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New upper bounds of n!
Journal of Inequalities and Applications volume 2012, Article number: 27 (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.
is one of the most widely known and used in asymptotics. In other words, we have
This formula provides an extremely accurate approximation of n! for large values of n. The first proofs of Stirling's formula was given by De Moivre (1730)  and Stirling (1730) . Both used what is now called the Euler-MacLaurin formula to approximate log 2 + log 3 + ... + log n. The first derivation of De Moivre did not explicity determine the constant . In 1731, Stirling determine this constant using Wallis' formula
where the numbers B i are called the Bernoulli numbers and are defined by
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!
In view of the relation (2), we begin with the two sequences K n and f n defined by
Then we have
Now define the sequence g n by
There are two cases. The first case if K n = a n such that
then we get and hence g n is strictly increasing function. But g n → 1 as n → ∞, then g n < 1. Hence which give us that fn+1< f n . Then f n is strictly decreasing function. Also, f n → 1 as n → ∞, then we obtain f n > 1. Then
The condition (11) means that the function a n - an+1-(n+ 1/2) ln is strictly increasing function also it tends to -1 as n → ∞. Then
The second case if K n = b n such that
Similarly, we can prove that f n < 1. Then
The condition (15) means that the function b n - bn+1- (n + 1/2) ln is strictly decreasing function also it tends to -1 as n → ∞. Then
From the well-known expansion
in which we substitute , so . Then
Now we obtain the following result
where the two sequences a n , b n → 0 as n → ∞ and satisfy
A q-analog of the inequality (20) was introduced in .
3 A new family of upper bounds of n!
By manipulating the series to find upper bounds, we get
Let's solve the following recurrence relation w.r.t n
which give us
where ζ(x) is the Riemann Zeta function. By using the relation 
where are Bernoulli's numbers. Then
Hence, we can choose
By using the relation
In the following Lemma, we will see that the upper bound will improved with increasing the value of m.
Proof. From (25), we get
The following Lemma show that is an improvement of the upper bound .
Proof. From Eq. (25), we have . By using (27), we obtain
which give us that
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.
Lemma 4.1. If (w n )n≥1is convergent to zero and there exists the limit
with k > 1, then there exists the limit:
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.
To measure the accuracy of approximation formula , define the sequence (w n )n≥1by the relation
This approximation formula will be better as (w n )n≥1converges faster to zero. Using the relation (30), we get
By using the relations (19) and (22), we have
Theorem 2. The rate of convergence of the sequence w n is equal to n-2m-3, since
In 2011, Mortici  shows by numerical computations that his formula
is much stronger than other known formulas such as:1
The following table shows numerically that our new formula has a superiority over the the Mortici's formula μ n .
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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.
The authors declare that they have no competing interests.
All authors carried out the proofs. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Cite this article
Mahmoud, M., Alghamdi, M.A. & Agarwal, R.P. New upper bounds of n!. J Inequal Appl 2012, 27 (2012). https://doi.org/10.1186/1029-242X-2012-27
- Stirling' formula
- Wallis' formula
- Bernoulli numbers
- Riemann Zeta function
- speed of convergence