求极限 n趋向于无穷大 lim n^2[k/n-1/(n+1)-1/(n+2)-……-1/(n+k)]

题目

求极限 n趋向于无穷大 lim n^2[k/n-1/(n+1)-1/(n+2)-……-1/(n+k)]

答案

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(n+1)] + 2 / [n (n+2)] + 3 /[n(n+3)] + .+ k / [n(n+k)]
原式 = lim(n->∞) n² { 1/[n(n+1)] + 2 / [n (n+2)] + 3 /[n(n+3)] + .+ k / [n(n+k)] }
= 1 + 2 + 3 + .+ k
= k(k+1)/2