Journal of Number Theory 172 (2017) 256–269
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Journal of Number Theory www.elsevier.com/locate/jnt
Wallis’ sequence estimated accurately using an alternating series Vito Lampret University of Ljubljana, Ljubljana, 386, Slovenia
a r t i c l e
i n f o
Article history: Received 4 May 2016 Accepted 25 August 2016 Available online 8 October 2016 Communicated by D. Goss MSC: primary 26D20, 40A25, 41A60 secondary 11Y60, 11Y99, 65B15 Keywords: Alternating Approximation Asymptotic Estimate Inequality π Rate of convergence Wallis
a b s t r a c t An asymptotic approximation of Wallis’ sequence m → m 4k2 is presented as Wm := 4k2 −1 k=1
Wm =
mπ exp 2σq (m) · exp rq (m) , 2m + 1
where q/2
σq (x) :=
i=1
1 − 4−i B2i i(2i − 1) · x2i−1
(Bk are the Bernoulli coefficients), and where |rq (m)| < rq∗ (m) :=
2π(q − 2)! , 3(2mπ)q−1
for any integers m ≥ 1 and q ≥ 2 . Parameters m and q control the error factor exp rq (m) . © 2016 Elsevier Inc. All rights reserved.
E-mail address:
[email protected]. http://dx.doi.org/10.1016/j.jnt.2016.08.014 0022-314X/© 2016 Elsevier Inc. All rights reserved.
Wallis’ sequence estimated accurately using an alternating series ABSTRACT Vito Lampret∗ University of Ljubljana Ljubljana, Slovenia 386, EU
Key words. alternating, approximation, asymptotic, estimate, inequality, π, rate of convergence, Wallis. MSC (2010). 26D20, 40A25, 41A60 (11Y60, 11Y99, 65B15).
1
Introduction
( ) The Wallis sequence Wn n≥1 defined as Wn :=
n ∏ k=1
4k 2 4k 2 − 1
(1)
is clearly strictly increasing and was used by English mathematician Wallis1 in 1655 [15, 16] to introduce π as a limit: π = lim Wn . (2) 2 n→∞ This is in the history the first presentation of π as a limit of an analytically given sequence. Wallis’ sequence was investigated by many authors since it is closely related to the constant π, see for example [2, 3, 4, 5, 15]. Although Wallis’ sequence was usually considered as unsuitable for numerical computation of π, it was shown in [8] and [13] that it is usable also for computation of some decimals of π. Moreover, knowing the value of π, it is possible to obtain rather good approximations of Wn . But Wn is closely related with (2n) 1 Catalan numbers cn := n+1 n which play important role in combinatorics and the theory of graphs, see e.g. [7]. The connection is given through the formula cn = ∗ 1
4n 1 √ ·√ Wn (n + 1) 2n + 1
e-mail address:
[email protected] John Wallis, 1616 – 1703
(n ∈ N).
All these and similar facts have attracted mathematicians to study Wallis’ sequence for a very long period of time. Consequently, during the time a great amount of articles about Wallis’ sequence have been published, recently [6, 8, 11, 13, 14]. In [6] are given the following three main results: [6, Theorem 1] For all n ∈ N, ( ) ( ) π 1 π 1 1− < Wn < 1− 2 4n + α 2 4n + β with the best possible constants α = 5/2 and β = 2.614 . . .. [6, Theorem 2] For all n ∈ N, π 2
( 1−
1 4n + 5/2
)λ
π < Wn < 2
( 1−
1 4n + 5/2
)µ
with the best possible constants λ = 1 and µ = 0.981 . . ( .. )2 ( ) Γ(x+1) 1 −1 1 π [6, Theorem 3] The function W (x) := 2 1 + 2x has the x Γ(x+1/2) property that W (n) ≡ Wn and has the following asymptotic expansion π W (x) ∼ 2
( 1−
1 4x + 5/2
j )∑∞ j=0 (rj /x )
, x → ∞,
with the coefficients rj given by some recurrence relation including the Bernoulli numbers. In [14] Mortici replaced the Wallis sequence n 7→ Wn , slowly converging to
π 2,
P (m,n) , where P (m, x) by a modified sequence n 7→ Wn′ (m) := Wn Q(m,n)
and Q(m, x) being polynomials of degree m and with leading coefficients equal to 1. Mortici optimized the rate of convergence of the sequences ( ) ( ) n 7→ ln (Wn′ (m) toward ln π2 for m ∈ {1, 2, 3}. He proved, for m = 1, 2, 3, that the fastest convergence has the speed n−3 , n−5 and n−7 , respectively. In the proofs Mortici used his lemma [12, p. 434], which is crucial for [14]. Lemma[12] If (ωn )n≥1 is convergent to zero and there exists the limit l := ( ) lim nk (ωn − ωn+1 ) with k > 1, then lim nk−1 ωn = l/(k − 1). n→∞ n→∞ ( ) This lemma was used for the sequence ωn (m) n≥1 defined by the equation ( ) Wn′ (m) · exp ωn (m) = π2 , for m, n ≥ 1. Mortici offered a posterior/experimental estimates of the discussed convergences. However, the knowledge of convergence speed is not enough for the estimate of the limit
π 2
or for the estimate of Wn . Only once Mortici
presented a prior sharp estimate of the convergence in question, namely in 2
[14, Theorem 3.1] he showed that, for n ≥ 1, ( ) ( ) π 3 9 π 3 9 1441 ′ exp − < Wn (1) < exp − + . 2 256n3 512n4 2 256n3 512n4 81920n5 It is worth mentioning that in [11, p. 777] an asymptotic approximation of Wallis’ sequence was presented in the form )2n+1 ( ) ( ) π ( 1 W (n) = e 1 − 2n+1 exp 2σr∗ (n) − 2σr∗ (n + 21 ) · exp δr (n) , 2 where B2 , B4 , B6 , . . . are the Bernoulli coefficients, σ0∗ (x) ≡ 0 and
σr∗ (x) =
r ∑ i=1
and |δr (n)|
0, m (m + 2)(t + x) tm (m + 2) tm+1 −
|x|m+2 |x|m+2 ( ) < r (x, t) < − , for x < 0. m (m + 2) tm+1 (m + 2) t − |x| tm
The asymptotic approximation given in [11] is able to compete, concerning the accuracy, with the approximations given in [14]. It is not clear why Mortici has not mentioned [11]? Perhaps due to the fact that the approximation given in [11] has a bit unattractive form due to the factors ( )2n+1 ( ) 1 1 − 2n+1 and exp 2σr (n) − 2σr (n + 21 ) . 3
In our contribution we shall improve, concerning the form and the accuracy, the majority of papers about the approximation of Wallis’ sequence. We shall use the Euler-Boole alternating summation formula, which, indeed, will give simple approximations of Wallis’ sequence having high accuracy and are aesthetically pleasing.
2
An alternating series
According to (1) we have Thus, for integers n ≥ m ≥ 1, we obtain2 ) ( 2 ) ( 2n−1 ∑ n (2m + 1) Wn = ln + 2 (−1)i ln(i) ln Wm m2 (2n + 1) i=2m ) ( 2 2n−1 ∑ n (2m + 1) − 2 = ln (−1)i+1 ln(i). m2 (2n + 1)
(3)
i=2m
3
Auxiliary results (Euler-Boole summation formula)
For the last series in (3) we shall use the Euler-Boole summation formula, which can be deduced from the Euler-Maclaurin formulas given also in [9] as Corollary 3.11. Lemma 1 (the Euler-Boole summation formula). For integers m < n and p ≥ 1, and for a function f ∈ C p [2m, 2n], the following summation formula holds: 2n−1 ∑
(−1)k+1 f (k) = −
p ∑ (
[ ] ) 2j − 1 Vj (0) f (j−1) (2n) − f (j−1) (2m)
j=1
k=2m
∫ p
n[
+ (−2)
m
] fp (x) − W fp (x − 1 ) f (p) (2x) dx. W 2
(4)
We note that in (4) the integral is occurring only in the remainder. Here we have Vk (x) ≡
Bk (x) k! ,
the standardized Bernoulli polynomials, where Bk (x) ∑ xt tk is the kth Bernoulli polynomial satisfying the identity etet −1 ≡ ∞ k=0 Bk (x) k! fk (x) is denoted the standardized k-th Bernoulli 1-periodic (|t| < 2π). By W function, that is the 1-periodic continuation of Vk (x) [0,1) (a restriction) fk (x) ≡ from the interval [0, 1) to R. More precisely3 , for k ≥ 2 we have W ∑ By definition we have n i=m xi = 0 for m > n. 3 The symbol ⌊x⌋ denotes the integer part of x. 2
4
f1 (x) = x − ⌊x⌋ − 1 , for x ∈ R r Z. Vk (x − ⌊x⌋), additionally W 2 For Bernoulli coefficients Bk := Bk (0) we have the well known equalities B1 = − 21 and B2j+1 = 0 for j ≥ 1. Therefore, [ V2k+1 (0) ] = 0 for k ∈ N and, ) ∑p ( j (j−1) consequently, the sum j=1 2 − 1 Vj (0) f (2n) figuring in (4) can be simplified. Moreover, referring to [1, items 23.16 and 23.2.24] we have, for q ≥ 2, vq := max |Vq (x)| = 0≤x≤1
1 2 2 1 max |Bq (x)| = ζ(q) ≤ ζ(2) = . q q 0≤x≤1 q! (2π) (2π) 12(2π)q−2 (5)
Additionally, v1 := max |V1 (x)| = max x − 0≤x 1, for m ∈ {2, 3, 4, . . . , 100}. Similarly we get L2 [1] > M2 [1], but L2 [m] < M2 [m] for m ∈ {2, 3, 4, . . . , 100}. It appears that the inequalities in Corollary 5 are more accurate than the estimates (17).
References [1] M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions, 9th edn, Dover Publications, New York, (1974). [2] D. H. Bailey, J. M. Borwein and P. B. Borwein Ramanujan, Modular Equations, and Approximations to Pi or How to compute One Billion Digits of Pi, Amer. Math. Monthly, 96(1989), 201 – 219. [3] P. Beckmann, A History of π; Golem Press, Boulder, Colorado, 1970, 1971, 1977, 1982; Hippocrene Books 1990; Marboro Books 1990; St. Martin’s Press 1971, 1976. [4] L. Berggren, J. Borwein and P. Borwein, Pi: A Source Book, Springer– Verlag, N.Y., 1997. [5] D. Blatner, The Joy of π, Walker & Co., 1999. [6] J.-E. Deng, T. Ban and C.-P. Chen;Sharp inequalities and asymptotic expansion associated with the Wallis sequence, J. Inequal. Appl., (2015), 2015:186. 8
[7] T. Koshy, Catalan numbers with applications, Oxford University Press, 2009; Oxford, NY. [8] V. Lampret, Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product π could be computed fairly accurately, Austral. Math. Soc. Gaz. 31(2004), 328-339. [9] V. Lampret, Constructing the Euler-Maclaurin formula – celebrating Euler’s 300th birthday International journal of mathematics and statistics, 2007, vol. 1, pp. 60-85. [10] V. Lampret, Approximating the powers with large exponents and bases close to unit, and the associated sequence of nested limits, Int. J. Contemp. Math. Sci. 6 (2011), no. 41-44, 2135–2145. [11] V. Lampret, An asymptotic approximation of Wallis’ sequence, Cent. Eur. J. Math. 10 (2012), no. 2, 775–787. [12] C. Mortici, Product approximations via asymptotic integration, Amer. Math. Monthly, 117 (2010), no. 5, 434–441. [13] C. Mortici, Refinements of Gurland’s formula for pi, Comput. Math. Appl. 62 (2011), 2616–2620. [14] C. Mortici, Optimizing the convergence rate of the Wallis sequence, Hacet. J. Math. Stat. 44 (2015), 101-109. [15] A. Sofo, Some Reprezentations of Pi, Austral. Math. Soc. Gaz., 31(2004), 184 – 189. [16] J. Wallis, Computation of π by Successive Interpolations, (1655) in: A Source Book in Mathematics, 1200 – 1800 (D.J. Struik, Ed.), Harvard University Press, Cambridge, MA, 1969, 224 – 253. [17] S. Wolfram, Mathematica, version 7.0, Wolfram Research, Inc., 1988– 2009.
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