tailieunhanh - Báo cáo hóa học: " Research Article A Sharp Double Inequality for Sums of Powers"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Sharp Double Inequality for Sums of Powers | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 721827 7 pages doi 2011 721827 Research Article A Sharp Double Inequality for Sums of Powers Vito Lampret Department of Mathematics and Physics KMF Faculty of Civil and Geodetic Engineering FGG University of Ljubljana UL 1000 Ljubljana Slovenia Correspondence should be addressed to Vito Lampret Received 26 September 2010 Revised 7 December 2010 Accepted 11 January 2011 Academic Editor Alberto Cabada Copyright 2011 Vito Lampret. 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. It is established that the sequences n S n yn 1 k n n and n n e e -1 - S n are strictly increasing and converge to e e -1 and e e 1 2 e -1 3 respectively. It is shown that there holds the sharp double inequality 1 e-1 - 1 n e e-1 -S n e e 1 2 e-1 3 - 1 n n e N . 1. Introduction The proof of the equality n n e n k 1 e -11 published recently in the form 1 n 1 e-11 n-1 n J k 1 was based on the equations n1-k n n -1 n - k 2 1 - 1 n 1 - 2 n 1 - k - 2 n 1 0 1 n with the false hypothesis that big O is independent of k see 1 pages 63-64 and 2 pages 54-55 . Deriving the author used the Euler-Maclaurin summation formula and a generating function for the Bernoulli numbers. 2 Journal of Inequalities and Applications Subsequently Spivey published the correction of his demonstration as the Letter to the Editor 2 . Additionally Holland 3 published two different derivations of in the same issue as Spivey s correction appeared. In this note using only elementary techniques we demonstrate that the sequence S n is strictly increasing and that holds in addition we establish a sharp estimate of the rate of convergence. 2. Monotone Convergence The formula is illustrated in Figure 1 where the .