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 ℓ, 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) [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 ζ function

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

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

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, Sîntămărian 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], 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

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

2 Sequences convergent to the Ioachimescu constant 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

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

(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-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:

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

(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 Lemma 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)

• (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

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

(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 Lemma 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 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

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

(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 Lemma 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

Theorem 2 provides some quasi-continued fraction sequences with a faster rate of convergence for the Ioachimescu constant.