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 |
| Published: |
SpringerOpen
2025-02-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-025-03253-2 |
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| Summary: | 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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| ISSN: | 1029-242X |