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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

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/27


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

© 2012 Mahmoud et al; licensee Springer.

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M1">View MathML</a>

We also proved that the approximation formula <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M2">View MathML</a> 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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M3">View MathML</a>

(1)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M4">View MathML</a>

(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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M5">View MathML</a>. In 1731, Stirling determine this constant using Wallis' formula

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M6">View MathML</a>

Over the years, there have been many different upper and lower bounds for n! by various authors [3-10]. Artin [11] show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M7">View MathML</a> lies between any two successive partial sums of the Stirling's series

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M8">View MathML</a>

(3)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M9">View MathML</a>

(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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M10">View MathML</a> 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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M11">View MathML</a>

(5)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M12">View MathML</a>

(6)

Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M13">View MathML</a>

(7)

Now define the sequence gn by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M14">View MathML</a>

(8)

which satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M15">View MathML</a>

(9)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M16">View MathML</a>

(10)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M17">View MathML</a>

(11)

then we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M18">View MathML</a> and hence gn is strictly increasing function. But gn → 1 as n → ∞, then gn < 1. Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M19">View MathML</a> which give us that fn+1 < fn. Then fn is strictly decreasing function. Also, fn → 1 as n → ∞, then we obtain fn > 1. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M20">View MathML</a>

(12)

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M21">View MathML</a>

(13)

The condition (11) means that the function an - an+1 -(n+ 1/2) ln <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M22">View MathML</a> is strictly increasing function also it tends to -1 as n → ∞. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M23">View MathML</a>

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M24">View MathML</a>

(14)

The second case if Kn = bn such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M25">View MathML</a>

(15)

Similarly, we can prove that fn < 1. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M26">View MathML</a>

(16)

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M27">View MathML</a>

(17)

The condition (15) means that the function bn - bn+1 - (n + 1/2) ln <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M22">View MathML</a> is strictly decreasing function also it tends to -1 as n → ∞. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M28">View MathML</a>

or

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M29">View MathML</a>

(18)

From the well-known expansion

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M30">View MathML</a>

in which we substitute <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M31">View MathML</a>, so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M32">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M33">View MathML</a>

(19)

Now we obtain the following result

Theorem 1.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M34">View MathML</a>

(20)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M35">View MathML</a>

(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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M36">View MathML</a> to find upper bounds, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M37">View MathML</a>

Let's solve the following recurrence relation w.r.t n

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M38">View MathML</a>

(22)

which give us

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M39">View MathML</a>

But

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M40">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M41">View MathML</a>

The series

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M42">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M43">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M44">View MathML</a> are Bernoulli's numbers. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M45">View MathML</a>

Hence, we can choose

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M46">View MathML</a>

(23)

which satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M47">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M48">View MathML</a>

(24)

By using the relation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M49">View MathML</a>

we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M50">View MathML</a>

(25)

In the following Lemma, we will see that the upper bound <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M51">View MathML</a> will improved with increasing the value of m.

Lemma 3.1.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M52">View MathML</a>

(26)

Proof. From (25), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M53">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M54">View MathML</a>

Now

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M55">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M56">View MathML</a>

(27)

and hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M57">View MathML</a>

The following Lemma show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M58">View MathML</a> is an improvement of the upper bound <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M59">View MathML</a>.

Lemma 3.2.

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M60">View MathML</a>

(28)

Proof. From Eq. (25), we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M61">View MathML</a>. By using (27), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M62">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M63">View MathML</a>

which give us that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M64">View MathML</a>

4 The speed of convergence of the approximation formula <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M65">View MathML</a> 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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M66">View MathML</a>

(29)

with k > 1, then there exists the limit:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M67">View MathML</a>

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M65">View MathML</a>, define the sequence (wn)n≥1 by the relation

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M68">View MathML</a>

(30)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M69">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M70">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M71">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M72">View MathML</a>

(31)

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M73">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M74">View MathML</a>

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is much stronger than other known formulas such as:1

The following table shows numerically that our new formula <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/27/mathml/M79">View MathML</a> 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.

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