A sharp double inequality for sums of powers (Q535537)

Let \(S(n)=\sum_{k=1}^n(k/n)^n\) and \(\Delta(n)=e/(e-1)-S(n)\). Using Tannery's theorem for complex series, the author first reproves that \(\Delta(n)\to 0\) as \(n\to\infty\). It is shown that sequences with general terms \(S(n)\) and \(n\Delta(n)\) are strictly increasing. Moreover, some inequalities concerning \(S(n)\), like (in terms of \(\Delta(n)\)) \(1/(2n)<\Delta(n)<1/n\) holding for \(n\geq 1\), are obtained.