A Sharp Double Inequality for Sums of Powers
© Vito Lampret. 2011
Received: 26 September 2010
Accepted: 11 January 2011
Published: 18 January 2011
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.
2. Monotone 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).
(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 ).
3. The Rate of Convergence
Open Question 3.
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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