ESTIMATING
FROM THE WALLIS SEQUENCE CRISTINEL MORTICI
Abstract. The aim of this paper is to de…ne new sequences related to the Wallis sequence having higher rates of convergence. Some sharp inequalities are established.
1. Introduction and motivation Perhaps one of the most known sequence related to the constant sequence n Y 4k 2 W (n) = ; 4k 2 1
is the Wallis
k=1
which converges to =2 with the convergence rate estimated by n 1 ; since 4 3 < W (n) < (n 3); 10n 2 10n see, e.g., [6, Rel. 2c]. As the Wallis sequence is slowly convergent towards its limit, it is not suitable for approximating the constant : In consequence many authors were preoccupied in the recent past to accelerate the Wallis sequence. See [1, p. 258], [2, p. 384], [3, p. 213], [4, p. 5, p. 47], [5, p. 384], [7, p. 14, p. 465], [11] and the references therein. In particular, Lampret [6, Rel. 2c] used the following version of the Stirling formula r 2 x x x (x) = exp (x > 0) (1.1) x e 12x where x 2 (0; 1) to prove the following representation = W (n) 2 +
1 n
e
1
1+
1 2n
2n
exp
0 n
6n + 3
n
6n
(n
1) :
(1.2)
As the Stirling formula (1.1) is now the …rst approximation of the following Stirling asymptotic series r x x 1 1 2 1 1 (x) exp + + ; 3 5 x e 12x 360x 1260x 1680x7 it results that x tends to 1; as x approaches in…nity. Motivated by Lampret’s representation (1.2), we prove that the best approximation of the form W (n) 2 +
1 n
e
1
1+
1 2n
2n
exp
a 6n + 3
b 6n
(a; b 2 [0; 1])
(1.3)
1991 Mathematics Subject Classi…cation. 40A05; 40A20; 40A25; 65B10; 65B15. Key words and phrases. Wallis product; rate of convergence; inequalities; asymptotic series. 1
2
CRISTINEL M ORTICI
is obtained for a = b = 1: For these priviledged values, (1.3) becomes W (n) 2 +
1 n
1
e
1+
2n
1 2n
1 12n2 + 6n
exp
;
but we prove that the approximation W (n) 2 +
1 n
1
e
2n
1 2n
1+
exp
12n2
1 + 6n +
6 5
gives better results. Moreover, we state and prove the following double inequality Theorem 1. For every integer n W (n) 2 +
1 n
< W (n) 2 +
1 n
e
1; we have
1
e
1+
1
1 2n
2n
2n
1 2n
1+
1 12n2 + 6n
exp
0 and g1 < 0: In this sense, we have f100 (x) =
(4x + 3) 60x + 148x2 + 144x3 + 48x4 + 9 3
3
3
3x3 (x + 1) (2x + 1) (2x + 3)
and g100 (x) =
(4x + 3) Q (x) 2
2
2
3;
3
x2 (x + 1) (2x + 1) (2x + 3) (25x + 10x2 + 16) (5x + 10x2 + 1)
where Q (x) = 12 288+291 072x+2962 853x2 +16 496 490x3 +55 483 705x4 +118 363 500x5 +163 202 000x6 + 144 648 000x7 + 79 458 000x8 + 24 600 000x9 + 3280 000x10 : Finally f1 is strictly convex, g1 is strictly concave, with f1 (1) = g1 (1) = 0; so f1 > 0 and g1 < 0: Now, in order to …nd the best approximation of type (1.6), we associate the relative error sequence zn by the relations W (n) =
2
1
1 2n + 1
exp
n
( ; ; ) ezn :
By making appeal again to Maple, we get zn +
zn+1 =
25 45 1 + + 32 16 8
3 3 + 16 4 19 24
7 32 1 + n6
1 + n4 45 32
3 8 75 16
3 2
1 7 + 20 16
2 41 + 5 32
1 +O n7
1 n5 1 n8
:
ESTIM ATING
The system
8 < :
FROM THE WALLIS SEQUENCE
3 16 3 8 25 32
+
3 4 3 2 45 16
5
7 32 = 0 1 7 20 + 16 = 0 1 19 +8 24 = 0
+ determined by the …rst three coe¢ cients of this power series gives the best values 23 = 11 = 60 ; = 0 and the corresponding approximation 30 ; ! 23 1 11n n : W (n) 1 exp + 2 2n + 1 4n2 1 30 (4n2 1)2 60n (4n2 1) Proof of Theorem 2. Denoting x 11x u (x) = 2 + 4x 1 30 (4x2 1)2 and v (x) = u (x) +
23 60x (4x2
493 1)
493 107520x3 (x2
2
107520x3 (x2
1)
2;
1)
we have to prove that 1 1 1 exp u (n) < W (n) < 1 exp v (n) : 2 2n + 1 2 2n + 1 In fact it su¢ ces to show that the sequence cn is strictly increasing and dn is strictly decreasing, where 1 cn = We have ln cn+1
1 2n+1
;
W (n)
dn =
ln cn = f2 (n) ; ln dn+1 2
f2 (x) = ln
4 (x + 1)
1 2
4 (x + 1) 2
g2 (x) = ln
4 (x + 1)
1 2
4 (x + 1)
1 2n+1
1
exp u (n)
+ ln + ln
exp v (n) :
W (n)
ln dn = g2 (n) ; where 1 1 1 1
1 2x+3 1 2x+1
+ u (x + 1)
u (x)
1 2x+3 1 2x+1
+ v (x + 1)
v (x) ;
with the derivatives f20 (x) =
P2 (x) 8960x4 (x g20 (x) =
3
4
3
3
1) (x + 1) (x + 2) (2x + 3) (4x2 Q2 (x)
60x2
2
3
(x + 1) (2x + 3) (4x2
3
1)
3;
1)
where P2 (x) = 26 622 + 115 362x +5343 092x6
906 057x2
2812 130x3 + 8973 969x4 + 32 403 996x5
67 006 464x7 60 727 776x8 + 23 468 480x9 + 57 479 616x10 +28 792 320x11 + 4798 720x12
and Q2 (x) = 621 864x + 7024x2 + 15 776x3 + 7888x4 : Since all the coe¤cients of the polynomials P2 (x + 1) and Q2 (x + 1) are positive we have f20 (x) < 0 and g20 (x) > 0, for x > 1. Now, f2 is strictly decreasing, g2 is strictly increasing, with f2 (1) = g2 (1) = 0; so f2 > 0 and g2 < 0 and the conclusion follows.
6
CRISTINEL M ORTICI
Finally, we are convinced that our new approach presented here can be useful for establishing further improvements and re…nements in other related problems. Acknowledgements. 1. The author thanks the anonymous referees for a careful reading of the initial manuscript and for useful comments and corrections. 2. This work was supported by a grant of the Romanian National Authority for Scienti…c Research, CNCS – UEFISCDI, project number PN-II-ID-PCE-20113-0087. References [1] M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions, 9th edn. (Dover Publications New York 1974). [2] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Harcourt/Academic Press 2001). [3] T. J. I’. A. Bromwich, An Introduction to the Theory of In…nite Series, 3rd ed. (Chelsea Publishing Company New York 1991). [4] P. Henrici, Applied and Computational Complex Analysis, Vol.2, 7th ed. (John Wiley & Sons Inc. 1991). [5] K. Knopp, Theory and Applications of In…nite Series (Hafner New York 1971). [6] V. Lampret, Wallis sequence estimated through the Euler–Maclaurin formula: even from the Wallis product could be computed fairly accurately, Gaz. Australian Math. Soc., 31 (2004), no. 5, 328-339. [7] J. Lewin and M. Lewin, An Introduction to Mathematical Analysis, 2nd ed. (McGraw-Hill Inc. 1993). [8] C. Mortici, Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), no. 5, 434-441. [9] C. Mortici, Completely monotonic functions associated with gamma function and applications, Carpathian J. Math., 25 (2009), no. 2, 186-191. [10] C. Mortici, Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant, Anal. Appl. (Singap.), 8 (2010), no. 1, 99-107. [11] A. Sofo, Some Representations of Pi, AustMS Gazette 31 (2004), 184–189. Valahia University of TârgoviS¸te, Department of Mathematics, Bd. Unirii 18, 130082 TârgoviS¸te, Romania E-mail address :
[email protected] URL: http://www.cristinelmortici.ro
