An asymptotic approximation of Wallis’ sequence

Open Mathematics (2012)

• Volume: 10, Issue: 2, page 775-787
• ISSN: 2391-5455

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Abstract

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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\left(n\right)·\left({a}_{n}+{b}_{n}\right)<\pi with ${a}_{n}=2+\frac{1}{2n+1}+\frac{2}{3{\left(2n+1\right)}^{2}}-\frac{1}{3n{\left(2n+1\right)}^{\text{'}}}{b}_{n}=\frac{2}{33{\left(n+1\right)}^{{2}^{\text{'}}}}{b}_{n}^{\text{'}}\frac{1}{13{n}^{{2}^{\text{'}}}}n\in ℕ$ .

How to cite

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Vito 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

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