Abstract
The aim of this paper is to introduce a new family of sequences which faster converge to the Euler-Mascheroni constant. Finally, numerical computations are given. Mathematical subject classification: 41A60, 41A25, 57Q55.
Euler-Mascheroni constant; speed of convergence
Fast convergences towards Euler-Mascheroni constant
Cristinel Mortici
Valahia University of Târgoviste, Department of Mathematics, Bd. Unirii 18, 130082 Târgoviste, Romania E-mails: cmortici@valahia.ro / cristinelmortici@yahoo.com
ABSTRACT
The aim of this paper is to introduce a new family of sequences which faster converge to the Euler-Mascheroni constant. Finally, numerical computations are given.
Mathematical subject classification: 41A60, 41A25, 57Q55.
Key words: Euler-Mascheroni constant; speed of convergence.
1 Introduction
One of the most important constants in mathematics is defined as the limit ofthe sequence
denoted γ = 0.57721566490153286.... It is now known as the Euler-Mascheroni constant, in honour of the Swiss mathematician Leonhard Euler (1707-1783) and of the Italian mathematician Lorenzo Mascheroni (1750-1800).
The sequence (γn)n > 1and the constant γ have numerous applications inmany areas of mathematics, such as analysis, theory of probability, special functions, and number theory. As a consequence, many authors are preoccupied to improve the speed of convergence of the sequence (γn) n > 1, which is very slowly, if we take into account that it converges toward its limit like n-1.
More precisely, we mention the following results related to the speed of convergence of the sequence (γn) n > 1:
(see, [14, 15, 28]). We also refer here to the papers [1, 2, 5-12, 20-27], where important improvements of the speed of convergence of γn were established.
The complete asymptotic expansion of the sequence (γn) n > 1 is
where the Bernoulli numbers B2k are defined by
DeTemple [3-4] introduced the sequence
which converges to γ like n-2, since
Recently, Mortici [17] introduced the sequences
and
which converges as n-3, since
See [17, Theorem 2.1]. Furthermore, the arithmetic mean of the sequences (un)n > 1 and (vn) n > 1,
converges to γ as n-4.
We open here a new direction to accelerate the sequence (γn) n > 1, that is to consider an additional term of the form
where P, Q are polynomials of the same degree, having the leading coefficient equal to one. Precisely, we introduce the sequences
whose speeds of convergence increase to n-3, respective n-5, since
Our study is based on the following result, which represents a powerful tool for constructing some asymptotic expansions, or to accelerate some convergences.
Lemma 1. If (ωn)n > 1 is convergent to zero and there exists the limit
wiht k > 1, then there exist the limit:
For proofs and further applications, see [13-19]. The sequences (1)-(2) were introduced in [17] using Lemma 1. Clearly the sequence (ωn) n > 1 converges more quickly when the value of k satisfying (3) is larger.
2 First degree term
In this section we define the sequence
to find the values a, b which provide the fastest sequence (ωn) n > 1. First
and we are concentrated to compute a limit of the form (3). In this sense, we used a computer software to obtain the following representation in power series:
We can state the following
Theorem 2.
ii) If
, then the speed of convergence of the sequence (ωn)n > 1 is n-2, since
iii) If
(equivalent with a = -1/12, b = 5/12), then the speed of convergence of the sequence (ωn) n > 1 is n-3, since
The proof of Theorem 2 easily follows from Lemma 1 and (4).
For a = -1/12, b = 5/12, the relation (4) becomes
and so, iii) is completely proved.
3 Second degree term
Now we define the sequence
to find the values a, b, c, d which provide the fastest sequence (λn) n > 1. First
and we are concentrated to compute a limit of the form (3).
In this sense, we used again the computer software to obtain the following representation in power series:
We can state the following
Theorem 3.
i) Let us denote the coefficients of (5) by
ii) If α ≠ 0, then the speed of convergence of the sequence (λn)n > 1 is n-1, since
iii) If α = 0 and β ≠ 0, then the speed of convergence of the sequence (λn) n > 1 is n-2, since
iv) If α = β = 0 and δ ≠ 0, then the speed of convergence of the sequence (λn) n > 1 is n-3, since
v) If α = β = δ = 0 and η ≠ 0, then the speed of convergence of the sequence (λn) n > 1 is n-4, since
vi) If a = β = δ = η = 0 (equivalent with a = 33/140, b = 37/1680, c = 103/140, d = 61/336), then the speed of convergence of the sequence (λn) n > 1 is n-5, since
The proof of Theorem 3 easily follows from Lemma 1 and (5).
For a = 33/140, b = 37/1680, c = 103/140, d = 61/336, the relation (5) becomes
and so, vi) is completely proved.
4 Concluding remarks
As least theoretically, further sequences of the form
can be defined, where deg P = deg Q = k > 3. As above,
and if we expand (6) into a power series of n-1, then the 2k coefficients of the polynomials P and Q are the unique solution of the system obtained by imposing that the first 2k coefficients of the power series (6) vanish. In this case,
with θ ≠ 0. By Lemma 1, (Mn)n > 1 tends to zero as n-(2k+1) , since
Finally, we offer some numerical computations to prove the superiority of our sequences (νn)n > 1and (µn) n > 1 over the classical sequence (γn) n > 1 and theDeTemple sequence (Rn) n > 1. Remark that already µ1 approximates γ with seven exact decimals.
Our approach works by small degrees k of P (t) and Q (t) , respectively, compared with the obtained order 2k+1 of the speed of convergence, but for larger values of k, it becomes quite difficult (even for computer softwares) to compute the coefficients of the polynomials and the limit θ /(2k+1) of the sequences n2k+1Mn.
We propose now a refined numerical method to compute the coefficients based on an explicit formula for the series expansion of Mn-Mn+1 around 1/n.
Starting with formula (6), we have
As in the cases studied above, we assume that the polynomials P and Q have rational coefficients. Let us consider the factorizations
where the roots α ν and βν are complex numbers. Then (7) takes the form
Next, we have for real numbers α, β that
Replacing α by α-1 and β by β-1, we get
and
Finally, using
it follows from (10)-(13):
For every d the corresponding term in brackets on the right-hand side of (14) is a polynomial in 2k variables, which is symmetric in α1, α2,...,αk and in β1,β2,...,βk. Therefore, the proof of the main theorem on symmetric polynomials involves an algorithm to express the polynomials (14) in terms of the elementary symmetric polynomials in α1, α2,...,αk and in β1,β2,...,βk. So, one obtains equations in terms of the coefficients a0,...,ak and b0,...,bk.
But, a second method to deduce the rational coefficients a0,...,ak and b0,...,bk from the (complex) solution of the system
is based on the numerical continued fraction algorithm. Provided that the coefficients a0,...,ak-1 and b0,...,bk-1 are rationals, the following pure numerical method works without using any computer algebra software in order to obtain the coefficients. Here, we first solve the system (15) numerically, which for large k is much simpler than to compute the coefficients a0,...,ak and b0,...,bk by a computer algebra system. We demonstrate the method by computing the rationals a0 ,a1,a2 and b0,b1,b2 for k = 3. The system
has (the unique) solution
It follows that
Next, the continued fraction algorithm applied to the above numerical values, gives
Then we have
with
It is to be noticed that these results were rediscovered by us using Lemma 1 presented in the first part of this paper. We omit the proof for sake of simplicity.
Finally, remark that if the polynomials P and Q of k-th degree are already determined, say using the previous numerical method, then the problem of verifiying the speed of convergence of the corresponding sequence Mn using Lemma 1 becomes a much easier task.
Acknowledgements. The author thanks the editors and the anonymous referee for useful ideas which improved much the initial form of this paper.
Received: 31/VIII/09
Accepted: 29/VII/10.
#CAM-122/09.
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Publication Dates
-
Publication in this collection
22 Nov 2010 -
Date of issue
2010
History
-
Accepted
29 July 2010 -
Received
31 Aug 2009
