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New upper bounds of n!

De
9 pages
In this article, we deduce a new family of upper bounds of n ! of the form n ! < 2 π n ( n / e ) n e M n [ m ] n ∈ â„• , M n [ m ] = 1 2 m + 3 1 4 n + ∑ k = 1 m 2 m - 2 k + 2 2 k + 1 2 - 2 k ζ ( 2 k , n + 1 / 2 ) m = 1 , 2 , 3 , . . In this article, we deduce a new family of upper bounds of n ! of the form n ! < 2 π n ( n / e ) n e M n [ m ] n ∈ â„• , M n [ m ] = 1 2 m + 3 1 4 n + ∑ k = 1 m 2 m - 2 k + 2 2 k + 1 2 - 2 k ζ ( 2 k , n + 1 / 2 ) m = 1 , 2 , 3 , . .
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Mahmoudet al.Journal of Inequalities and Applications2012,2012:27 http://www.journalofinequalitiesandapplications.com/content/2012/1/27
R E S E A R C H
New upper bounds ofn! 1,3* 1 2 Mansour Mahmoud , Mohammed A Alghamdi and Ravi P Agarwal
* Correspondence: mansour@mans. edu.eg 1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia Full list of author information is available at the end of the article
Open Access
Abstract In this article, we deduce a new family of upper bounds ofn! of the form [m] n Mn n!<2πn(n/e)e nN,   m 1 2m2k+ 2 [m]12k Mn2= + ζ(2k,n+ 1/2)m= 1, 2, 3, .... 2m+ 3 4n2k+ 1 k=1
[m] We also proved that the approximation formula for big factorials n Mn 2πn(n/e)e 2m3 has a speed of convergence equal tonform= 1,2,3,..., which give us a superiority over other known formulas by a suitable choice ofm. Mathematics Subject Classification (2000): 41A60; 41A25; 57Q55; 33B15; 26D07. Keywords:Stirling?? formula, Wallis?? formula, Bernoulli numbers, Riemann Zeta function, speed of convergence
1 Introduction Stirlingformula
n!
2nπn/e
(1)
is one of the most widely known and used in asymptotics. In other words, we have
lim n→∞
n n!e = 1. n 2πn n
(2)
This formula provides an extremely accurate approximation ofn! for large values of n. The first proofs of Stirlings formula was given by De Moivre (1730) [1] and Stirling (1730) [2]. Both used what is now called the EulerMacLaurin formula to approximate log 2 + log 3 + ... + logn. The first derivation of De Moivre did not explicity deter mine the constant . In 1731, Stirling determine this constant using Wallisfor 2π mula
2n2 2 (n!) lim n→∞ (2n)!
1 = n
π.
Over the years, there have been many different upper and lower bounds forn! by n n!e various authors [310]. Artin [11] show thatµ(n) = lnlies between any two n n2πn successive partial sums of the Stirlings series
© 2012 Mahmoud et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.