5-)++74)6--561)6-5.
,-,1+)6-,1++)51.60-%;-)45.24.-5540)415418)56)8)

Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 4(2010), Pages 137-139.

ON SOME ACCURATE ESTIMATES OF 𝜋

(DEDICATED IN OCCASION OF THE 70-YEARS OF PROFESSOR HARI M. SRIVASTAVA)

CRISTINEL MORTICI

Abstract. The aim of this paper is to establish some inequalities related to an accurate approximation formula of 𝜋. Being practically difficult, the computations arising in this problem were made using computer softwares such as Maple.

1. Introduction

Maybe the best known example of infinite product for estimating the constant 𝜋is the Wallis product [4]

𝜋 2 =2

1 ⋅2 3 ⋅4

3⋅ 4 5⋅ 6

5⋅6 7⋅ ⋅ ⋅=

∞

∏

𝑛=1

4𝑛²

4𝑛²−1, (1.1)

which is related to Euler’s gamma function Γ,since

𝑛

∏

𝑘=1

4𝑘²

4𝑘²−1 = 16^𝑛(Γ (𝑛+ 1))⁴

(2𝑛+ 1) (Γ (2𝑛+ 1))². (1.2) Although it has a nice form, (1.1) is very slow, so it is not suitable for approximating the constant𝜋.

A possible starting point for accelerating (1.1) is the work of Fields [1] who shown that

Γ (𝑧+𝑎)

Γ (𝑧+𝑏) ∼ (𝑧+𝑎−𝜌)^𝑎−𝑏

𝑁−1

∑

𝑘=0

𝐵_2𝑘^(2𝜌)(𝜌) (𝑏−𝑎)_2𝑘(𝑧+𝑎−𝜌)^−2𝑘

(2𝑘)! (1.3)

+ (𝑧+𝑎−𝜌)^𝑎−𝑏𝑂(

(𝑧+𝑎−𝜌)^−2𝑁) , 2𝜌 = 1 +𝑎−𝑏, ∣arg (𝑧+𝑎)∣ ≤𝜋−𝜀, 𝜀 >0.

where the symbols𝐵_2𝑘^(2𝜌)stand for the generalized Bernoulli polynomials [2, 5].

2000Mathematics Subject Classification. 33B15, 26D15.

Key words and phrases. Wallis formula inequalities; asymptotic series; approximations.

c

⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.

Submitted October 21, 2010. Published November 22, 2010.

137

138 C. MORTICI

With𝑧=𝑛, 𝑎=−𝑥, 𝑏= 1,and𝜌=−𝑥/2 in (1.3), we get ( 𝑥

𝑛 )

∼ (−1)^𝑛(𝑛−𝑥/2)^−(𝑥+1) Γ (−𝑥)

∞

∑

𝑘=0

𝐵_2𝑘^(2𝜌)(𝜌) (𝑥+ 1)_2𝑘 (2𝑘)! (𝑛−𝑥/2)^2𝑘

, 𝜌=−𝑥 2 , ( 𝑥

𝑛 )

∼(−1)^𝑛(𝑛−𝑥/2)^−(𝑥+1) Γ (−𝑥)

[

1 + (𝑥)₃

24 (𝑛−𝑥/2)²+ (𝑥)₅(5𝑥−2)

5760 (𝑛−𝑥/2)⁴ (1.4) +(𝑥)₇(

35𝑥²−42𝑥+ 16) 2903040 (𝑛−𝑥/2)⁶ +⋅ ⋅ ⋅

] . See also [3, p. 142], where the following identity is stated

( 𝑥 𝑛

)

= (−1)^𝑛 Γ (−𝑥)

Γ (𝑛−𝑥) Γ (𝑛+ 1).

Further, with𝑥=−1/2 in (1.4),we obtain the following formula 𝜋 = 4 (Γ (𝑛+ 1))⁴16^𝑛

(4𝑛+ 1) (Γ (2𝑛+ 1))² [

1− 1

4 (4𝑛+ 1)² + 21

32 (4𝑛+ 1)⁴ (1.5)

− 671

128 (4𝑛+ 1)⁶ + 180323

2048 (4𝑛+ 1)⁸ − 20898423 8192 (4𝑛+ 1)¹⁰ + 7426362705

65536 (4𝑛+ 1)¹² − 1874409465055 262144 (4𝑛+ 1)¹⁴ +𝑂

( 1 𝑛¹⁶

)]2

.

The idea of expressing𝜋using the asymptotic expansion of the ratio ^Γ(𝑛+1/2)_Γ(𝑛+1) was introduced by Tricomi and Erd´elyi in [3, p. 142, Rel. 23]. Here we make use of the asymptotic expansion for ^Γ(𝑛+1/2)_Γ(𝑛+1) given in [1] to improve the results of Tricomi and Erd´elyi [3].

2. The results

By truncation of series (1.5), increasingly accurate approximations for𝜋can be derived. As example, if𝑛= 10, use of the first five terms in (1.5) gives𝜋with an error of 1.1639×10⁻¹²,while use of the first six terms in (1.5) gives𝜋with an error of 3.0431×10⁻¹⁴.

We prove the following

Theorem 2.1. For every integer𝑛≥1, we have 4 (Γ (𝑛+ 1))⁴16^𝑛

(4𝑛+ 1) (Γ (2𝑛+ 1))²𝑎(𝑛)< 𝜋 < 4 (Γ (𝑛+ 1))⁴16^𝑛

(4𝑛+ 1) (Γ (2𝑛+ 1))²𝑏(𝑛), where

𝑎(𝑛) = (

1− 1

4 (4𝑛+ 1)²+ 21

32 (4𝑛+ 1)⁴ − 671

128 (4𝑛+ 1)⁶+ 180323 2048 (4𝑛+ 1)⁸

)2

and 𝑏(𝑛) =

(

1− 1

4 (4𝑛+ 1)²+ 21

32 (4𝑛+ 1)⁴ − 671

128 (4𝑛+ 1)⁶+ 180323

2048 (4𝑛+ 1)⁸ − 20898423 8192 (4𝑛+ 1)¹⁰

)² .

ON SOME ACCURATE ESTIMATES OF𝜋 139

Proof. The sequences

𝑥𝑛 = 4 (Γ (𝑛+ 1))⁴16^𝑛

(4𝑛+ 1) (Γ (2𝑛+ 1))²𝑎(𝑛) , 𝑦𝑛= 4 (Γ (𝑛+ 1))⁴16^𝑛 (4𝑛+ 1) (Γ (2𝑛+ 1))²𝑏(𝑛) converge to𝜋and it suffices to show that𝑥_𝑛 is strictly increasing and𝑦_𝑛 is strictly decreasing. In this sense, we have

𝑥𝑛+1

𝑥_𝑛 −1 = 4 (4𝑛+ 1) (𝑛+ 1)² (4𝑛+ 5) (2𝑛+ 1)²

𝑎(𝑛+ 1) 𝑎(𝑛) −1

= − 𝑃(𝑛)

(4𝑛+ 5)¹⁷(2𝑛+ 1)²(134217728𝑛⁸+⋅ ⋅ ⋅+ 172467)², where the polynomial

𝑃(𝑛) = 60235603222675842001797120𝑛²⁴+⋅ ⋅ ⋅+ 22691018044772336786409 has all coefficients positive. In consequence,𝑥𝑛 is strictly increasing, convergent to 𝜋,so 𝑥𝑛< 𝜋.

Then 𝑦𝑛+1

𝑦𝑛

−1 = 4 (4𝑛+ 1) (𝑛+ 1)² (4𝑛+ 5) (2𝑛+ 1)²

𝑏(𝑛+ 1) 𝑏(𝑛) −1

= 𝑄(𝑛)

(2𝑛+ 1)²(4𝑛+ 5)¹²(8589934592𝑛¹⁰+⋅ ⋅ ⋅ −20208555)², where the polynomial

𝑄(𝑛+ 1) = 210420037966350927549442377646080𝑛³⁰

+⋅ ⋅ ⋅+ 1909672653415578833630022434217112437351 has all coefficients positive. In consequence,𝑦𝑛 is strictly decreasing, convergent to

𝜋,so 𝑦𝑛> 𝜋. □

Remark. The computations in this paper were made using Maple software.

Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.

References

[1] J. L. Fields,A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edin- burgh Math. Soc.152 (1966) 43-45.

[2] N.E. Nørlund,Vorksungen ¨uber Differenzenrechnung, Springer (Berlin), 1924.

[3] F. G. Tricomi, A. Erd´elyi,The asymptotic expansion of a ratio of gamma functions, Pacific J. Math.,11 (1951) 133-142.

[4] J. Wallis,Computation of 𝜋by successive interpolations, (1655) in: A Source Book in Math- ematics, 1200-1800 (D. J. Struik, Ed.), Harvard University Press, Cambridge, MA, (1969), 244-253.

[5] N.M. Temme,Special Functions, Wiley (NewYork), 1996.

Cristinel Mortici, Valahia University of Tˆargovis¸te, Department of Mathematics, Bd. Unirii 18, 130082 Tˆargovis¸te, Romania

E-mail address:[email protected]; [email protected] URL:http://www.cristinelmortici.ro