Open Access

A Sharp Double Inequality for Sums of Powers

Journal of Inequalities and Applications20112011:721827

https://doi.org/10.1155/2011/721827

Received: 26 September 2010

Accepted: 11 January 2011

Published: 18 January 2011

Abstract

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 .

1. Introduction

The proof of the equality
(1.1a)
published recently in the form [1]
(1.1b)

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 [2]. Additionally, Holland [3] 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.
Figure 1

The graph of the sequence .

To prove that the sequence is strictly increasing, we change the order of summation
(2.1)

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 [5] 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
(2.2)

Proof.

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
(2.3)
Now, using (2.1) and putting and into Tannery's Lemma, we obtain
(2.4)

3. The Rate of Convergence

Referring to Figure 1, the convergence of the sequence appears to be rather slow. The difference
(3.1)
determines the sequence . Its graph, shown in Figure 2, suggests it is monotonic increasing, which we will prove first.
Figure 2

The graph of the sequence .

Indeed, according to (3.1) and (2.1), we have
(3.2)
where
(3.3)
and, for , the sequence is strictly decreasing and converges to zero [4, (4)]. Thus, we have
(3.4)
with
(3.5)
To examine the monotonicity of the sequence , we study, using (3.3), (3.4), and (3.5), the difference , which is equal to
(3.6)
Hence:
(3.7)
Next, we examine also the question of convergence of the above sequence. First, referring to (3.3), (3.5), and [4, page 29, equation (16)], there exists the limit
(3.8)
Moreover, according to (3.3), (3.5), and [4, (15)], the estimates
(3.9)
hold true for . Additionally, , due to (3.3) and (3.5). Thus, the estimate
(3.10)
is being valid for and with
(3.11)
According to (3.8) and differentiating the appropriate power series resulting from the geometric series, we obtain
(3.12)
Now, referring to (3.4) and (3.8)–(3.12), and applying Tannery's Lemma–-equation (2.2), with , we obtain the result
(3.13)
Therefore, using (3.1) and (3.7), we find the following sharp inequality
(3.14)
true for every . In addition, we have also the estimate
(3.15)

valid for every such that .

We have , and for the function we calculate , and . This way we obtain simple and rather accurate estimates
(3.16)
Consequently, we get, for example, a simple double inequality
(3.17)

Open Question 3.

Are the sequences and strictly concave?

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
Department of Mathematics and Physics (KMF), Faculty of Civil and Geodetic Engineering (FGG), University of Ljubljana (UL)

References

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  2. Spivey MZ: Letter to the editor. Mathematics Magazine 2010, 83: 54–55. 10.4169/002557010X485896View ArticleGoogle Scholar
  3. Holland F: . Mathematics Magazine 2010, 83: 51–54. 10.4169/002557010X485111MATHView ArticleGoogle Scholar
  4. Lampret V: Estimating powers with base close to unity and large exponents. Divulgaciones Matematicas 2005,13(1):21–34.MATHMathSciNetGoogle Scholar
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Copyright

© Vito Lampret. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.