An asymptotic approximation of Wallis’ sequence
Open Mathematics (2012)
- Volume: 10, Issue: 2, page 775-787
- ISSN: 2391-5455
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topVito Lampret. "An asymptotic approximation of Wallis’ sequence." Open Mathematics 10.2 (2012): 775-787. <http://eudml.org/doc/269161>.
@article{VitoLampret2012,
abstract = {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, $W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b^\{\prime \}_n )$ with $a_n = 2 + \frac\{1\}\{\{2n + 1\}\} + \frac\{2\}\{\{3(2n + 1)^2 \}\} - \frac\{1\}\{\{3n(2n + 1)^\{\prime \}\}\}b_n = \frac\{2\}\{\{33(n + 1)^\{2^\{\prime \}\} \}\}b^\{\prime \}_n \frac\{1\}\{\{13n^\{2^\{\prime \}\} \}\}n \in \mathbb \{N\} $ .},
author = {Vito Lampret},
journal = {Open Mathematics},
keywords = {Approximation; Estimate; Inequality; π; Rate of convergence; Wallis’ ratio; Wallis’ sequence; approximation; estimate; rate of convergence; Wallis' ratio},
language = {eng},
number = {2},
pages = {775-787},
title = {An asymptotic approximation of Wallis’ sequence},
url = {http://eudml.org/doc/269161},
volume = {10},
year = {2012},
}
TY - JOUR
AU - Vito Lampret
TI - An asymptotic approximation of Wallis’ sequence
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 775
EP - 787
AB - 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, $W(n) \cdot (a_n + b_n ) < \pi < W(n) \cdot (a_n + b^{\prime }_n )$ with $a_n = 2 + \frac{1}{{2n + 1}} + \frac{2}{{3(2n + 1)^2 }} - \frac{1}{{3n(2n + 1)^{\prime }}}b_n = \frac{2}{{33(n + 1)^{2^{\prime }} }}b^{\prime }_n \frac{1}{{13n^{2^{\prime }} }}n \in \mathbb {N} $ .
LA - eng
KW - Approximation; Estimate; Inequality; π; Rate of convergence; Wallis’ ratio; Wallis’ sequence; approximation; estimate; rate of convergence; Wallis' ratio
UR - http://eudml.org/doc/269161
ER -
References
top- [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 Zbl0672.10017
- [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 Zbl1049.05006
- [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 Zbl0106.27203
- [9] Henrici P., Applied and Computational Complex Analysis. II, Wiley Classics Lib., John Wiley & Sons, New York, 1991 Zbl0925.30003
- [10] Hirschhorn M.D., Comments on the paper “Wallis’ sequence …” by Lampret, Austral. Math. Soc. Gaz., 2005, 32(3), 194 Zbl1111.40003
- [11] Kazarinoff D.K., On Wallis’ formula, Edinburgh Math. Notes, 1956, 40, 19–21 http://dx.doi.org/10.1017/S095018430000029X Zbl0072.28401
- [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 Zbl1132.65002
- [15] Lewin J., Lewin M., An Introduction to Mathematical Analysis, Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1993 Zbl1091.26001
- [16] Mortici C., A new method for establishing and proving accurate bounds for the Wallis ratio, Math. Inequal. Appl., 2010, 13(4), 803–815 Zbl1206.33006
- [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 Zbl1201.33003
- [18] Mortici C., On some accurate estimates of π, Bull. Math. Anal. Appl., 2010, 2(4), 137–139 Zbl1312.11095
- [19] Mortici C., Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Soc. Simon Stevin, 2010, 17(5), 929–936 Zbl1209.26026
- [20] Păltănea E., On the rate of convergence of Wallis’ sequence, Austral. Math. Soc. Gaz., 2007, 34(1), 34–38 Zbl1185.26004
- [21] Sofo A., Some representations of π, Austral. Math. Soc. Gaz., 2004, 31(3), 184–189 Zbl1119.11324
- [22] Sun J.-S., Qu C.-M., Alternative proof of the best bounds of Wallis’ inequality, Commun. Math. Anal., 2007, 2(1), 23–27 Zbl1160.05301
- [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
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