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A Sharp Double Inequality for Sums of Powers
Journal of Inequalities and Applications volume 2011, Article number: 721827 (2011)
It is established that the sequences and are strictly increasing and converge to and , respectively. It is shown that there holds the sharp double inequality .
The proof of the equality
published recently in the form 
was based on the equations with the false hypothesis that big is independent of (see [1, pages 63-64] and [2, pages 54-55]). Deriving (1.1b) the author used the Euler-Maclaurin summation formula and a generating function for the Bernoulli numbers.
Subsequently, Spivey published the correction of his demonstration as the Letter to the Editor . Additionally, Holland  published two different derivations of (1.1a) in the same issue as Spivey's correction appeared.
In this note, using only elementary techniques, we demonstrate that the sequence is strictly increasing and that (1.1a) holds; in addition, we establish a sharp estimate of the rate of convergence.
2. Monotone Convergence
The formula (1.1a) is illustrated in Figure 1, where the sequence is depicted. Its monotonicity is seen very clearly.
To prove that the sequence is strictly increasing, we change the order of summation
Now, consider the function which is, for , strictly increasing on the open interval and , for any [4, page 42]. Consequently, the sequence is strictly increasing. We use Tannery's theorem for series (see  or [6, item 49, page 136]) to determine its limit.
Lemma 2.1 (Tannery).
Let a double sequence of complex numbers satisfy the following conditions:
(1)The finite limit exists for every fixed .
(2)There exists a sequence of positive constants such that for every satisfying the estimate , and the series converges. (In [6, item 49, page 136], we have the stronger supposition that for all ).
Then we have
Let all the conditions of the Lemma be satisfied and be given. Then we estimate for and for some . Moreover, for any , also for at some . Thus, for , we estimate
Now, using (2.1) and putting and into Tannery's Lemma, we obtain
3. The Rate of Convergence
Referring to Figure 1, the convergence of the sequence appears to be rather slow. The difference
determines the sequence . Its graph, shown in Figure 2, suggests it is monotonic increasing, which we will prove first.
Indeed, according to (3.1) and (2.1), we have
and, for , the sequence is strictly decreasing and converges to zero [4, (4)]. Thus, we have
To examine the monotonicity of the sequence , we study, using (3.3), (3.4), and (3.5), the difference , which is equal to
Moreover, according to (3.3), (3.5), and [4, (15)], the estimates
hold true for . Additionally, , due to (3.3) and (3.5). Thus, the estimate
is being valid for and with
According to (3.8) and differentiating the appropriate power series resulting from the geometric series, we obtain
Now, referring to (3.4) and (3.8)–(3.12), and applying Tannery's Lemma–-equation (2.2), with , we obtain the result
Therefore, using (3.1) and (3.7), we find the following sharp inequality
true for every . In addition, we have also the estimate
valid for every such that .
We have , and for the function we calculate , and . This way we obtain simple and rather accurate estimates
Consequently, we get, for example, a simple double inequality
Open Question 3.
Are the sequences and strictly concave?
Spivey MZ: The Euler-Maclaurin formula and sums of powers. Mathematics Magazine 2006, 79: 61–65. 10.2307/27642905
Spivey MZ: Letter to the editor. Mathematics Magazine 2010, 83: 54–55. 10.4169/002557010X485896
Holland F: . Mathematics Magazine 2010, 83: 51–54. 10.4169/002557010X485111
Lampret V: Estimating powers with base close to unity and large exponents. Divulgaciones Matematicas 2005,13(1):21–34.
Tannery J: Introduction à la Théorie des Fonctions d'une Variable. Hermann, Paris, France; 1886.
Bromwich T: Introduction to the Theory of Infinite Series. 3rd edition. , Chelsea, NY, USA; 1991.
The author wishes to express his sincere thanks to Prof. I. Vidav for some useful suggestions and also to the referee whose comments made possible a significant improvement of the paper.
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Lampret, V. A Sharp Double Inequality for Sums of Powers. J Inequal Appl 2011, 721827 (2011). https://doi.org/10.1155/2011/721827
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