A sharp double inequality for sums of powers revisited

The proof of the monotonicity of the sequence $n\mapsto n\Delta (n)$ , presented in the 2011 article “A Sharp double inequality for sums of powers” by V. Lampret, is corrected. Namely, it is demonstrated that, for $S(n):=\sum _{k=1}^{n}\left (\frac{k}{n}\right )^{n}=\sum _{j=0}^{n} \left (1-\frac{j}{n}\right )^{n}$ , the sequence $n\mapsto n\left (\frac{e}{e-1}-S(n)\right )$ is strictly increasing.