New upper bounds of n!
© Mahmoud et al; licensee Springer. 2012
Received: 15 October 2011
Accepted: 13 February 2012
Published: 13 February 2012
In this article, we deduce a new family of upper bounds of n! of the form
We also proved that the approximation formula for big factorials has a speed of convergence equal to n-2m- 3for 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.
KeywordsStirling' formula Wallis' formula Bernoulli numbers Riemann Zeta function speed of convergence
We can't take infinite sum of the series (3) because the series diverges. Also, Impens  deduce Artin result with different proof and show that the Bernoulli numbers in this series cannot be improved by any method whatsoever.
The organization of this article is as follows. In Section 2, we deduce a general double inequality of n!, which already obtained in  with different proof. Section 3 is devoted to getting a new family of upper bounds of n! different from the partial sums of the Stirling's series. In Section 4, we measure the speed of convergence of our approximation formula for big factorials. Also, we offer some numerical computations to prove the superiority of our formula over other known formulas.
2 A double inequality of n!
Now we obtain the following result
A q-analog of the inequality (20) was introduced in .
3 A new family of upper bounds of n!
In the following Lemma, we will see that the upper bound will improved with increasing the value of m.
The following Lemma show that is an improvement of the upper bound .
4 The speed of convergence of the approximation formula for big factorials
In what follows, we need the following result, which represents a powerful tool to measure the rate of convergence.
This Lemma was first used by Mortici for constructing asymptotic expansions, or to accelerate some convergences [15–21]. By using Lemma (4.1), clearly the sequence (w n )n≥1converges more quickly when the value of k satisfying (29) is larger.
The following table shows numerically that our new formula has a superiority over the the Mortici's formula μ n .
|n! -μ n |
|n! - λn,1|
1.1 × 1015
2.8 × 1014
6.8 × 1052
1.7 × 1052
6.5 × 10144
1.4 × 10144
This project was founded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. 448/130/1431. The authors, therefore, acknowledge with thanks DSR support for Scientific Research.
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