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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| Format: | Article |
| Language: | English |
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SpringerOpen
2025-02-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-025-03253-2 |
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| _version_ | 1823861456482861056 |
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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. |
| format | Article |
| id | doaj-art-331572b411e840cd90a551ca0a95d2e2 |
| institution | Kabale University |
| issn | 1029-242X |
| language | English |
| publishDate | 2025-02-01 |
| publisher | SpringerOpen |
| record_format | Article |
| 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 |
| work_keys_str_mv | AT vitolampret asharpdoubleinequalityforsumsofpowersrevisited AT vitolampret sharpdoubleinequalityforsumsofpowersrevisited |