Abstract
We establish more accurate formulas for approximating π which refine some known results due to Gurland and Mortici.
MSC:33B15, 26D07, 41A60.
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1 Introduction
Gurland [1] proved that for all integers ,
Recently, Mortici [[2], Theorem 2] improved Gurland’s result and obtained the following inequality:
where
and
In this paper, we establish more accurate formulas for approximating π which refine the results due to Gurland and Mortici.
Before stating and proving the main theorems, we first introduce the gamma function and some known results.
The familiar gamma function defined by Euler,
is one of the most important functions in mathematical analysis and applications in various diverse areas. The logarithmic derivative of , denoted by , is called the psi (or digamma) function.
The following lemmas are required in the sequel.
If the sequence converges to zero and if the following limit:
exists, then
where ℝ denotes the set of real numbers.
Lemma 1.1 is useful for accelerating some convergences or in constructing some better asymptotic expansions.
Lemma 1.2 For ,
Proof The lower bound in (1.5) is obtained by considering the function defined for by
Using the following representations:
in [[5], p.259, 6.3.21] and
in [[5], p.255, 6.1.1], we find (for and ) that
with
so that (1.8) implies for . Hence, the first inequality in (1.5) holds for .
The upper bound in (1.5) is obtained by considering the function defined for by
Using the above representations (1.6) and (1.7), we find that
with
so that (1.9) implies for . Hence, the second inequality in (1.5) holds for . This completes the proof of Lemma 1.2. □
Remark 1.3 A function f is said to be completely monotonic on an interval I if it has derivatives of all orders on I and satisfies the following inequality:
Dubourdieu [[6], p.98] pointed out that if a non-constant function f is completely monotonic on , then a strict inequality holds true in (1.10). See also [7] for a simpler proof of this result.
From (1.8) and (1.9), we obtain
and
Hence, the functions and are both completely monotonic on .
2 Main results
The famous Wallis sequence is defined by
Wallis (1655) showed that .
The convergence of is very slow, so it is not suitable for approximating π. The Wallis sequence can be expressed as (see [11–13])
Now we define the sequence by
We are interested in finding fixed parameters a, b, c, p, q and r such that converges as fast as possible to the constant π. Our study is based on Lemma 1.1.
Theorem 2.1 Let the sequence be defined by (2.2). Then for
we have
The speed of convergence of the sequence is given by the order estimate .
Proof We write the difference as the following power series in :
The fastest sequence is obtained when the first six coefficients of this power series vanish. In this case, , , , , and , we have
Finally, by using Lemma 1.1, we obtain assertion (2.4) of Theorem 2.1. □
Solutions (2.3) provide the following approximation for π:
This fact motivated us to observe the following theorem.
Theorem 2.2 For all , we have
where
and
Proof Inequality (2.6) can be rewritten as
where
and
The lower bound in (2.9) is obtained by considering the function defined for by
Using the asymptotic expansion [[5], p.257, 6.1.41]
we find
Differentiating and applying the second inequality in (1.5), we find that, for ,
Consequently, the sequence is strictly decreasing. This leads to
which means that the first inequality in (2.9) is valid for .
The upper bound in (2.9) is obtained by considering the function defined for by
We conclude from the asymptotic expansion (2.10) that
Differentiating and applying the first inequality in (1.5) yields, for ,
Consequently, the sequence is strictly increasing. This leads to
which means that the second inequality in (2.9) is valid for . The proof of Theorem 2.2 is complete. □
Remark 2.3 Let , , and be defined by (1.3), (1.4), (2.7) and (2.8), respectively. Direct computation would yield
and
which show that inequality (2.6) is sharper than inequality (1.2).
By using Lemma 1.1, we find that
Among sequences , , and , the sequence is the best in the sense that it is the fastest sequence which would approximate the constant π.
The logarithm of the gamma function has the asymptotic expansion (see [[14], p.32]):
Here denote the Bernoulli polynomials defined by the following generating function:
Note that the Bernoulli numbers are defined by in (2.12).
From (2.11) we easily obtain
Taking in (2.13) and noting that
(see [[5], p.805]), we obtain
namely,
From (2.15) we imply
which implies the following asymptotic expansion for π:
The formula (2.17) motivated us to observe the following theorem.
Theorem 2.4 For all , we have
where
and
Proof Inequality (2.18) can be rewritten as
where
and
The lower bound in (2.21) is obtained by considering the function defined for by
Differentiating and applying the first inequality in (1.5) yields, for ,
with
Hence, for , and therefore, the sequence is strictly increasing. This leads to
which means that the first inequality in (2.21) is valid for .
The upper bound in (2.21) is obtained by considering the function defined for by
We conclude from the asymptotic expansion (2.10) that
Differentiating and applying the first inequality in (1.5) yields, for ,
with
Hence, for , and therefore, the sequence is strictly increasing. This leads to
which means that the second inequality in (2.21) is valid for . The proof of Theorem 2.4 is complete. □
Remark 2.5 The following numerical computations (see Table 1) would show that, for , inequality (2.18) is sharper than inequality (2.6).
By using Lemma 1.1, we find that
which provide the higher-order estimates for the constant π.
Remark 2.6 Some calculations in this work were performed by using the Maple software for symbolic calculations.
References
Gurland J: On Wallis’ formula. Am. Math. Mon. 1956, 63: 643–645. 10.2307/2310591
Mortici C: Refinements of Gurland’s formula for pi. Comput. Math. Appl. 2011, 62: 2616–2620. 10.1016/j.camwa.2011.07.073
Mortici C: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 2010, 23: 97–100. 10.1016/j.aml.2009.08.012
Mortici C: Product approximations via asymptotic integration. Am. Math. Mon. 2010, 117: 434–441. 10.4169/000298910X485950
Abramowitz M, Stegun IA (Eds): Applied Mathematics Series 55 In Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th edition. National Bureau of Standards, Washington; 1972.
Dubourdieu J: Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes. Compos. Math. 1939, 7: 96–111. (in French)
van Haeringen H: Completely monotonic and related functions. J. Math. Anal. Appl. 1996, 204: 389–408. 10.1006/jmaa.1996.0443
Hirschhorn MD: Comments on the paper “Wallis’ sequence…” by Lampret. Aust. Math. Soc. Gaz. 2005, 32: 194.
Lampret V: Wallis sequence estimated through the Euler-Maclaurin formula: even from the Wallis product π could be computed fairly accurately. Aust. Math. Soc. Gaz. 2004, 31: 328–339.
Păltănea E: On the rate of convergence of Wallis’ sequence. Aust. Math. Soc. Gaz. 2007, 34: 34–38.
Chen C-P, Qi F: The best bounds in Wallis’ inequality. Proc. Am. Math. Soc. 2005, 133: 397–401. 10.1090/S0002-9939-04-07499-4
Mortic C: New approximation formulas for evaluating the ratio of gamma functions. Math. Comput. Model. 2010, 52: 425–433. 10.1016/j.mcm.2010.03.013
Mortic C: Completely monotone functions and the Wallis ratio. Appl. Math. Lett. 2012, 25: 717–722. 10.1016/j.aml.2011.10.008
Luke YL: The Special Functions and Their Approximations, vol. I. Academic Press, New York; 1969.
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Lin, L. Further refinements of Gurland’s formula for π. J Inequal Appl 2013, 48 (2013). https://doi.org/10.1186/1029-242X-2013-48
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DOI: https://doi.org/10.1186/1029-242X-2013-48