An asymptotic approximation of Wallis’ sequence

• Autorzy: Vito Lampret
• Języki publikacji: EN

Abstrakty

EN

An asymptotic approximation of Wallis’ sequence W(n) = Πk=1n 4k 2/(4k 2 − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Πk=1n(2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example, $W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b'_n )$ with $a_n = 2 + \frac{1} {{2n + 1}} + \frac{2} {{3(2n + 1)^2 }} - \frac{1} {{3n(2n + 1)'}}b_n = \frac{2} {{33(n + 1)^{2'} }}b'_n \frac{1} {{13n^{2'} }}n \in \mathbb{N} $ .

Słowa kluczowe

EN

Approximation; Estimate; Inequality; π; Rate of convergence; Wallis’ ratio; Wallis’ sequence

Kategorie tematyczne

• 11Y60: Evaluation of constants
• 40A25: Approximation to limiting values (summation of series, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 775-787

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Vito Lampret

• University of Ljubljana

Bibliografia

• [1] Abramowitz M., Stegun I.A. (Eds.), Handbook of Mathematical Functions, Dover, New York, 1974
• [2] Beckmann P., A History of π, St. Martin’s Griffin, New York, 1974
• [3] Berggren L., Borwein J., Borwein P., Pi: A Source Book, Springer, New York-Berlin-Heidelberg-Hong Kong-London-Milan-Paris-Tokyo, 2004
• [4] Blatner D, The Joy of π, Walker & Co., New York, 1997
• [5] Borwein J.M., Borwein P.B., Bailey D.H., Ramanujan, modular equations, and approximations to Pi or how to compute one billion digits of Pi, Amer. Math. Monthly, 1989, 96(3), 201–219 http://dx.doi.org/10.2307/2325206
• [6] Bromwich T.J.I’A., An Introduction to the Theory of Infinite Series, Chelsea, Providence, 1991
• [7] Chen C.-P., Qi F., The best bounds in Wallis’ inequality, Proc. Amer. Math. Soc., 2005, 133(2), 397–401 http://dx.doi.org/10.1090/S0002-9939-04-07499-4
• [8] Chu J.T., A modified Wallis product and some applications, Amer. Math. Monthly, 1962, 69(5), 402–404 http://dx.doi.org/10.2307/2312135
• [9] Henrici P., Applied and Computational Complex Analysis. II, Wiley Classics Lib., John Wiley & Sons, New York, 1991
• [10] Hirschhorn M.D., Comments on the paper “Wallis’ sequence …” by Lampret, Austral. Math. Soc. Gaz., 2005, 32(3), 194
• [11] Kazarinoff D.K., On Wallis’ formula, Edinburgh Math. Notes, 1956, 40, 19–21 http://dx.doi.org/10.1017/S095018430000029X
• [12] Knopp K., Theory and Applications of Infinite Series, Hafner, New York, 1971
• [13] Lampret V., Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product π could be computed fairly accurately, Austral. Math. Soc. Gaz., 2004, 31(5), 328–339
• [14] Lampret V., Constructing the Euler-Maclaurin formula (Celebrating Euler’s 300th birthday), Int. J. Math. Stat., 2007, 1(A07), 60–85
• [15] Lewin J., Lewin M., An Introduction to Mathematical Analysis, Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1993
• [16] Mortici C., A new method for establishing and proving accurate bounds for the Wallis ratio, Math. Inequal. Appl., 2010, 13(4), 803–815
• [17] Mortici C., New approximation formulas for evaluating the ratio of gamma functions, Math. Comput. Modelling, 2010, 52(1–2), 425–433 http://dx.doi.org/10.1016/j.mcm.2010.03.013
• [18] Mortici C., On some accurate estimates of π, Bull. Math. Anal. Appl., 2010, 2(4), 137–139
• [19] Mortici C., Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Soc. Simon Stevin, 2010, 17(5), 929–936
• [20] Păltănea E., On the rate of convergence of Wallis’ sequence, Austral. Math. Soc. Gaz., 2007, 34(1), 34–38
• [21] Sofo A., Some representations of π, Austral. Math. Soc. Gaz., 2004, 31(3), 184–189
• [22] Sun J.-S., Qu C.-M., Alternative proof of the best bounds of Wallis’ inequality, Commun. Math. Anal., 2007, 2(1), 23–27
• [23] Wallis J., Computation of π by successive interpolations, Arithmetica Infinitorum, 1655; In: A Source Book in Mathematics, 1200–1800, Harvard University Press, Cambridge, 1969, 224–253
• [24] Wolfram S., Mathematica, v. 7.0, Wolfram Research, 1988–2009

Identyfikatory

DOI

10.2478/s11533-011-0138-4