- Research Article
- Open Access
A Sharp Double Inequality for Sums of Powers
© Vito Lampret. 2011
- Received: 26 September 2010
- Accepted: 11 January 2011
- Published: 18 January 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 .
- Generate Function
- Power Series
- Accurate Estimate
- Open Interval
- Summation Formula
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.
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 ).
valid for every such that .
Open Question 3.
Are the sequences and strictly concave?
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.
- Spivey MZ: The Euler-Maclaurin formula and sums of powers. Mathematics Magazine 2006, 79: 61–65. 10.2307/27642905MATHView ArticleGoogle Scholar
- Spivey MZ: Letter to the editor. Mathematics Magazine 2010, 83: 54–55. 10.4169/002557010X485896View ArticleGoogle Scholar
- Holland F: . Mathematics Magazine 2010, 83: 51–54. 10.4169/002557010X485111MATHView ArticleGoogle Scholar
- Lampret V: Estimating powers with base close to unity and large exponents. Divulgaciones Matematicas 2005,13(1):21–34.MATHMathSciNetGoogle Scholar
- Tannery J: Introduction à la Théorie des Fonctions d'une Variable. Hermann, Paris, France; 1886.Google Scholar
- Bromwich T: Introduction to the Theory of Infinite Series. 3rd edition. , Chelsea, NY, USA; 1991.MATHGoogle Scholar
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.