Abstract
The purpose of this paper is to give some sequences that converge quickly to Ioachimescu’s constant related to Ramanujan’s formula by the multiple-correction method.
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1 Introduction
In 1895, Ioachimescu (see [1]) introduced a constant \(\ell \), which today bears his names, as the limit of the sequence defined by
The sequence \(I(n)_{n\ge 1}\) has attracted much attention lately and several generalizations have been given (see, e.g., [2, 3]). Recently, Chen et al. [4] have obtained the complete asymptotic expansion of Ioachimescu’s sequence,
where \(\mathbf {b}_n\) denotes the nth Bernoulli number.
One easily obtains the following representations of the Ioachimescu’s constant:
and
A representation of the Ioachimescu’s constant has also been given by Ramanujan (1915) [20]:
From this, one easily obtains a representation of the Ioachimescu’s constant in terms of the extended \(\zeta \) function
From [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 [9, 10], Sîntămărian has obtained some new sequences that converge to \(\ell (a,s)\) with the rate of convergence of \(n^{-s-15}\). Other results regarding \(\ell (a,s)\) can be found in [6,7,8] and some of the references therein. In this paper, we will give some sequences that converge quickly to Ioachimescu’s constant \(\ell \) by multiple-correction method [11,12,13], based on the sequence
This method could be used to solve other problems, such as Euler–Mascheroni constant, Glaisher–Kinkelin’s and Bendersky–Adamchik’s constants, and Somos’ quadratic recurrence constant [14,15,16,17].
2 Main result
The following lemma gives a method for measuring the rate of convergence; for its proof, see Mortici [18, 19].
Lemma 2.1
If the sequence \((x_n)_{n\in \mathbb {N}}\) is convergent to zero and the limit
exists when \(s>1,\) then
Now we apply the multiple-correction method to study sequences with faster rate of convergence for Ioachimescu’s constant.
Theorem 2.2
For Ioachimescu’s constant, we have the following convergent sequence:
where
Proof
(Step 1) The initial-correction. We choose \(\eta _0(n)=0\), and let
Developing the expression (2.4) into power series expansion in 1 / n, we easily obtain
By Lemma 2.1, we get the rate of convergence of the \(\left( I_0(n)-\ell \right) _{n\in \mathbb {N}}\) as \(n^{-\frac{1}{2}}\), since
(Step 2) The first-correction. Ramanujan [20] made the claim (without proof) for the gamma function
where \(\theta _x\rightarrow 1\) as \(x\rightarrow +\infty \) and \(\frac{3}{10}< \theta _x < 1\). This open problem was solved by Karatsuba [21]. This formula provides a more accurate estimation for the factorial function. Motivated by his idea, we let
and define
Developing (2.7) into power series expansion in 1 / n, we have
-
(i)
If \(a\ne -\frac{1}{2}\), then the rate of convergence of the \(\left( I_1(n)-\ell \right) _{n\in \mathbb {N}}\) is \(n^{-\frac{1}{2}}\), since
$$\begin{aligned} \lim _{n\rightarrow \infty }n^{\frac{1}{2}}\left( I_1(n)-\ell \right) =\frac{1}{2}(2a+1)\ne 0. \end{aligned}$$ -
(ii)
If \(a_1=-\frac{1}{2}, b_2=\frac{1}{2}, b_1=\frac{7}{48}\), and \(b_0=\frac{1}{864}\), from (2.8) we have
$$\begin{aligned} I_1(n)-I_1(n+1)=\frac{7}{1536}\frac{1}{n^\frac{11}{2}}+O\left( \frac{1}{n^\frac{13}{2}}\right) . \end{aligned}$$Hence the rate of convergence of the \(\left( I_1(n)-\ell \right) _{n\in \mathbb {N}}\) is \(n^{-^\frac{9}{2}}\), since
$$\begin{aligned} \lim _{n\rightarrow \infty }n^{\frac{9}{2}}\left( I_1(n)-\ell \right) =\frac{7}{6912}. \end{aligned}$$
(Step 3) The second-correction. We set the second-correction function in the form of
and define
Developing (2.10) into power series expansion in 1 / n, we have
By the same method as above, we find \(u_1=-\frac{7}{576}, v_1=\frac{23}{42}.\)
Applying Lemma 2.1 again, one has
(Step 4) The third-correction. Similarly, we set the third-correction function in the form of
and define
By the same method as above, we find \(u_2=\frac{67483}{84672}, v_2=-\frac{735611}{14171430}.\)
Applying Lemma 1 again, one has
Similarly, by repeating the above approach for Ioachimescu’s constant (the details are omitted here), we can prove Theorem 2.2. \(\square \)
Remark 2.3
It is worth to point out that Theorem 2.2 provides some sequences with faster rate of convergence for Ioachimescu’s constant related to Ramanujan’s formula by multiple-correction method. Similarly, by repeating the above approach step by step, we can get some sequences with faster rate of convergence for Ioachimescu’s constant. Meanwhile, parameters that need to be calculated are also greatly increased, this will lead to dramatic increase in computing.
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The author was supported by the National Natural Science Foundation of China under Grant Number 61403034.
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You, X. Some sequences converging towards Ioachimescu’s constant related to Ramanujan’s formula. Ramanujan J 45, 391–397 (2018). https://doi.org/10.1007/s11139-016-9848-3
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DOI: https://doi.org/10.1007/s11139-016-9848-3