New upper bounds of n!

Mahmoud Mansour, Alghamdi, Agarwal

Description

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 , . .

Sujets

• Bernoulli number
• Riemann zeta function
• Rate of convergence

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 11
• Langue: English

Extrait

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 M n n!<2πn(n/e)e n∈ N ,   m  1 2m−2k+ 2 [m] 1 −2k M n 2 = + 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 M n 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 Stirling’formula

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 Stirling’s 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 Wallis’for 2π mula

2n 2 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 that µ(n) = ln√ lies between any two n n2πn successive partial sums of the Stirling’s 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.