Abstract
In this article, we deduce a new family of upper bounds of n! of the form
We also proved that the approximation formula
Mathematics Subject Classification (2000): 41A60; 41A25; 57Q55; 33B15; 26D07.
Keywords:
Stirling' formula; Wallis' formula; Bernoulli numbers; Riemann Zeta function; speed of convergence1 Introduction
Stirling' formula
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) [1] and Stirling (1730) [2]. Both used what is now called the EulerMacLaurin formula to approximate log 2 +
log 3 + ... + log n. The first derivation of De Moivre did not explicity determine the constant
Over the years, there have been many different upper and lower bounds for n! by various authors [310]. Artin [11] show that
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 [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
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
where
Then we have
Now define the sequence g_{n }by
which satisfies
and
There are two cases. The first case if K_{n }= a_{n }such that
then we get
or
The condition (11) means that the function a_{n } a_{n+1 }(n+ 1/2) ln
or
The second case if K_{n }= b_{n }such that
Similarly, we can prove that f_{n }< 1. Then
or
The condition (15) means that the function b_{n } b_{n+1 } (n + 1/2) ln
or
From the wellknown expansion
in which we substitute
Now we obtain the following result
Theorem 1.
where the two sequences a_{n}, b_{n }→ 0 as n → ∞ and satisfy
A qanalog of the inequality (20) was introduced in [13].
3 A new family of upper bounds of n!
By manipulating the series
Let's solve the following recurrence relation w.r.t n
which give us
But
Then
The series
where ζ(x) is the Riemann Zeta function. By using the relation [14]
where
Hence, we can choose
which satisfies
Then
By using the relation
we get
In the following Lemma, we will see that the upper bound
Lemma 3.1.
Proof. From (25), we get
Then
Now
Then
and hence
The following Lemma show that
Lemma 3.2.
Proof. From Eq. (25), we have
Then
which give us that
4 The speed of convergence of the approximation formula
2
π
n
(
n
/
e
)
n
e
M
n
[
m
]
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≥1 }is 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 [1521]. By using Lemma (4.1), clearly the sequence (w_{n})_{n≥1 }converges more quickly when the value of k satisfying (29) is larger.
To measure the accuracy of approximation formula
This approximation formula will be better as (w_{n})_{n≥1 }converges faster to zero. Using the relation (30), we get
Then
By using the relations (19) and (22), we have
Then
Theorem 2. The rate of convergence of the sequence w_{n }is equal to n^{2m3}, since
In 2011, Mortici [22] 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
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

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

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

Cesàro, E: Elements Lehrbuch der algebraischen analysis und der Infinitesimalrechnung. Leipzig, Teubner (1922)

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

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

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

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

Beesack, PR: Improvement of Stirling's formula by elementary methods. Univ. Beograd, Publ. Elektrotenhn. Fak Ser Mat Fiz. 17–21 (1969) No. 274301

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

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

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

Mansour, M, Obaid, MA: Bounds of qfactorial [n]_{q}!. Ars Combinatoria CII. 313–319 (2011)

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

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

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

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

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

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

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

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

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

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

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

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

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