# 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

 Received: 15 October 2011 Accepted: 13 February 2012 Published: 13 February 2012

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.

### Abstract

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-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

### 1 Introduction

Stirling' formula

(1)

is one of the most widely known and used in asymptotics. In other words, we have

(2)

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) [1] and Stirling (1730) [2]. 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

Over the years, there have been many different upper and lower bounds for n! by various authors [3-10]. Artin [11] show that lies between any two successive partial sums of the Stirling's series

(3)

where the numbers Bi are called the Bernoulli numbers and are defined by

(4)

We can't take infinite sum of the series (3) because the series diverges. Also, Impens [12] 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 [9] 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 Kn and fn defined by

(5)

where

(6)

Then we have

(7)

Now define the sequence gn by

(8)

which satisfies

(9)

and

(10)

There are two cases. The first case if Kn = an such that

(11)

then we get and hence gn is strictly increasing function. But gn → 1 as n → ∞, then gn < 1. Hence which give us that fn+1 < fn. Then fn is strictly decreasing function. Also, fn → 1 as n → ∞, then we obtain fn > 1. Then

(12)

or

(13)

The condition (11) means that the function an - an+1 -(n+ 1/2) ln is strictly increasing function also it tends to -1 as n → ∞. Then

or

(14)

The second case if Kn = bn such that

(15)

Similarly, we can prove that fn < 1. Then

(16)

or

(17)

The condition (15) means that the function bn - bn+1 - (n + 1/2) ln is strictly decreasing function also it tends to -1 as n → ∞. Then

or

(18)

From the well-known expansion

in which we substitute , so . Then

(19)

Now we obtain the following result

Theorem 1.

(20)

where the two sequences an, bn → 0 as n → ∞ and satisfy

(21)

A q-analog of the inequality (20) was introduced in [13].

### 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

(22)

which give us

But

Then

The series

where ζ(x) is the Riemann Zeta function. By using the relation [14]

where are Bernoulli's numbers. Then

Hence, we can choose

(23)

which satisfies

Then

(24)

By using the relation

we get

(25)

In the following Lemma, we will see that the upper bound will improved with increasing the value of m.

Lemma 3.1.

(26)

Proof. From (25), we get

Then

Now

Then

(27)

and hence

The following Lemma show that is an improvement of the upper bound .

Lemma 3.2.

(28)

Proof. From Eq. (25), we have . By using (27), we obtain

Then

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 (wn)n≥1 is convergent to zero and there exists the limit

(29)

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 (wn)n≥1 converges more quickly when the value of k satisfying (29) is larger.

To measure the accuracy of approximation formula , define the sequence (wn)n≥1 by the relation

(30)

This approximation formula will be better as (wn)n≥1 converges faster to zero. Using the relation (30), we get

Then

By using the relations (19) and (22), we have

Then

(31)

Theorem 2. The rate of convergence of the sequence wn is equal to n-2m-3, since

In 2011, Mortici [22] shows by numerical computations that his formula

(32)

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 μn.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

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.

### Acknowledgements

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.

### References

1. De Moivre, A: Miscellanea Analytica de Seriebus et Quadraturis. London (1730)

2. Stirling, J: Methodus Differentialis. London (1730)

3. Hummel, PM: A note on Stirling's formoula. Am Math Monthly. 42(2), 97–99 (1940)

4. Cesàro, E: Elements Lehrbuch der algebraischen analysis und der In-finitesimalrechnung. Leipzig, Teubner (1922)

5. Uspensky, JV: Introduction to Mathematical Probability. McGraw Hill, New York (1937)

6. Robbins, H: A remark on Stirling's formula. Am Math Monthly. 62, 26–29 (1955). Publisher Full Text

7. Nanjundiah, TS: Note on Stirling's formula. Am Math Monthly. 66, 701–703 (1965)

8. Maria, AJ: A remark on Striling's formula. Am Math Monthly. 72, 1096–1098 (1965). Publisher Full Text

9. Beesack, PR: Improvement of Stirling's formula by elementary methods. Univ. Beograd, Publ. Elektrotenhn. Fak Ser Mat Fiz. 17–21 (1969) No. 274-301

10. Michel, R: On Stirling's formula. Am. Math. Monthly. 109(4), 388–390 (2002)

11. Artin, E: The Gamma Function (translated by M Butler). Holt, Rinehart and Winston. New York (1964)

12. Impens, C: Stirling's series made easy. Am Math Monthly. 110(8), 730–735 (2003). Publisher Full Text

13. Mansour, M, Obaid, MA: Bounds of q-factorial [n]q!. Ars Combinatoria CII. 313–319 (2011)

14. Andrews, GE, Askey, R, Roy, R: Special Functions. Cambridge University Press, Cambridge (1999)

15. Mortici, C: An ultimate extremely accurate formula for approximation of the factorial function. Archiv der Mathematik (Basel). 93(1), 37–45 (2009). Publisher Full Text

16. Mortici, C: Product approximations via asymptotic integration. Am Math Monthly. 117, 434–441 (2010). Publisher Full Text

17. Mortici, C: New approximations of the gamma function in terms of the digamma function. Appl Math Lett. 23, 97–100 (2010). Publisher Full Text

18. Mortici, C: New improvements of the Stirling formula. Appl Math Comput. 217(2), 699–704 (2010). Publisher Full Text

19. Mortici, C: Best estimates of the generalized Stirling formula. Appl Math Comput. 215(11), 4044–4048 (2010). Publisher Full Text

20. Mortici, C: A class of integral approximations for the factorial function. Comput Math Appl. 59(6), 2053–2058 (2010). Publisher Full Text

21. Mortici, C: Ramanujan formula for the generalized Stirling approximation. Appl Math Comput. 217(6), 2579–2585 (2010). Publisher Full Text

22. Mortici, C: A new Stirling series as continued fraction. Numer Algorithm. 56(1), 17–26 (2011). Publisher Full Text

23. Burnside, W: A rapidly convergent series for Log N!. Messenger Math. 46, 157–159 (1917)

24. Batir, N: Sharp inequalities for factorial n. Proyecciones. 27(1), 97–102 (2008)

25. Gosper, RW: Decision procedure for indefinite hypergeometric summation. Proc Natl Acad Sci USA. 75, 40–42 (1978). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

26. Ramanujan, S: The Lost Notebook and Other Unpublished Papers (Introduced by Andrews, GE). Narosa Publishing House, New Delhi (1988)