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^{2m3 }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 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 . 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 [310]. Artin [11] show that lies between any two successive partial sums of the Stirling's series
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 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
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 and hence g_{n }is strictly increasing function. But g_{n }→ 1 as n → ∞, then g_{n }< 1. Hence which give us that f_{n+1 }< f_{n}. Then f_{n }is strictly decreasing function. Also, f_{n }→ 1 as n → ∞, then we obtain f_{n }> 1. Then
or
The condition (11) means that the function a_{n } a_{n+1 }(n+ 1/2) ln is strictly increasing function also it tends to 1 as n → ∞. Then
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 is strictly decreasing function also it tends to 1 as n → ∞. Then
or
From the wellknown expansion
in which we substitute , so . Then
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 to find upper bounds, we get
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 are Bernoulli's numbers. Then
Hence, we can choose
which satisfies
Then
By using the relation
we get
In the following Lemma, we will see that the upper bound will improved with increasing the value of m.
Lemma 3.1.
Proof. From (25), we get
Then
Now
Then
and hence
The following Lemma show that is an improvement of the upper bound .
Lemma 3.2.
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 (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 , define the sequence (w_{n})_{n≥1 }by the relation
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 has a superiority over the the Mortici's formula μ_{n}.
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.
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