1 Introduction

In 1895, Ioachimescu (see [1]) introduced a constant \(\ell \), which today bears his names, as the limit of the sequence defined by

$$\begin{aligned} I_n=1+\frac{1}{\sqrt{2}}+\cdots +\frac{1}{\sqrt{n}}-2(\sqrt{n}-1), n\in \mathbb {N}. \end{aligned}$$

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,

$$\begin{aligned} I_n\sim \ell +\frac{1}{2\sqrt{n}}-\sum _{k=1}^{\infty }\frac{\mathbf {b}_{2k}}{(2k)!}\frac{(4k-3)!!}{2^{2k-1}n^{2k-1/2}}, n\in \mathbb {N}, \end{aligned}$$

where \(\mathbf {b}_n\) denotes the nth Bernoulli number.

One easily obtains the following representations of the Ioachimescu’s constant:

$$\begin{aligned} \ell =\int _0^{\infty }\frac{1-x+\lfloor x \rfloor }{2(1+x)^{3/2}}dx \end{aligned}$$

and

$$\begin{aligned} \ell =2-\sum _{k=1}^{\infty }\frac{1}{\left( \sqrt{k}+\sqrt{k-1}\right) ^2\sqrt{k}}. \end{aligned}$$

A representation of the Ioachimescu’s constant has also been given by Ramanujan (1915) [20]:

$$\begin{aligned} \ell =2-\left( \sqrt{2}+1\right) \sum _{k=1}^{\infty }\frac{(-1)^{k+1}}{\sqrt{k}}. \end{aligned}$$

From this, one easily obtains a representation of the Ioachimescu’s constant in terms of the extended \(\zeta \) function

$$\begin{aligned} \ell =\zeta \left( \frac{1}{2}\right) +2. \end{aligned}$$

From [2], we have \(\ell =0.539645491\ldots .\)

Let \(a\in (0,+\infty )\) and \(s\in (0,1)\). The sequence

$$\begin{aligned} y_n(a,s)=\frac{1}{a^s}+\frac{1}{(a+1)^s}+\cdots \frac{1}{(a+n-1)^s}-\frac{1}{1-s} \left[ (a+n-1)^{1-s}-a^{1-s}\right] , n\in \mathbb {N}, \end{aligned}$$

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

$$\begin{aligned} \lim _{n\rightarrow \infty } n^s \big (y_n(a,s)-\ell (a,s)\big )=\frac{1}{2}. \end{aligned}$$

Also in [3], considering the sequence

$$\begin{aligned} u_n(a,s)=y_n(a,s)-\frac{1}{2(a+n-1)^s}, \end{aligned}$$

she has proved that

$$\begin{aligned} \lim _{n\rightarrow \infty } n^{s+1} \big (\ell (a,s)-u_n(a,s)\big )=\frac{s}{12} \end{aligned}$$

and, for the sequence

$$\begin{aligned} \alpha _n(a,s)=&\frac{1}{a^s}+\frac{1}{(a+1)^s}+\cdots \frac{1}{(a+n-1)^s}\\&\quad -\,\frac{1}{1-s} \left( \left( a+n-\frac{1}{2}\right) ^{1-s}-a^{1-s}\right) , n\in \mathbb {N}, \end{aligned}$$

she has proved that

$$\begin{aligned} \lim _{n\rightarrow \infty } n^{s+1} \big (\alpha _n(a,s)-\ell (a,s)\big )=\frac{s}{24}. \end{aligned}$$

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

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

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

$$\begin{aligned} \lim _{n\rightarrow +\infty }n^s(x_n-x_{n+1})=l\in [-\infty ,+\infty ], \end{aligned}$$
(2.1)

exists when \(s>1,\) then

$$\begin{aligned} \lim _{n\rightarrow +\infty }n^{s-1}x_n=\frac{l}{s-1}. \end{aligned}$$
(2.2)

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:

$$\begin{aligned} I_i(n)&=\sum _{k=1}^{n}{\frac{1}{\sqrt{k}}} -2\left( \sqrt{n} -1\right) \nonumber \\&\qquad +\,\frac{a}{\root 6 \of {n^3+b_2 n^2+b_1 n+b_0+\frac{u_1}{n+v_1+\frac{u_2}{n+v_2+\frac{u_3}{n+v_3+\frac{u_4}{n+v_4+\cdots }}}}}}, \end{aligned}$$
(2.3)

where

$$\begin{aligned}&a=-\frac{1}{2}, \quad b_2=\frac{1}{2}, \quad b_1=\frac{7}{48}, \quad b_0=\frac{1}{864};\quad u_1=-\frac{7}{576}, \quad v_1=\frac{23}{42}; \quad u_2=\frac{67483}{84672},\\&\quad v_2=-\frac{735611}{14171430};\\&\quad u_3=\frac{106772389611377}{196730868484800}, \quad v_3=\frac{5732111704318866731}{10293315954492220130};\\&\quad u_4=\frac{3960720843020595280578879811}{2093940584692105796281439289},\\&\quad v_4=-\frac{330844832640429778096837755246211177}{ 53714560364565453879762494412621881220};\ldots . \end{aligned}$$

Proof

(Step 1) The initial-correction. We choose \(\eta _0(n)=0\), and let

$$\begin{aligned} I_0(n):=I(n)+\eta _0(n)=\sum _{k=1}^{n}{\frac{1}{\sqrt{k}}}-2\left( \sqrt{n} -1\right) +\eta _0(n). \end{aligned}$$
(2.4)

Developing the expression (2.4) into power series expansion in 1 / n, we easily obtain

$$\begin{aligned} I_0(n)-I_0(n+1)=\frac{1}{4}\frac{1}{n^\frac{3}{2}}+O\left( \frac{1}{n^\frac{5}{2}}\right) . \end{aligned}$$
(2.5)

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

$$\begin{aligned} \lim _{n\rightarrow \infty }n^{\frac{1}{2}}\left( I_0(n)-\ell \right) =\frac{1}{2}. \end{aligned}$$

(Step 2) The first-correction. Ramanujan [20] made the claim (without proof) for the gamma function

$$\begin{aligned} \Gamma (x+1)=\sqrt{\pi }\left( \frac{x}{e}\right) ^x \left( 8x^3+4x^2+x+\frac{\theta _x}{30}\right) ^{\frac{1}{6}}, \end{aligned}$$

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

$$\begin{aligned} \eta _1(n)=\frac{a}{\root 6 \of {n^3+b_2 n^2+b_1 n+b_0}} \end{aligned}$$
(2.6)

and define

$$\begin{aligned} I_1(n):=\sum _{k=1}^{n}{\frac{1}{\sqrt{k}}} -2\left( \sqrt{n} -1\right) +\eta _1(n). \end{aligned}$$
(2.7)

Developing (2.7) into power series expansion in 1 / n, we have

$$\begin{aligned} I_1(n)-I_1(n+1)=&\frac{1}{4}(2a+1)\frac{1}{n^\frac{3}{2}}-\frac{1}{8} \big (2+a(3+2b_2)\big )\frac{1}{n^\frac{5}{2}}\nonumber \\&\quad +\frac{5}{576} \big (27 + 4 a (9 - 12 b_1 + 9 b_2 + 7 b_2^2)\big )\frac{1}{n^\frac{7}{2}}\nonumber \\&\quad -\frac{7}{10368}\Big (324 + a \big (405 + 864 b0 + 540 b2 + 630 b2^2\nonumber \\&\quad + 364 b2^3 -72 b1 (15 + 14 b2)\big )\Big )\frac{1}{n^\frac{9}{2}}\nonumber \\&\quad +\frac{7}{13824} \Big (405 + 2 a \big (243 + 432 b_1^2 + 405 b_2 + 630 b_2^2\nonumber \\&\quad + 546 b_2^3 + 247 b_2^4 + 432 b_0 (3 + 2 b_2)\nonumber \\&\quad - 72 b_1 (15 + 21 b_2 + 13 b_2^2)\big )\Big )\frac{1}{n^\frac{11}{2}} +O\left( \frac{1}{n^\frac{13}{2}}\right) . \end{aligned}$$
(2.8)
  1. (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

    $$\begin{aligned} \lim _{n\rightarrow \infty }n^{\frac{1}{2}}\left( I_1(n)-\ell \right) =\frac{1}{2}(2a+1)\ne 0. \end{aligned}$$
  2. (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

$$\begin{aligned} \eta _2(n)=\frac{a}{\root 6 \of {n^3+b_2 n^2+b_1 n+b_0+\frac{u_1}{n+v_1}}} \end{aligned}$$
(2.9)

and define

$$\begin{aligned} I_2(n):=\sum _{k=1}^{n}{\frac{1}{\sqrt{k}}} -2\left( \sqrt{n} -1\right) +\eta _2(n). \end{aligned}$$
(2.10)

Developing (2.10) into power series expansion in 1 / n, we have

$$\begin{aligned} I_2(n)-I_2(n+1)=&\left( \frac{7}{1536}+\frac{3u_1}{8}\right) \frac{1}{n^\frac{11}{2}}-\frac{11}{41472}\big (71 + 288 u_1 (17 + 6 v_1)\big )\frac{1}{n^\frac{13}{2}}\\&\quad +\left( \frac{1132703}{23887872} +\frac{13}{576} u_1 (141 + 80 v_1 + 24 v_1^2)\right) \frac{1}{n^\frac{15}{2}}+O\left( \frac{1}{n^\frac{17}{2}}\right) .\nonumber \end{aligned}$$
(2.11)

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

$$\begin{aligned}&\lim _{n\rightarrow \infty } n^{\frac{15}{2}} \big (I_2(n)-I_2(n+1)\big )=-\frac{877279}{167215104},\end{aligned}$$
(2.12)
$$\begin{aligned}&\lim _{n\rightarrow \infty } n^{\frac{13}{2}} \big (I_2(n)-\ell \big )=-\frac{67483}{83607552}. \end{aligned}$$
(2.13)

(Step 4) The third-correction. Similarly, we set the third-correction function in the form of

$$\begin{aligned} \eta _3(n)=\frac{a}{\root 6 \of {n^3+b_2 n^2+b_1 n+b_0+\frac{u_1}{n+v_1+\frac{u_2}{n+v_2}}}} \end{aligned}$$
(2.14)

and define

$$\begin{aligned} I_3(n):=\sum _{k=1}^{n}{\frac{1}{\sqrt{k}}} -2\left( \sqrt{n} -1\right) +\eta _3(n). \end{aligned}$$
(2.15)

By the same method as above, we find \(u_2=\frac{67483}{84672}, v_2=-\frac{735611}{14171430}.\)

Applying Lemma 1 again, one has

$$\begin{aligned}&\lim _{n\rightarrow \infty } n^{\frac{19}{2}} \big (I_3(n)-I_3(n+1)\big )=\frac{259304374770487 }{69639491498803200},\end{aligned}$$
(2.16)
$$\begin{aligned}&\lim _{n\rightarrow \infty } n^{\frac{17}{2}} \big (I_3(n)-\ell \big )=\frac{15253198515911}{34819745749401600}. \end{aligned}$$
(2.17)

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.