题目

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)

提问时间：2021-01-28

答案

n^2{(k/n)-(1/n+1)-(1/n+2)-.-(1/n+k)}
=nn{(1/n-1/n+1)+(1/n-1/n+2)+……+(1/n-1/n+k)}
=nn{1/n(n+1)+2/n(n+2)+……+k/n(n+k)}
=n{1/(n+1)+2/(n+2)+……+k/(n+k)}
lim(n{1/(n+1)+2/(n+2)+……+k/(n+k)})
=
lim(n{1/(n+k)+2/(n+k)+……+k/(n+k)})
=k(k+1)/2*lim(n/(n+k))
=k(k+1)/2
=>
lim n^2{(k/n)-(1/n+1)-(1/n+2)-.-(1/n+k)}
=k(k+1)/2