Abstract
The purpose of this paper is to give some sequences that converge quickly to the Ioachimescu constant by a multiple-correction method.
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1 Introduction
In 1895, Ioachimescu (see [1]) introduced a constant ℓ, which today bears his name, as the limit of the sequence defined by
The sequence \(I(n)_{n\geq1}\) has attracted much attention lately and several generalizations have been given (see, e.g., [2, 3]). Recently, Chen, Li and Xu [4] have obtained the complete asymptotic expansion of the Ioachimescu sequence,
where \(\mathbf{b}_{n}\) denotes the nth Bernoulli number.
One easily obtains the following representations of the Ioachimescu constant:
and
A representation of the Ioachimescu constant has also been given by Ramanujan (1915) [5],
From it one easily obtains a representation of the Ioachimescu constant in terms of the extended ζ function
As a result of [2], we have \(\ell=0.539645491\ldots\) .
Let \(a\in(0,+\infty)\) and \(s\in(0,1)\), the sequence
is convergent [3] and its limit is a generalized Euler constant denoted by \(\ell(a,s)\). Clearly, \(\ell(1,1/2)=\ell\). Furthermore, Sîntămărian has proved that
Also in [3], considering the sequence
she has proved that
and, for the sequence
she has proved that
In [6, 7], Sîntămărian has obtained some new sequences that convergence to \(\ell(a,s)\) with the rate of convergence \(n^{-s-15}\). Other results regarding \(\ell(a,s)\) can be found in [8–10] and some of the references therein.
In our paper, we will give some sequences that converge quickly to the Ioachimescu constant ℓ by a multiple-correction method [11–13], based on the sequence
2 Sequences convergent to the Ioachimescu constant ℓ
The following lemma gives a method for measuring the rate of convergence; for its proof see Mortici [14, 15].
Lemma 1
If the sequence \((x_{n})_{n\in\mathbb{N}}\) is convergent to zero and there exists the limit
with \(s>1\), then
Now we apply multiple-correction method to study faster convergence sequences for the Ioachimescu constant, and this method could be used to solve other problems, such as the Euler-Mascheroni constant, Glaisher-Kinkelin’s and Bendersky-Adamchik’s constants, Somos’ quadratic recurrence constant, and so on [16–19].
Theorem 1
For the Ioachimescu constant, we have the following convergent sequence:
where
Proof
(Step 1) The initial correction. We choose \(\eta_{0}^{(1)}(n)=0\), and let
Developing equation (2.6) into a power series expansion in \(1/n\), we have
By Lemma 1, the rate of convergence of \((I_{0}^{(1)}(n)-\ell )_{n\in\mathbb{N}}\) is \(n^{-\frac{1}{2}}\), since
(Step 2) The first correction. Let
and define
Developing (2.9) into power series expansion in \(1/n\), we obtain
-
(i)
If \(a_{1}\neq-\frac{1}{2}\), the rate of convergence of the \((I_{1}^{(1)}(n)-\ell)_{n\in\mathbb{N}}\) is \(n^{-\frac{1}{2}}\), since
$$\lim_{n\rightarrow\infty}n^{\frac{1}{2}} \bigl(I_{1}^{(1)}(n)- \ell \bigr)=\frac{2a_{1}+1}{4}\neq0. $$ -
(ii)
If \(a_{1}=-\frac{1}{2}\) and \(b_{0}=\frac{1}{6}\), from (2.10) we obtain
$$I_{1}^{(1)}(n)-I_{1}^{(1)}(n+1)=- \frac{5}{384}\frac{1}{n^{\frac{7}{2}}}+O \biggl(\frac{1}{n^{\frac{9}{2}}} \biggr). $$Then the rate of convergence of the \((I_{1}^{(1)}(n)-\ell)_{n\in\mathbb{N}}\) is \(n^{-\frac{5}{2}}\), since
$$\lim_{n\rightarrow\infty}n^{\frac{5}{2}} \bigl(I_{1}^{(1)}(n)- \ell \bigr)=-\frac{1}{192}. $$
(Step 3) The second correction. Similarly, set the second-correction function
and define
By the same method as above, we get \(a_{2}=\frac{1}{192}\), \(b_{4}=\frac {23}{18}\), \(b_{3}=\frac{341}{288}\), \(b_{2}=\frac{27{,}833}{46{,}656}\), \(b_{1}=\frac {726{,}647}{26{,}873{,}856}\), \(b_{0}=-\frac{9{,}196{,}141}{806{,}215{,}680}\).
Applying Lemma 1 again, one has
Repeating the above approach for the Ioachimescu constant, we can prove Theorem 1. □
3 Other sequences convergent to the Ioachimescu constant ℓ
In this section, we provide some other approximation for the Ioachimescu constants by a multiple-correction method. The initial correction is the same as above, we change the correction function from step 2.
(Step 2) The first-correction. Let the second-correction function be
and define
By the same method as above, we find \(a=-\frac{1}{2}\), \(u_{1}=\frac{1}{6}\), \(v_{1}=-\frac{1}{8}\).
Applying Lemma 1, one has
Repeating the above approach for the Ioachimescu constant, we can prove the following theorem.
Theorem 2
For the Ioachimescu constant, we have the following convergent sequence:
where
Remark 1
Theorem 2 provides some quasi-continued fraction sequences with a faster rate of convergence for the Ioachimescu constant.
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Acknowledgements
We are grateful to the editor and anonymous reviewers for their valuable comments and corrections that helped improve the original version of this paper. The research was supported by the National Natural Science Foundation of China under grant no. 61403034, 11571267, and 91538112.
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You, X., Chen, DR. & Shi, H. Some new sequences that converge to the Ioachimescu constant. J Inequal Appl 2016, 148 (2016). https://doi.org/10.1186/s13660-016-1089-x
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DOI: https://doi.org/10.1186/s13660-016-1089-x