Abstract
Let denote the Euler-Mascheroni constant, and let the sequences
The main aim of this paper is to find the values r, s, t, a, b, c and d which provide the fastest sequences and approximating the Euler-Mascheroni constant. Also, we give the upper and lower bounds for in terms of .
MSC: 11Y60, 40A05, 33B15.
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1 Introduction
The Euler-Mascheroni constant is defined as the limit of the sequence
where denotes the n th harmonic number defined for by
Several bounds for have been given in the literature [1–7]. For example, the following bounds for were established in [3, 7]:
The convergence of the sequence to γ is very slow. Some quicker approximations to the Euler-Mascheroni constant were established in [8–21]. For example, Cesàro [8] proved that for every positive integer , there exists a number such that the following approximation is valid:
Entry 9 of Chapter 38 of Berndt’s edition of Ramanujan’s Notebooks [[22], p.521] reads,
‘Let , where n is a positive integer. Then, as n approaches infinity,
For the history and the development of Ramanujan’s formula, see [20].
Recently, by changing the logarithmic term in (1.1), DeTemple [15], Negoi [18] and Chen et al. [14] have presented, respectively, faster and faster asymptotic formulas as follows:
Chen and Mortici [13] provided a faster asymptotic formula than those in (1.2) to (1.4),
and posed the following natural question.
Open problem For a given positive integer p, find the constants () such that
is the sequence which would converge to γ in the fastest way.
Very recently, Yang [21] published the solution of the open problem (1.6) by using logarithmic-type Bell polynomials.
For all , let
and
Chen and Li [12] proved that for all integers ,
and
Now we define the sequences
and
respectively. Our Theorems 1 and 2 are to find the values r, s, t, a, b, c and d which provide the fastest sequences and approximating the Euler-Mascheroni constant.
Theorem 1 Let be defined by (1.9). For
we have
and
The speed of convergence of the sequence is .
Theorem 2 Let be defined by (1.10). For
we have
The speed of convergence of the sequence is .
Our Theorems 3 and 4 establish the bounds for in terms of .
Theorem 3 Let be defined by (1.7). Then
Theorem 4 Let be defined by (1.7). Then
Remark 1 The inequality (1.14) is sharper than (1.8), while the inequality (1.13) is sharper than (1.14).
2 Lemmas
Before we prove the main theorems, let us give some preliminary results.
The constant γ is deeply related to the gamma function thanks to the Weierstrass formula:
The logarithmic derivative of the gamma function
is known as the psi (or digamma) function. The successive derivatives of the psi function
are called the polygamma functions.
The following recurrence and asymptotic formulas are well known for the psi function:
(see [[23], p.258]), and
(see [[23], p.259]). From (2.1) and (2.2), we get
It is also known [[23], p.258] that
If is convergent to zero and there exists the limit
with , then there exists the limit
Lemma 1 gives a method for measuring the speed of convergence.
Lemma 2 [[26], Theorem 9]
Let and be integers. Then, for all real numbers ,
where
and () are Bernoulli numbers defined by
It follows from (2.4) that for ,
from which we imply that for ,
3 Proofs of Theorems 1-4
Proof of Theorem 1 By using the Maple software, we write the difference as a power series in :
According to Lemma 1, we have three parameters r, s and t which produce the fastest convergence of the sequence from (3.1)
namely if
Thus, we have
By using Lemma 1, we obtain the assertion of Theorem 1. □
Proof of Theorem 2 By using the Maple software, we write the difference as a power series in :
According to Lemma 1, we have four parameters a, b, c and d which produce the fastest convergence of the sequence from (3.2)
namely if
Thus, we have
By using Lemma 1, we obtain the assertion of Theorem 2. □
Proof of Theorem 3 Here we only prove the second inequality in (1.13). The proof of the first inequality in (1.13) is similar. The upper bound of (1.13) is obtained by considering the function F for defined by
Differentiation and applying the right-hand inequality of (2.5) yield
where
Therefore, for .
For , we compute directly:
Hence, the sequence is strictly increasing. This leads to
by using the asymptotic formula (2.3). This completes the proof of the second inequality in (1.13). □
Proof of Theorem 4 Here we only prove the first inequality in (1.14). The proof of the second inequality in (1.14) is similar. The lower bound of (1.14) is obtained by considering the function G for defined by
Differentiation and applying the left-hand inequality of (2.5) yield
where
Therefore, for .
For , we compute directly:
Hence, the sequence is strictly increasing. This leads to
by using the asymptotic formula (2.3). This completes the proof of the first inequality in (1.14). □
Remark 2 Some calculations in this work were performed by using the Maple software for symbolic calculations.
Remark 3 The work of the first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.
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Dedicated to Professor Hari M Srivastava.
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Authors’ contributions
CM proposed the sequence . CPC proposed the sequence . CM proposed to solve the problems using Lemma 1, while CPC used Lemma 2 in evaluations. Both authors made the computations and verified their corectedness. The authors read and approved the final manuscript.
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Mortici, C., Chen, CP. On the harmonic number expansion by Ramanujan. J Inequal Appl 2013, 222 (2013). https://doi.org/10.1186/1029-242X-2013-222
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DOI: https://doi.org/10.1186/1029-242X-2013-222