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Open Access Research

New upper bounds of n!

Mansour Mahmoud13*, Mohammed A Alghamdi1 and Ravi P Agarwal2

Author Affiliations

1 Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

2 Department of Mathematics, Texas A&M University, Kingsville 700 University Blvd, Kingsville, TX 78363-8202, USA

3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

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Journal of Inequalities and Applications 2012, 2012:27  doi:10.1186/1029-242X-2012-27

Published: 13 February 2012

Abstract

In this article, we deduce a new family of upper bounds of n! of the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M1">View MathML</a>

We also proved that the approximation formula <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M2">View MathML</a> for big factorials has a speed of convergence equal to n-2m-3 for m = 1,2,3,..., which give us a superiority over other known formulas by a suitable choice of m.

Mathematics Subject Classification (2000): 41A60; 41A25; 57Q55; 33B15; 26D07.

Keywords:
Stirling' formula; Wallis' formula; Bernoulli numbers; Riemann Zeta function; speed of convergence