(1/2+2/1)+(2/3+3/2)+.(2000/2001+2001/2000)
=(1-1/2+1+1/1)+(1-1/3+1+1/2)+...(1-1/2001+1+1/2000)
=(2-1/2+1/1)+(2-1/3+1/2)+...(2-1/2001+1/2000)
=2*2000+1-1/2001
=4001-1/2001
=4000又2000/2001