A sharp double inequality for sums of powers revisited

Abstract The proof of the monotonicity of the sequence n ↦ n Δ ( n ) $n\mapsto n\Delta (n)$ , 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 ) : = ∑ k = 1 n ( k n ) n = ∑ j = 0 n ( 1 − j n ) n $S(n)...

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Main Author: Vito Lampret
Format: Article
Language:English
Published: SpringerOpen 2025-02-01
Series:Journal of Inequalities and Applications
Subjects:
Online Access:https://doi.org/10.1186/s13660-025-03253-2
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author Vito Lampret
author_facet Vito Lampret
author_sort Vito Lampret
collection DOAJ
description Abstract The proof of the monotonicity of the sequence n ↦ n Δ ( n ) $n\mapsto n\Delta (n)$ , 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 ) : = ∑ k = 1 n ( k n ) n = ∑ j = 0 n ( 1 − j n ) n $S(n):=\sum _{k=1}^{n}\left (\frac{k}{n}\right )^{n}=\sum _{j=0}^{n} \left (1-\frac{j}{n}\right )^{n}$ , the sequence n ↦ n ( e e − 1 − S ( n ) ) $n\mapsto n\left (\frac{e}{e-1}-S(n)\right )$ is strictly increasing.
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publishDate 2025-02-01
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series Journal of Inequalities and Applications
spelling doaj-art-331572b411e840cd90a551ca0a95d2e22025-02-09T12:59:34ZengSpringerOpenJournal of Inequalities and Applications1029-242X2025-02-012025111110.1186/s13660-025-03253-2A sharp double inequality for sums of powers revisitedVito Lampret0Faculty of Civil and Geodetic Engineering, University of LjubljanaAbstract The proof of the monotonicity of the sequence n ↦ n Δ ( n ) $n\mapsto n\Delta (n)$ , 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 ) : = ∑ k = 1 n ( k n ) n = ∑ j = 0 n ( 1 − j n ) n $S(n):=\sum _{k=1}^{n}\left (\frac{k}{n}\right )^{n}=\sum _{j=0}^{n} \left (1-\frac{j}{n}\right )^{n}$ , the sequence n ↦ n ( e e − 1 − S ( n ) ) $n\mapsto n\left (\frac{e}{e-1}-S(n)\right )$ is strictly increasing.https://doi.org/10.1186/s13660-025-03253-2EstimateEuler’s numberInequalityLimitMonotone sequenceRate of convergence
spellingShingle Vito Lampret
A sharp double inequality for sums of powers revisited
Journal of Inequalities and Applications
Estimate
Euler’s number
Inequality
Limit
Monotone sequence
Rate of convergence
title A sharp double inequality for sums of powers revisited
title_full A sharp double inequality for sums of powers revisited
title_fullStr A sharp double inequality for sums of powers revisited
title_full_unstemmed A sharp double inequality for sums of powers revisited
title_short A sharp double inequality for sums of powers revisited
title_sort sharp double inequality for sums of powers revisited
topic Estimate
Euler’s number
Inequality
Limit
Monotone sequence
Rate of convergence
url https://doi.org/10.1186/s13660-025-03253-2
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