A sharp double inequality for sums of powers revisited
The proof of the monotonicity of the sequence n ↦ n Δ ( 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 , the sequence n ↦ n ( e e − 1 − S (...
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| Published in | Journal of inequalities and applications Vol. 2025; no. 1; pp. 14 - 11 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
06.02.2025
Springer Nature B.V SpringerOpen |
| Subjects | |
| Online Access | Get full text |
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| Abstract | The proof of the monotonicity of the sequence
n
↦
n
Δ
(
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
, the sequence
n
↦
n
(
e
e
−
1
−
S
(
n
)
)
is strictly increasing. |
|---|---|
| AbstractList | The proof of the monotonicity of the sequence n↦nΔ(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=1n(kn)n=∑j=0n(1−jn)n, the sequence n↦n(ee−1−S(n)) is strictly increasing. The proof of the monotonicity of the sequence n ↦ n Δ ( 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 , the sequence n ↦ n ( e e − 1 − S ( n ) ) is strictly increasing. 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. |
| ArticleNumber | 14 |
| Author | Lampret, Vito |
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| Cites_doi | 10.1155/2011/721827 |
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| Keywords | Estimate Euler’s number Monotone sequence Sums of powers 40A05 Limit 40A25 Rate of convergence Inequality |
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| References | V. Lampret (3253_CR1) 2011; 2011 3253_CR2 |
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| Snippet | The proof of the monotonicity of the sequence
n
↦
n
Δ
(
n
)
, presented in the 2011 article “A Sharp double inequality for sums of powers” by V. Lampret, is... The proof of the monotonicity of the sequence n↦nΔ(n), presented in the 2011 article “A Sharp double inequality for sums of powers” by V. Lampret, is... 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... |
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| SubjectTerms | Analysis Applications of Mathematics Estimate Euler’s number Graphs Inequality Limit Mathematics Mathematics and Statistics Monotone sequence Rate of convergence |
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| Title | A sharp double inequality for sums of powers revisited |
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