On new sequences converging towards the Ioachimescu’s constant. (English) Zbl 1421.11110
From the introduction: In 1895, A. G. Ioachimescu [Problem 16. Gaz. Mat. 1, No. 2, 39 (1895)] introduced a constant \(\ell\), which today bears his
names, as the limit of the sequence defined by
\[ I_n = 1+ \frac 1{\sqrt 2}+ \ldots +\frac 1{\sqrt n} - 2(\sqrt{n} - 1),\quad n\in\mathbb N. \]
The sequence \(I(n)_{\ge1}\) has attracted much attention lately and several generalizations have been given. Recently, Chen, Li and Xu [C. Chen et al., Proc. Jangjeon Math. Soc. 13, No. 3, 299–304 (2010; Zbl 1245.11128)] have obtained the complete asymptotic expansion of the Ioachimescu sequence,
\[ I_n \sim \ell + \frac 1{2\sqrt n} - \sum_{k=1}^\infty \frac{B_{2k}}{(2k)!} \frac {(4k - 3)!!}{2^{2k-1}n^{2k-1/2}}, \quad n\in\mathbb N, \]
where \(B_n\) denotes the \(n\)-th Bernoulli number.
The purpose of this paper is to give some sequences that converge quickly to Ioachimescu’s constant related to Ramanujan formula by multiple-correction method. This method could be used to solve other problems, such as Euler-Mascheroni constant, Glaisher-Kinkelin’s and Bendersky-Adamchik’s constants, Somos’ quadratic recurrence constant, and so on.
\[ I_n = 1+ \frac 1{\sqrt 2}+ \ldots +\frac 1{\sqrt n} - 2(\sqrt{n} - 1),\quad n\in\mathbb N. \]
The sequence \(I(n)_{\ge1}\) has attracted much attention lately and several generalizations have been given. Recently, Chen, Li and Xu [C. Chen et al., Proc. Jangjeon Math. Soc. 13, No. 3, 299–304 (2010; Zbl 1245.11128)] have obtained the complete asymptotic expansion of the Ioachimescu sequence,
\[ I_n \sim \ell + \frac 1{2\sqrt n} - \sum_{k=1}^\infty \frac{B_{2k}}{(2k)!} \frac {(4k - 3)!!}{2^{2k-1}n^{2k-1/2}}, \quad n\in\mathbb N, \]
where \(B_n\) denotes the \(n\)-th Bernoulli number.
The purpose of this paper is to give some sequences that converge quickly to Ioachimescu’s constant related to Ramanujan formula by multiple-correction method. This method could be used to solve other problems, such as Euler-Mascheroni constant, Glaisher-Kinkelin’s and Bendersky-Adamchik’s constants, Somos’ quadratic recurrence constant, and so on.
Reviewer: Olaf Ninnemann (Uffing am Staffelsee)
MSC:
| 11Y60 | Evaluation of number-theoretic constants |
| 11A55 | Continued fractions |
| 41A25 | Rate of convergence, degree of approximation |
Citations:
Zbl 1245.11128References:
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