Abstract
The aim of this paper is to establish new inequalities for the Euler-Mascheroni constant by the continued fraction method.
MSC:11Y60, 41A25, 41A20.
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1 Introduction
The Euler-Mascheroni constant was first introduced by Leonhard Euler (1707-1783) in 1734 as the limit of the sequence
There are many famous unsolved problems about the nature of this constant (see, e.g., the survey papers or books of Brent and Zimmermann [1], Dence and Dence [2], Havil [3] and Lagarias [4]). For example, it is a long-standing open problem if the Euler-Mascheroni constant is a rational number. A good part of its mystery comes from the fact that the known algorithms converging to γ are not very fast, at least, when they are compared to similar algorithms for π and e.
The sequence converges very slowly toward γ, like . Up to now, many authors have been preoccupied with improving its rate of convergence (see, e.g., [2, 5–22] and the references therein). We list some main results as follows:
Recently, Mortici and Chen [14] provided a very interesting sequence,
and proved
Hence the rate of convergence of the sequence is .
Very recently, by inserting the continued fraction term in (1.1), Lu [9] introduced a class of sequences (see Theorem 1) and showed
In fact, Lu [9] also found without proof. In general, the continued fraction method could provide a better approximation than others, and has less numerical computations.
First, we will prove the following theorem.
Theorem 1 For the Euler-Mascheroni constant, we have the following convergent sequence:
where , and for .
Let
(see the Appendix for their simple expressions) and
For , we have
where
Open problem For every , we have .
The main aim of this paper is to improve (1.3) and (1.4). We establish the following more precise inequalities.
Theorem 2 Let , , and be defined in Theorem 1, then
Remark 1 In fact, Theorem 2 implies that is a strictly increasing function of n, whereas is a strictly decreasing function of n. Certainly, it has similar inequalities for (), we leave these for readers to verify. It is also should be noted that (1.4) cannot deduce the monotonicity of .
Remark 2 It is worth to point out that Theorem 2 provides sharp bounds for a harmonic sequence which are superior to Theorems 3 and 4 of Mortici and Chen [14].
2 The proof of Theorem 1
The following lemma gives a method for measuring the rate of convergence. This lemma was first used by Mortici [23, 24] for constructing asymptotic expansions or to accelerate some convergences. For proof and other details, see, e.g., [24].
Lemma 1 If the sequence is convergent to zero and there exists the limit
with , then there exists the limit
In the sequel, we always assume .
We need to find the value which produces the most accurate approximation of the form
here we note . To measure the accuracy of this approximation, we usually say that approximation (2.3) is better as faster converges to zero. Clearly,
It is well known that for ,
Develo** expression (2.4) into power series expansion in , we obtain
From Lemma 1, we see that the rate of convergence of the sequence is even higher than the value s satisfying (2.1). By Lemma 1, we have
-
(i)
If , then the rate of convergence of is since
-
(ii)
If , from (2.5) we have
Hence the rate of convergence of is since
We also observe that the fastest possible sequence is obtained only for .
Just as Lu [9] did, we may repeat the above approach to determine to step by step. However, the computations become very difficult when . In this paper we use Mathematica software to manipulate symbolic computations.
Let
then
It is easy to get the following power series:
Hence the key step is to expand into power series in . Here we use some examples to explain our method.
Step 1: For example, given to , find . Define
By using Mathematica software (Mathematica Program is very similar to the one given in Remark 3; however, it has a parameter ), we obtain
Substituting (2.8) and (2.10) into (2.7), we get
The fastest possible sequence is obtained only for . At the same time, it follows from (2.11) that
the rate of convergence of is since
We can use the above approach to find (). Unfortunately, it does not work well for . Since , and . So, we may conjecture . Now let us check it carefully.
Step 2: Check to .
Let and be defined in Theorem 1. Applying Mathematica software, we obtain
which is the desired result. Substituting (2.8) and (2.13) into (2.7), we get
the rate of convergence of is since
Next, we can use Step 1 to find , and Step 2 to check and . It should be noted that Theorem 2 will provide the other proofs for and . So we omit the details here.
Finally, we check .
Substituting (2.8) and (2.15) into (2.7), one has
Since
thus the rate of convergence of is .
This completes the proof of Theorem 1.
Remark 3 In fact, if the assertion holds, then the other values () must be true. The following Mathematica Program will generate into power series in with order 16: . .
Remark 4 It is a very interesting question to find for . However, it seems impossible by the above method.
3 The proof of Theorem 2
Before we prove Theorem 2, let us give a simple inequality by the Hermite-Hadamard inequality, which plays an important role in the proof.
Lemma 2 Let f be twice derivable with continuous. If , then
In the sequel, the notation means a polynomial of degree k in x with all of its non-zero coefficients positive, which may be different at each occurrence.
Let us begin to prove Theorem 2. Note , it is easy to see
where
Let . By using Mathematica software, we have
and
Hence, we get the following inequalities for :
Applying , (3.3) and Lemma 2, we get
From (3.1) and (3.4) we obtain
Similarly, we also have
and
Combining (3.5) and (3.6) completes the proof of (1.6).
Note , it is easy to deduce
where
We write . By using Mathematica software, we have
and
Hence, for ,
Applying , (3.8) and (3.1), we get
It follows from (3.7) and (3.9) that
Finally,
and
Combining (3.10) and (3.11) completes the proof of (1.7).
Remark 5 As an example, we give Mathematica Program for the proof of the left-hand side of (3.3):
-
(i)
Together ;
-
(ii)
Take out the numerator of the above rational function, then manipulate the program: Apart .
Appendix
For the reader’s convenience, we rewrite () with minimal denominators as follows.
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Acknowledgements
The authors would like to thank the referees and Prof **aodong Cao for their careful reading of the manuscript and insightful comments. Research of this paper was supported by the National Natural Science Foundation of China (Grant No.11171344) and the Natural Science Foundation of Bei**g (Grant No.1112010).
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Xu, H., You, X. Continued fraction inequalities for the Euler-Mascheroni constant. J Inequal Appl 2014, 343 (2014). https://doi.org/10.1186/1029-242X-2014-343
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DOI: https://doi.org/10.1186/1029-242X-2014-343