Limit[Sum[k/((k-1) n),{k,2,n+1}],n->Infinity]
Assuming limit refers to a continuous limit | Use the instead

Limit

lim_(n->∞) sum_(k=2)^(n + 1) k/((k - 1) n) = 1

Series expansion at n=∞

1 + (log(n) + gamma )/n + 1/(2 n^2) - 1/(12 n^3) + 1/(120 n^5) + O((1/n)^6) (generalized Puiseux series)
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