lim n^2{(k/n)-(1/n+1)-(1/n+2)-....-(1/n+k)}(其中K是与N无关的常数)

{(k/n)-(1/n+1)-(1/n+2)-....-(1/n+k)}
=[1/n-1/(n+1)]+[1/n+1/(n+2)]+[1/n+1/(n+3)]+...+[1/n+1/(n+k)]
=1/[n^2+n]+2/[n^2+2n]+3/[n^2+3n]+...+k/[n^2+kn]
所以n^2{(k/n)-(1/n+1)-(1/n+2)-....-(1/n+k)}
=n^2/[n^2+n]+2n^2/[n^2+2n]+3n^2/[n^2+3n]+...+kn^2/[n^2+kn]
=1/[1+1/n]+2/[1+2/n]+3/[1+3/n]+...+k/[1+k/n]
所以当n取无穷大的时候，1/n,2/n,3/n,...k/n都为0
此时原式=1+2+3...+k=k(k+1)/2
所以lim n^2{(k/n)-(1/n+1)-(1/n+2)-....-(1/n+k)}=k(k+1)/2
【此处我假设了n趋向于无穷大】