【超难】【难】数列极限

1)原式=lim[(k^2+1)/(n^2+1)]=lim[(k/n)^2+1/n^2]/(1+1/n^2)=(0+0)/(1+0)=0
2)原式=limn^2*{[1/n-1/(n+1)]+[1/n-1/(n+2)]+……+[1/n-1/(n+k)]}
=limn^2*[1/(n^2+n)+2/(n^2+2n)++……+k/(n^2+kn)]
=lim[1/(1+1/n)+2/(1+2/n)+……+k/(1+k/n)]
=1+2+……+k
=k(k+1)/2