A sharp double inequality for sums of powers

Abstract

It is established that the sequences n S (n): = ∑k = 1 n (k / n)n and n n (e / (e - 1) - S (n)) are strictly increasing and converge to e / (e - 1) and e (e + 1) / 2 (e - 1) 3, respectively. It is shown that there holds the sharp double inequality (1 / (e - 1)) (1 / n)≤ e / (e - 1) - S (n)