Some new sequences that converge to the Ioachimescu constant

Abstract

The purpose of this paper is to give some sequences that converge quickly to the Ioachimescu constant by a multiple-correction method.

1 Introduction

In 1895, Ioachimescu (see [1]) introduced a constant â„“, which today bears his name, as the limit of the sequence defined by

$$I_{n}=1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt {n}}-2(\sqrt {n}-1),\quad n\in\mathbb{N}.$$

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,

$$I_{n}\sim\ell+\frac{1}{2\sqrt {n}}-\sum_{k=1}^{\infty} \frac{\mathbf{b}_{2k}}{(2k)!}\frac{(4k-3)!!}{2^{2k-1}n^{2k-1/2}},\quad n\in\mathbb{N},$$

where $$\mathbf{b}_{n}$$ denotes the nth Bernoulli number.

One easily obtains the following representations of the Ioachimescu constant:

$$\ell= \int_{0}^{\infty}\frac{1-x+\lfloor x \rfloor}{2(1+x)^{3/2}}\,dx$$

and

$$\ell=2-\sum_{k=1}^{\infty}\frac{1}{(\sqrt {k}+\sqrt{k-1})^{2}\sqrt {k}}.$$

A representation of the Ioachimescu constant has also been given by Ramanujan (1915)Â [5],

$$\ell=2-(\sqrt{2}+1)\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{\sqrt {k}}.$$

From it one easily obtains a representation of the Ioachimescu constant in terms of the extended Î¶ function

$$\ell=\zeta \biggl(\frac{1}{2} \biggr)+2.$$

As a result of [2], we have $$\ell=0.539645491\ldots$$â€‰.

Let $$a\in(0,+\infty)$$ and $$s\in(0,1)$$, the sequence

$$y_{n}(a,s)=\frac{1}{a^{s}}+\frac{1}{(a+1)^{s}}+\cdots+ \frac{1}{(a+n-1)^{s}}-\frac{1}{1-s} \bigl[(a+n-1)^{1-s}-a^{1-s} \bigr],\quad n\in\mathbb{N},$$

is convergent [3] and its limit is a generalized Euler constant denoted by $$\ell(a,s)$$. Clearly, $$\ell(1,1/2)=\ell$$. Furthermore, SÃ®ntÄƒmÄƒrian has proved that

$$\lim_{n\rightarrow\infty} n^{s} \bigl(y_{n}(a,s)- \ell(a,s) \bigr)=\frac{1}{2}.$$

Also in [3], considering the sequence

$$u_{n}(a,s)=y_{n}(a,s)-\frac{1}{2(a+n-1)^{s}},$$

she has proved that

$$\lim_{n\rightarrow\infty} n^{s+1} \bigl(\ell(a,s)-u_{n}(a,s) \bigr)=\frac{s}{12}$$

and, for the sequence

$$\alpha_{n}(a,s)=\frac{1}{a^{s}}+\frac{1}{(a+1)^{s}}+\cdots+ \frac{1}{ (a+n-1)^{s}}-\frac{1}{1-s} \biggl( \biggl(a+n-\frac{1}{2} \biggr)^{1-s}-a^{1-s} \biggr),\quad n\in\mathbb{N},$$

she has proved that

$$\lim_{n\rightarrow\infty} n^{s+1} \bigl(\alpha_{n}(a,s)- \ell(a,s) \bigr)=\frac{s}{24}.$$

In [6, 7], SÃ®ntÄƒmÄƒrian 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 [8â€“10] and some of the references therein.

In our paper, we will give some sequences that converge quickly to the Ioachimescu constant â„“ by a multiple-correction method [11â€“13], based on the sequence

$$I(n)=1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt {n}}-2(\sqrt {n}-1),\quad n\in \mathbb{N}.$$

2 Sequences convergent to the Ioachimescu constant â„“

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

$$\lim_{n\rightarrow+\infty}n^{s}(x_{n}-x_{n+1})=l \in[-\infty,+\infty],$$
(2.1)

with $$s>1$$, then

$$\lim_{n\rightarrow+\infty}n^{s-1}x_{n}= \frac{l}{s-1}.$$
(2.2)

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-Kinkelinâ€™s and Bendersky-Adamchikâ€™s constants, Somosâ€™ quadratic recurrence constant, and so on [16â€“19].

Theorem 1

For the Ioachimescu constant, we have the following convergent sequence:

$$I_{i}^{(1)}(n)=\sum_{k=1}^{n}{ \frac{1}{\sqrt {k}}} -2 (\sqrt {n} -1 )+\eta_{0}^{(1)}(n)+ \eta_{1}^{(1)}(n)+\cdots+\eta_{i}^{(1)}(n),$$
(2.3)

where

\begin{aligned}& \eta_{0}^{(1)}(n) = 0,\qquad\eta_{1}^{(1)}(n)= \frac{-\frac{1}{2}}{\sqrt{n+\frac{1}{6}}}, \end{aligned}
(2.4)
\begin{aligned}& \eta_{2}^{(1)}(n) = \frac{\frac{1}{192}}{\sqrt{n^{5}+\frac {23}{18}n^{4}+\frac {341}{288}n^{3}+\frac{27{,}833}{46{,}656}n^{2}+\frac{726{,}647}{26{,}873{,}856}n-\frac {9{,}196{,}141}{806{,}215{,}680}}},\ldots. \end{aligned}
(2.5)

Proof

(Step 1) The initial correction. We choose $$\eta_{0}^{(1)}(n)=0$$, and let

$$I_{0}^{(1)}(n):=I(n)+\eta_{0}^{(1)}(n)= \sum_{k=1}^{n}{\frac{1}{\sqrt {k}}}-2 (\sqrt {n} -1 )+\eta_{0}^{(1)}(n).$$
(2.6)

Developing equation (2.6) into a power series expansion in $$1/n$$, we have

$$I_{0}^{(1)}(n)-I_{0}^{(1)}(n+1)= \frac{1}{4}\frac{1}{n^{\frac{3}{2}}}+O \biggl(\frac{1}{n^{\frac{5}{2}}} \biggr).$$
(2.7)

By LemmaÂ 1, the rate of convergence of $$(I_{0}^{(1)}(n)-\ell )_{n\in\mathbb{N}}$$ is $$n^{-\frac{1}{2}}$$, since

$$\lim_{n\rightarrow\infty}n^{\frac{1}{2}} \bigl(I_{0}^{(1)}(n)- \ell \bigr)=\frac{1}{2}.$$

(Step 2) The first correction. Let

$$\eta_{1}^{(1)}(n)=\frac{a_{1}}{\sqrt{n+b_{0}}}$$
(2.8)

and define

$$I_{1}^{(1)}(n):=\sum_{k=1}^{n}{ \frac{1}{\sqrt {k}}} -2 (\sqrt {n} -1 )+\eta_{0}^{(1)}(n)+ \eta_{1}^{(1)}(n).$$
(2.9)

Developing (2.9) into power series expansion in $$1/n$$, we obtain

\begin{aligned} I_{1}^{(1)}(n)-I_{1}^{(1)}(n+1) =& \frac{2a_{1}+1}{4}\frac{1}{n^{\frac{3}{2}}}+\frac{-2-3a_{1}(1+2b_{0})}{8}\frac{1}{n^{\frac{5}{2}}} \\ &{}+\frac{5}{64} \bigl(3+4a_{1} \bigl(1+3b_{0}+3b_{0}^{2} \bigr) \bigr)\frac{1}{n^{\frac{7}{2}}}+O \biggl(\frac {1}{n^{\frac{9}{2}}} \biggr). \end{aligned}
(2.10)
1. (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

$$\lim_{n\rightarrow\infty}n^{\frac{1}{2}} \bigl(I_{1}^{(1)}(n)- \ell \bigr)=\frac{2a_{1}+1}{4}\neq0.$$
2. (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

$$\eta_{2}^{(1)}(n)=\frac{a_{2}}{\sqrt{n^{5}+b_{4} n^{4}+b_{3} n^{3}+b_{2} n^{2}+b_{1} n+b_{0}}}$$
(2.11)

and define

$$I_{2}^{(1)}(n):=\sum_{k=1}^{n}{ \frac{1}{\sqrt {k}}} -2 (\sqrt {n} -1 )+\eta_{0}^{(1)}(n)+ \eta_{1}^{(1)}(n)+\eta_{2}^{(1)}(n).$$
(2.12)

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 LemmaÂ 1 again, one has

\begin{aligned}& \lim_{n\rightarrow\infty} n^{\frac{19}{2}} \bigl(I_{2}^{(1)}(n)-I_{2}^{(1)}(n+1) \bigr)= \frac {1{,}287{,}793{,}943{,}249}{267{,}483{,}013{,}447{,}680}, \end{aligned}
(2.13)
\begin{aligned}& \lim_{n\rightarrow\infty} n^{\frac{17}{2}} \bigl(I_{2}^{(1)}(n)- \ell \bigr)=\frac{75{,}752{,}584{,}897}{133{,}741{,}506{,}723{,}840}. \end{aligned}
(2.14)

Repeating the above approach for the Ioachimescu constant, we can prove TheoremÂ 1.â€ƒâ–¡

3 Other sequences convergent to the Ioachimescu constant â„“

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

$$\eta_{1}^{(2)}(n)=\frac{a}{\sqrt {n}\sqrt{1+\frac{u_{1}}{n+v_{1}}}}$$
(3.1)

and define

$$I_{1}^{(2)}(n):=\sum_{k=1}^{n}{ \frac{1}{\sqrt {k}}} -2 (\sqrt {n} -1 )+\frac{a}{\sqrt {n}\sqrt{1+\frac {u_{1}}{n+v_{1}}}}.$$
(3.2)

By the same method as above, we find $$a=-\frac{1}{2}$$, $$u_{1}=\frac{1}{6}$$, $$v_{1}=-\frac{1}{8}$$.

Applying LemmaÂ 1, one has

\begin{aligned}& \lim_{n\rightarrow\infty} n^{\frac{9}{2}} \bigl(I_{1}^{(2)}(n)-I_{1}^{(2)}(n+1) \bigr)=\frac{259}{27{,}648}, \end{aligned}
(3.3)
\begin{aligned}& \lim_{n\rightarrow\infty} n^{\frac{7}{2}} \bigl(I_{1}^{(2)}(n)- \ell \bigr) = \frac{37}{13{,}824}. \end{aligned}
(3.4)

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:

$$I_{i}(n)=\sum_{k=1}^{n}{ \frac{1}{\sqrt {k}}} -2 (\sqrt {n} -1 )+\frac {a}{\sqrt {n}\sqrt{1+\frac{u_{1}}{n+v_{1}+\frac{u_{2}}{n+v_{2}+\frac {u_{3}}{n+v_{3}+\ddots+\frac{u_{i}}{n+v_{i}}}}}}},$$
(3.5)

where

\begin{aligned}& a = -\frac{1}{2},\qquad u_{1}=\frac{1}{6}, \qquad v_{1}=-\frac{1}{8};\qquad u_{2}=\frac{37}{576}, \qquad v_{2}=-\frac{1}{888}; \end{aligned}
(3.6)
\begin{aligned}& u_{3} = \frac{837}{2{,}738},\qquad v_{3}=\frac{33}{18{,}352}; \end{aligned}
(3.7)
\begin{aligned}& u_{4} = \frac{2{,}311{,}279}{3{,}690{,}240},\qquad v_{4}=- \frac{162{,}349}{92{,}950{,}896};\qquad u_{5}=\frac {393{,}826{,}357{,}519}{351{,}191{,}348{,}010}, \end{aligned}
(3.8)
\begin{aligned}& v_{5} = \frac {5{,}022{,}056{,}744{,}279}{2{,}720{,}864{,}635{,}038{,}456};\qquad\cdots. \end{aligned}
(3.9)

Remark 1

TheoremÂ 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