Skip to main content

A new sequence related to the Euler–Mascheroni constant

Abstract

In this paper, we provide a new quicker sequence convergent to the Euler–Mascheroni constant using an approximation of Padé type. Our sequence has a relatively simple form and higher speed of convergence. Moreover, we establish lower and upper bound estimates for the difference between the sequence and the Euler–Mascheroni constant.

Introduction

The Euler–Mascheroni constant

$$ \gamma =0.5772156649015328\ldots $$

is one of the most famous constants in analysis and number theory. It is the limit of the sequence

$$ \gamma_{n}=1+\frac{1}{2}+\cdots +\frac{1}{n}-\log n. $$
(1.1)

There are many famous problems related to the properties of this constant; for example, it is not known yet whether the Euler–Mascheroni constant is a rational number. In recent years, many researchers made great efforts in the area of concerning the rate of convergence of the sequence \((\gamma_{n})_{n\geq 1}\) and establishing sequences converging faster to the Euler–Mascheroni constant γ.

We begin with a brief overview of the relevant research.

To reveal the speed of convergence of the sequence \((\gamma_{n})_{n \geq 1}\), Boas [5] and Mortici and Vernescu [20, 21] established the following double inequality for the difference between the sequence and the Euler–Mascheroni constant:

$$ \frac{1}{2n+1} < \gamma_{n}-\gamma < \frac{1}{2n}. $$
(1.2)

DeTemple [12] modified the logarithmic term of \(\gamma_{n}\) and showed that the sequence

$$ R_{n}=1+\frac{1}{2}+\cdots +\frac{1}{n}-\log \biggl( n+ \frac{1}{2} \biggr) $$
(1.3)

converges to γ with rate of convergence \(n^{-2}\), since

$$ \frac{1}{24(n+1)^{2}} < R_{n}-\gamma < \frac{1}{24n^{2}}. $$
(1.4)

Vernescu [28] provided the sequence

$$ V_{n}=1+\frac{1}{2}+\cdots +\frac{1}{n-1}+\frac{1}{2n}- \log n, $$
(1.5)

which also converges to γ with rate of convergence \(n^{-2}\), since

$$ \frac{1}{12(n+1)^{2}}< \gamma -V_{n}< \frac{1}{12n^{2}}. $$
(1.6)

Cristea and Mortici [11] introduced the family of sequences

$$ v_{n}(a,b)=1+\frac{1}{2}+\cdots +\frac{1}{n-2}+ \frac{an+b}{n(n-1)}- \log n, $$
(1.7)

where a, b are real parameters. Furthermore, they proved that, among the sequences \((v_{n}(a,b))_{n\geq 1}\), the privileged one \(( v _{n} ( \frac{3}{2},-\frac{5}{12} ) ) _{n\geq 1}\) offers the best approximation to γ, since it has the rate of convergence \(n^{-3}\). More precisely, for

$$ v_{n} \biggl( \frac{3}{2},-\frac{5}{12} \biggr) =1+ \frac{1}{2}+\cdots +\frac{1}{n-2}+\frac{13}{12(n-1)}+ \frac{5}{12n}-\log n, $$
(1.8)

they obtained the bounds

$$ \frac{1}{12n^{3}}+\frac{11}{120n^{4}}< v_{n} \biggl( \frac{3}{2},- \frac{5}{12} \biggr) -\gamma < \frac{1}{12n^{3}}+ \frac{13}{120n^{4}} \quad (n\geq 9). $$
(1.9)

Lu [16] used continued fraction approximation to obtain the following faster sequence converging to the Euler–Mascheroni constant:

$$ r_{n}^{(2)}{ \biggl( \frac{1}{2},\frac{1}{6} \biggr) }=1+\frac{1}{2}+ \cdots +\frac{1}{n}-\frac{3}{6n+1}-\log n, $$
(1.10)

which satisfies

$$ \frac{1}{72(n+1)^{3}}< \gamma -r_{n}^{(2)}{ \biggl( \frac{1}{2}, \frac{1}{6} \biggr) }< \frac{1}{72n^{3}}. $$
(1.11)

Recently, Wu and Bercu [29] constructed the new sequence

$$ \omega_{n}=\sum_{k=1}^{n}{ \frac{1+(2b_{1}-1)(-1)^{k-1}}{k}}-\log \biggl[ n+{\frac{(-1)^{n-1}(2b_{1}-1)+1}{2}} \biggr] -(2b_{1}-1)\log 2, $$
(1.12)

which converges to γ with rate of convergence \(n^{-2}\).

For more detail about the approximation of the Euler–Mascheroni constant with very high accuracy, we mention the works of Lu [1618], Sweeney [27], Bailey [2], Crînganu [10], and Alzer and Koumandos [1]. We also mention the excellent survey by Lagarias [15]. Hu and Mortici [13, 14, 19] provided some similar methods to deal with approximation of the constant e.

In this paper, starting from the sequence \((\gamma_{n})_{n\geq 1}\), we use an approximation of Padé type and provide a new convergent sequence for Euler–Mascheroni constant.

The Padé approximant is the best approximation of a function by a rational function and often gives better approximation of the function than truncating its Taylor series. For these reasons, Padé approximants are also used in computer calculations (see [3, 30]).

Recall the Padé approximant of \(P(n)\) of order \([1/2]\):

$$ P_{[1/2]}(n)=\frac{\alpha_{0}+\alpha_{1}n}{1+\beta_{1}n+\beta_{2}n ^{2}}=\frac{a_{1}}{n+b_{1}}+\frac{a_{2}}{n+b_{2}}. $$
(1.13)

We will use this Padé approximant \(p_{[1/2]}(n)\) as an additional term to establish a new quicker sequence converging to the Euler–Mascheroni constant. More precisely, we consider the following sequence:

$$ \Gamma_{n}^{(2)}=1+\frac{1}{2}+\cdots + \frac{1}{n}-\log n-\frac{a _{1}}{n+b_{1}}-\frac{a_{2}}{n+b_{2}}. $$
(1.14)

Furthermore, we will provide lower and upper bound estimates for the difference between the sequence and the Euler–Mascheroni constant.

Main results

Our main results are stated in the following theorem.

Theorem 2.1

Let

$$ \Gamma_{n}^{(2)}=1+\frac{1}{2}+\cdots + \frac{1}{n}-\log n-\frac{a _{1}}{n+b_{1}}-\frac{a_{2}}{n+b_{2}}, $$

and let

$$\begin{aligned}& a_{1}=\frac{1}{24b_{1}(1-3b_{1})}, \\& a_{2}=-\frac{(6b_{1}-1)^{2}}{24b _{1}(1-3b_{1})}, \\& b_{2}=\frac{b_{1}}{6b_{1}-1},\quad b_{1}\in \biggl( \frac{1}{6},\frac{1}{3} \biggr)\cup \biggl(\frac{1}{3},+ \infty \biggr). \end{aligned}$$

Then we have the asymptotic expansion

$$ \Gamma_{n}^{(2)}=\gamma +\sum_{k=4}^{m} \biggl( -\frac{B_{k}}{k}+(-1)^{k} \bigl(a _{1}b_{1}^{k-1}+a_{2}b_{2}^{k-1} \bigr) \biggr) \frac{1}{n^{k}}+O \biggl( \frac{1}{n ^{m+1}} \biggr) $$
(2.1)

as \(n\rightarrow \infty \), where \(B_{k}\) are Bernoulli numbers. More explicitly, we have

$$\begin{aligned} \Gamma_{n}^{(2)}&=\gamma +\frac{1-10p}{120}\cdot \frac{1}{n^{4}}+\frac{p ^{2}}{2}\cdot \frac{1}{n^{5}}+ \biggl( \frac{p^{2}-36p^{3}}{12}- \frac{1}{252} \biggr) \frac{1}{n^{6}}+{p^{3}}(18p-1) \frac{1}{n^{7}} \\ &\quad {}+\cdots + \biggl( \frac{(3-\sqrt{9-p^{-1}})^{m-3}-(3+ \sqrt{9-p^{-1}})^{m-3}}{24(-1)^{m}p^{3-m}\sqrt{9-p^{-1}}}-\frac{B _{m}}{m} \biggr) \frac{1}{n^{m}}+O \biggl( \frac{1}{n^{m+1}} \biggr) \end{aligned}$$
(2.2)

as \(n\rightarrow \infty \), where \(p=b_{1}^{2}/(6b_{1}-1)\).

Furthermore, we have the following double inequality:

$$\begin{aligned} \frac{1-10p}{120}\cdot \frac{1}{n^{4}}< \Gamma_{n}^{(2)}- \gamma < \frac{1-10p}{120}\cdot \frac{1}{n^{4}}+\frac{p^{2}}{2}\cdot \frac{1}{n ^{5}}. \end{aligned}$$
(2.3)

Proof

Using the representation of the harmonic sum in terms of digamma function (see [4])

$$ 1+\frac{1}{2}+\cdots +\frac{1}{n}=\gamma +\frac{1}{n}+\Psi (n) $$
(2.4)

and the asymptotic formula

$$\begin{aligned} \Psi (z)&=\log z-\frac{1}{2z}-\sum_{k=2}^{m} \frac{B_{2k-2}}{(2k-2)z ^{2k-2}}+O \biggl( \frac{1}{z^{2m}} \biggr) \\ &=\log z-\frac{1}{2z}-\frac{1}{12z^{2}}+\frac{1}{120z^{4}}- \frac{1}{252z ^{6}}+\cdots +\frac{-B_{2m-2}}{(2m-2)z^{2m-2}}+O \biggl( \frac{1}{z^{2m}} \biggr) , \end{aligned}$$
(2.5)

we obtain

$$\begin{aligned} &1+\frac{1}{2}+\cdots +\frac{1}{n}-\log n \\ &\quad =\gamma +\frac{1}{n}+ \Psi (n)- \log n \\ &\quad =\gamma +\frac{1}{2n}-\frac{1}{12n^{2}}+\frac{1}{120n^{4}}- \frac{1}{252n ^{6}}+\cdots +\frac{-B_{2m-2}}{(2m-2)n^{2m-2}}+O \biggl( \frac{1}{n^{2m}} \biggr) . \end{aligned}$$

Hence

$$\begin{aligned} \Gamma_{n}^{(2)} & =1+\frac{1}{2}+\cdots + \frac{1}{n}-\log n-\frac{a _{1}}{n+b_{1}}-\frac{a_{2}}{n+b_{2}} \\ & =\gamma -\frac{a_{1}}{n+b_{1}}-\frac{a_{2}}{n+b_{2}}+ \frac{1}{2n}- \frac{1}{12n^{2}}+\frac{1}{120n^{4}} \\ & \quad {}-\frac{1}{252n^{6}}+\cdots +\frac{-B_{2m-2}}{(2m-2)n^{2m-2}}+O \biggl( \frac{1}{n^{2m}} \biggr) . \end{aligned}$$

Using the power series expansion gives

$$\begin{aligned} \frac{a_{1}}{n+b_{1}} & =\frac{a_{1}}{n} \biggl( \frac{1}{1+\frac{b _{1}}{n}} \biggr) \\ &= \frac{a_{1}}{n} \biggl(1-\frac{b_{1}}{n}+\frac{b_{1}^{2}}{n ^{2}}- \frac{b_{1}^{3}}{n^{3}}+\cdots +(-1)^{m}\frac{b_{1}^{m}}{n^{m}} \biggr)+O \biggl( \frac{1}{n^{m+2}} \biggr) \\ & =\frac{a_{1}}{n}-\frac{a_{1}b_{1}}{n^{2}}+\frac{a_{1}b_{1}^{2}}{n ^{3}}-\frac{a_{1}b_{1}^{3}}{n^{4}}+ \cdots +(-1)^{m}\frac{a_{1}b_{1} ^{m}}{n^{m+1}}+O \biggl( \frac{1}{n^{m+2}} \biggr) \end{aligned}$$

and

$$ \frac{a_{2}}{n+b_{2}}=\frac{a_{2}}{n}-\frac{a_{2}b_{2}}{n^{2}}+\frac{a _{2}b_{2}^{2}}{n^{3}}- \frac{a_{2}b_{2}^{3}}{n^{4}}+\cdots +(-1)^{m}\frac{a _{2}b_{2}^{m}}{n^{m+1}}+O \biggl( \frac{1}{n^{m+2}} \biggr) $$

as \(n\rightarrow \infty \). Thus we obtain

$$\begin{aligned} \Gamma_{n}^{(2)} & = \gamma + \biggl( \frac{1}{2}-a_{1}-a_{2} \biggr) \frac{1}{n}+ \biggl( - \frac{1}{12}+a_{1}b_{1}+a_{2}b_{2} \biggr) \frac{1}{n^{2}}- \bigl( a _{1}b_{1}^{2}+a_{2}b_{2}^{2} \bigr) \frac{1}{n^{3}} \\ & \quad {}+ \biggl( \frac{1}{120}+a_{1}b_{1}^{3}+a_{2}b_{2}^{3} \biggr) \frac{1}{n ^{4}}- \bigl( a_{1}b_{1}^{4}+a_{2}b_{2}^{4} \bigr) \frac{1}{n^{5}}+ \biggl( -\frac{1}{252}+a_{1}b_{1}^{5}+a_{2}b_{2}^{5} \biggr) \frac{1}{n ^{6}} \\ & \quad {} +\cdots + \biggl( \frac{-B_{2m-2}}{2m-2}+a_{1}b_{1}^{2m-3}+a_{2}b _{2}^{2m-3} \biggr) \frac{1}{n^{2m-2}} \\ & \quad {} + \bigl( -a_{1}b_{1}^{2m-2}-a_{2}b_{2}^{2m-2} \bigr) \frac{1}{n^{2m-1}}+O \biggl( \frac{1}{n^{2m}} \biggr) . \end{aligned}$$

From the assumption conditions

$$\begin{aligned}& a_{1}=\frac{1}{24b_{1}(1-3b_{1})}, \\& a_{2}=-\frac{(6b_{1}-1)^{2}}{24b _{1}(1-3b_{1})}, \\& b_{2}=\frac{b_{1}}{6b_{1}-1},\quad b_{1}\in \biggl( \frac{1}{6},\frac{1}{3} \biggr)\cup \biggl(\frac{1}{3},+ \infty \biggr), \end{aligned}$$

we have

$$ \frac{1}{2}-a_{1}-a_{2}=0,\qquad -\frac{1}{12}+a_{1}b_{1}+a_{2}b_{2}=0,\qquad a_{1}b_{1}^{2}+a_{2}b_{2}^{2}=0. $$

Therefore

$$\begin{aligned} \Gamma_{n}^{(2)}&=\gamma+ \biggl( \frac{1}{120}+a_{1}b_{1}^{3}+a_{2}b_{2}^{3} \biggr) \frac{1}{n ^{4}}- \bigl( a_{1}b_{1}^{4}+a_{2}b_{2}^{4} \bigr) \frac{1}{n^{5}}+ \biggl( -\frac{1}{252}+a_{1}b_{1}^{5}+a_{2}b_{2}^{5} \biggr) \frac{1}{n ^{6}} \\ & \quad {}+\cdots + \biggl( \frac{-B_{2m-2}}{2m-2}+a_{1}b_{1}^{2m-3}+a_{2}b _{2}^{2m-3} \biggr) \frac{1}{n^{2m-2}} \\ & \quad {}+ \bigl(-a_{1}b_{1}^{2m-2}-a_{2}b_{2}^{2m-2} \bigr)\frac{1}{n^{2m-1}}+O \biggl( \frac{1}{n ^{2m}} \biggr) . \end{aligned}$$

Note that, for all odd Bernoulli numbers \(B_{2m-1}=0\) \((m\geq 2)\), the last expression can be rewritten as

$$\begin{aligned} \Gamma_{n}^{(2)}&=\gamma+ \biggl( \frac{1}{120}+a_{1}b_{1}^{3}+a_{2}b_{2}^{3} \biggr) \frac{1}{n ^{4}}+\cdots + \biggl( \frac{-B_{2m-2}}{2m-2}+a_{1}b_{1}^{2m-3}+a_{2}b _{2}^{2m-3} \biggr) \frac{1}{n^{2m-2}} \\ & \quad {} + \biggl( \frac{-B_{2m-1}}{2m-1}-a_{1}b_{1}^{2m-2}-a_{2}b_{2}^{2m-2} \biggr) \frac{1}{n ^{2m-1}}+O \biggl( \frac{1}{n^{2m}} \biggr) \end{aligned}$$

and

$$\begin{aligned} \Gamma_{n}^{(2)} &=\gamma + \biggl(\frac{1}{120}+a_{1}b_{1}^{3}+a_{2}b _{2}^{3} \biggr)\frac{1}{n^{4}}+\cdots + \biggl( \frac{-B_{2m-3}}{2m-3}+a _{1}b_{1}^{2m-4}+a_{2}b_{2}^{2m-4} \biggr)\frac{1}{n^{2m-3}} \\ & \quad {}+ \biggl(\frac{-B_{2m-2}}{2m-2}+a_{1}b_{1}^{2m-3}+a_{2}b_{2}^{2m-3} \biggr)\frac{1}{n^{2m-2}}+O \biggl(\frac{1}{n^{2m-1}} \biggr), \end{aligned}$$

that is,

$$ \Gamma_{n}^{(2)}=\gamma +\sum_{k=4}^{m} \biggl( -\frac{B_{k}}{k}+(-1)^{k} \bigl(a _{1}b_{1}^{k-1}+a_{2}b_{2}^{k-1} \bigr) \biggr) \frac{1}{n^{k}}+O \biggl( \frac{1}{n ^{m+1}} \biggr) , $$
(2.6)

which is the desired Eq. (2.1) in Theorem 2.1.

On the other hand, from

$$ p=\frac{b_{1}^{2}}{6b_{1}-1},\qquad b_{2}=\frac{b_{1}}{6b_{1}-1},\quad b_{1}\in \biggl(\frac{1}{6},\frac{1}{3} \biggr)\cup \biggl(\frac{1}{3},+ \infty \biggr), $$

we have \(b_{1}b_{2}=p\), \(b_{1}+b_{2}=6p\) (\(p>\frac{1}{9}\)), which implies that \(b_{1}\) and \(b_{2}\) are the roots of the equation \(x^{2}-6px+p=0\). Therefore,

$$ b_{1,2}=3p\pm \sqrt{9p^{2}-p}. $$

It is easy to observe that

$$\begin{aligned} a_{1} & =\frac{1}{24b_{1}(1-3b_{1})}=\frac{b_{2}}{12b_{1}(b_{2}-b _{1})}, \\ a_{2} & =-\frac{(6b_{1}-1)^{2}}{24b_{1}(1-3b_{1})}=\frac{-b_{1}}{12b _{2}(b_{2}-b_{1})}, \end{aligned}$$

and thus

$$\begin{aligned} a_{1}b_{1}^{k-1}+a_{2}b_{2}^{k-1} & =\frac{b_{2}b_{1}^{k-1}}{12b_{1}(b _{2}-b_{1})}-\frac{b_{1}b_{2}^{k-1}}{12b_{2}(b_{2}-b_{1})} \\ & =-\frac{b_{1}b_{2}(b_{1}^{k-3}-b_{2}^{k-3})}{12(b_{1}-b_{2})} \\ & =\frac{(3-\sqrt{9-p^{-1}})^{k-3}-(3+\sqrt{9-p^{-1}})^{k-3}}{24p ^{3-k}\sqrt{9-p^{-1}}}. \end{aligned}$$

It follows from (2.6) that

$$\begin{aligned} \Gamma_{n}^{(2)}& =\gamma +\sum_{k=4}^{m} \biggl( -\frac{B_{k}}{k}+(-1)^{k} \bigl(a _{1}b_{1}^{k-1}+a_{2}b_{2}^{k-1} \bigr) \biggr) \frac{1}{n^{k}}+O \biggl( \frac{1}{n ^{m+1}} \biggr) \\ & =\gamma +\sum_{k=4}^{m} \biggl( - \frac{B_{k}}{k}+\frac{(3-\sqrt{9-p ^{-1}})^{k-3}-(3+\sqrt{9-p^{-1}})^{k-3}}{24(-1)^{k}p^{3-k}\sqrt{9-p ^{-1}}} \biggr) \frac{1}{n^{k}}+O \biggl( \frac{1}{n^{m+1}} \biggr) , \end{aligned}$$

which implies the desired Eq. (2.2) in Theorem 2.1.

Next, we will show the double inequality (2.3). We define the sequences \((z_{n})_{n\geq 1}\) and \((u_{n})_{n\geq 1}\) by

$$ z_{n}=\Gamma_{n}^{(2)}-\gamma -\frac{1-10p}{120} \cdot \frac{1}{n^{4}} $$

and

$$ u_{n}=\Gamma_{n}^{(2)}-\gamma -\frac{1-10p}{120} \cdot \frac{1}{n ^{4}} -\frac{p^{2}}{2}\cdot \frac{1}{n^{5}}. $$

It follows from (2.2) that

$$\begin{aligned}& \Gamma_{n}^{(2)}-\gamma -\frac{1-10p}{120}\cdot \frac{1}{n^{4}}=O \biggl( \frac{1}{n^{5}} \biggr) , \\& \Gamma_{n}^{(2)}-\gamma -\frac{1-10p}{120}\cdot \frac{1}{n^{4}}-\frac{p ^{2}}{2}\cdot \frac{1}{n^{5}}=O \biggl( \frac{1}{n^{6}} \biggr) , \end{aligned}$$

and thus we have

$$ \lim_{n\rightarrow \infty }z_{n}=0\quad \text{and }\quad \lim _{n\rightarrow \infty }u_{n}=0. $$

To prove that \(z_{n}>0\) and \(u_{n}<0\) for \(n\geq 1\), it suffices to show that \((z_{n})_{n\geq 1}\) is decreasing and \((u_{n})_{n\geq 1}\) is increasing.

Let

$$\begin{aligned} z_{n+1}-z_{n} & =f(n), \\ u_{n+1}-u_{n} & =g(n), \end{aligned}$$

where

$$\begin{aligned}& f(x) =\frac{1}{x+1}+\log x-\log (x+1)+a_{1} \biggl( \frac{1}{x+b _{1}}-\frac{1}{x+b_{1}+1} \biggr) \\& \hphantom{f(x) =}{}+a_{2} \biggl( \frac{1}{x+b_{2}}-\frac{1}{x+b_{2}+1} \biggr) + \frac{10p-1}{120} \biggl( \frac{1}{(x+1)^{4}}-\frac{1}{x^{4}} \biggr) , \\& g(x)=f(x)-\frac{p^{2}}{2} \biggl( \frac{1}{(x+1)^{5}}- \frac{1}{x^{5}} \biggr) ,\quad x\in {}[ 1,+\infty ). \end{aligned}$$

It is easy to verify that

$$ \frac{a_{1}}{x+b_{1}}+\frac{a_{2}}{x+b_{2}}=\frac{6x+36p-1}{12(x ^{2}+6px+p)} $$

and

$$ \frac{a_{1}}{x+b_{1}+1}+\frac{a_{2}}{x+b_{2}+1}=\frac{6x+36p+5}{12(x ^{2}+2x++6px+1+7p)}. $$

Hence

$$\begin{aligned} f(x) & =\frac{1}{x+1}+\log x-\log (x+1)+ \frac{6x+36p-1}{12(x^{2}+6px+p)} \\ & \quad {}-\frac{6x+36p+5}{12(x^{2}+ 2x++6px+1+7p)}+\frac{10p-1}{120} \biggl( \frac{1}{(x+1)^{4}}- \frac{1}{x ^{4}} \biggr) . \end{aligned}$$

Differentiating \(f(x)\) with respect to x gives

$$ f^{\prime }(x)=\frac{P(x)}{30x^{5} ( x+1 ) ^{5} ( x^{2}+6px+p ) ^{2} ( x ^{2}+ 2x+6px+1+7p ) ^{2}}, $$

where

$$\begin{aligned} P(x)& =450p^{2}x^{11}+ \biggl( 7020p^{2} \biggl( p-\frac{1}{9} \biggr)+3360p \biggl( p-\frac{1}{9} \biggr) + \frac{3360}{9} \biggl( p-\frac{1}{9} \biggr) +\frac{2955}{81} \biggr) x^{10} \\ & \quad {}+ \biggl( 36\text{,}720p^{3} \biggl( p-\frac{1}{9} \biggr) +41\text{,}820p^{2} \biggl( p- \frac{1}{9} \biggr) +\frac{102\text{,}570}{9}p \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +\frac{92\text{,}850}{81} \biggl( p-\frac{1}{9} \biggr) + \frac{74\text{,}625}{729} \biggr) x^{9}+ \biggl( 64\text{,}800p^{4} \biggl( p- \frac{1}{9} \biggr) +191\text{,}520p^{3} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +\frac{1\text{,}018\text{,}260}{9}p^{2} \biggl( p- \frac{1}{9} \biggr) + \frac{1\text{,}798\text{,}290}{81}p \biggl( p-\frac{1}{9} \biggr) + \frac{1\text{,}390\text{,}050}{729} \biggl( p-\frac{1}{9} \biggr) \\ &\quad {}+ \frac{1\text{,}055\text{,}439}{6561} \biggr) x^{8} \\ & \quad {} + \biggl( 302\text{,}400p^{4} \biggl( p-\frac{1}{9} \biggr) +446\text{,}910p^{3} \biggl( p-\frac{1}{9} \biggr) + \frac{1\text{,}619\text{,}250}{9}p^{2} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +\frac{2\text{,}158\text{,}710}{81}p \biggl( p-\frac{1}{9} \biggr) + \frac{1\text{,}383\text{,}054}{729} \biggl( p-\frac{1}{9} \biggr) + \frac{1\text{,}028\text{,}760}{6561} \biggr) x^{7} \\ & \quad {} + \biggl( 615\text{,}600p^{4} \biggl( p-\frac{1}{9} \biggr) +612\text{,}290p^{3} \biggl( p-\frac{1}{9} \biggr) + \frac{1\text{,}632\text{,}350}{9}p^{2} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +\frac{1\text{,}634\text{,}294}{81}p \biggl( p-\frac{1}{9} \biggr) + \frac{863\text{,}012}{729} \biggl( p-\frac{1}{9} \biggr) + \frac{659\text{,}621}{6561} \biggr) x^{6} \\ & \quad {} + \biggl( 724\text{,}800p^{4} \biggl( p-\frac{1}{9} \biggr)+ \frac{4\text{,}803\text{,}690}{9}p^{3} \biggl( p-\frac{1}{9} \biggr) + \frac{9\text{,}579\text{,}126}{81}p^{2} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +\frac{7\text{,}355\text{,}676}{729}p \biggl( p-\frac{1}{9} \biggr) + \frac{3\text{,}484\text{,}686}{6561} \biggl( p-\frac{1}{9} \biggr) + \frac{2\text{,}953\text{,}245}{59\text{,}049} \biggr) x^{5} \\ & \quad {} + \biggl( 536\text{,}210p^{4} \biggl( p-\frac{1}{9} \biggr) + \frac{2\text{,}700\text{,}431}{9}p^{3} \biggl( p-\frac{1}{9} \biggr) + \frac{4\text{,}053\text{,}779}{81}p^{2} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {}+\frac{2\text{,}169\text{,}314}{729}p \biggl( p-\frac{1}{9} \biggr) + \frac{962\text{,}090}{6561} \biggl( p-\frac{1}{9} \biggr) + \frac{903\text{,}041}{59\text{,}049} \biggr) x^{4} \\ & \quad {} + \biggl( 247\text{,}060p^{4} \biggl( p-\frac{1}{9} \biggr) +\frac{968\text{,}266}{9}p ^{3} \biggl( p-\frac{1}{9} \biggr) + \frac{1\text{,}113\text{,}742}{81}p^{2} \biggl( p- \frac{1}{9} \biggr) \\ & \quad {} +\frac{437\text{,}230}{729}p \biggl( p-\frac{1}{9} \biggr) +\frac{240\text{,}400}{6561} \biggl( p-\frac{1}{9} \biggr) +\frac{240\text{,}400}{59\text{,}049} \biggr) x^{3}+ \biggl( 66\text{,}580p^{4} \biggl( p-\frac{1}{9}\biggr) \\ & \quad {} +\frac{213\text{,}658}{9}p^{3} \biggl( p- \frac{1}{9} \biggr) + \frac{195\text{,}838}{81}p^{2} \biggl( p- \frac{1}{9} \biggr) + \frac{58\text{,}786}{729}p \biggl( p-\frac{1}{9} \biggr) \\ &\quad {}+\frac{45\text{,}664}{6561} \biggl( p-\frac{1}{9} \biggr)+ \frac{45\text{,}664}{59\text{,}049} \biggr) x^{2} \\ & \quad {} + \biggl(9170p^{4} \biggl( p- \frac{1}{9} \biggr) +\frac{25\text{,}937}{9}p^{3} \biggl( p-\frac{1}{9} \biggr) +\frac{20\text{,}429}{81}p^{2} \biggl( p-\frac{1}{9} \biggr) + \frac{5120}{729}p^{2} \biggr) x \\ & \quad {} + \biggl( 490p^{4} \biggl( p-\frac{1}{9} \biggr) +\frac{1309}{9}p^{3} \biggl( p-\frac{1}{9} \biggr) +\frac{985}{81}p^{2} \biggl( p- \frac{1}{9} \biggr) +\frac{256}{729}p^{2} \biggr) . \end{aligned}$$

Since \(p>\frac{1}{9}\), we have \(P(x)>0\), which implies that \(f^{\prime }(x)>0\) for \({{x\in {}[ 1,+\infty ).}}\) Hence \(f(x)\) is strictly increasing on \([1,+\infty )\). It follows from \(\lim_{x\rightarrow +\infty }f(x)=0\) that \(f(x)<0\) for \({{x\in {}[ 1,+\infty ).}}\) This yields \(z_{n+1}-z_{n}=f(n)<0\), so that \((z_{n})_{n\geq 1}\) is strictly decreasing, which, along with \(\lim_{n\rightarrow \infty }z_{n}=0\), leads us to \(z_{n}>0\). The left-hand inequality of (2.3) is proved.

Similarly, differentiating \(g(x)\) with respect to x, we obtain

$$ g^{\prime }(x)=\frac{-Q(x)}{ 30x^{6} ( x+1 ) ^{6} ( x^{2}+6px+p ) ^{2} ( x ^{2}+ 2x+6px+1+7p ) ^{2} }, $$

where

$$\begin{aligned} Q(x) & = \biggl( 3780p^{2} \biggl( p-\frac{1}{9} \biggr) + \frac{2835}{9}p ^{2}+5 \biggr) x^{12}+60\text{,}480p^{3} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +28\text{,}560p^{2} \biggl( p-\frac{1}{9} \biggr) + \frac{22\text{,}890}{9}p ^{2}+120p+30 ) x^{11}+ \biggl( 324\text{,}000p^{4} \biggl( p-\frac{1}{9} \biggr) \\ & \quad {} +381\text{,}960p^{3} \biggl( p-\frac{1}{9} \biggr) + \frac{916\text{,}560}{9}p ^{2} \biggl( p-\frac{1}{9} \biggr) + \frac{859\text{,}455}{81}p^{2}+680p \\ & \quad {} +76 \biggr) x^{10}+ \bigl( 583\text{,}200p^{6}+1\text{,}771\text{,}200p^{5}+927\text{,}870p ^{4}+105\text{,}180p^{3} \\ & \quad {} +2610p^{2}+1624p+105 \bigr) x^{9}+ \bigl( 3\text{,}013\text{,}200p^{6}+4\text{,}492\text{,}800p ^{5} \\ & \quad {} +1\text{,}555\text{,}150p^{4}+138\text{,}450p^{3}+9066p^{2}+2122p+85 \bigr) x^{8}+ \bigl( 7\text{,}095\text{,}600p^{6} \\ & \quad {} +7\text{,}020\text{,}600p^{5}+1\text{,}794\text{,}700p^{4}+138\text{,}504p^{3}+12\text{,}426p^{2}+1648p \\ & \quad {} +40 \bigr) x^{7}+ \bigl( 10\text{,}103\text{,}400p^{6}+7\text{,}393\text{,}390p^{5}+1\text{,}467\text{,}221p ^{4}+100\text{,}436p^{3} \\ & \quad {} +9010p^{2}+774p+10 \bigr) x^{6}+ \bigl( 9\text{,}461\text{,}250p^{6}+5\text{,}365\text{,}230p ^{5}+846\text{,}847p^{4} \\ & \quad {} +49\text{,}296p^{3}+4263p^{2}+214p+1 \bigr) x^{5}+ \bigl( 5\text{,}874\text{,}525p ^{6}+2\text{,}667\text{,}310p^{5} \\ & \quad {} +337\text{,}249p^{4}+15\text{,}224p^{3}+1191p^{2}+32p \bigr) x^{4}+ \bigl( 2\text{,}352\text{,}300p ^{6}+883\text{,}050p^{5} \\ & \quad {} +89\text{,}495p^{4}+2688p^{3}+209p^{2}+2p \bigr) x^{3}+ \bigl( 568\text{,}125p ^{6}+183\text{,}690p^{5} \\ & \quad {} +15\text{,}221p^{4}+222p^{3}+22p^{2} \bigr) x^{2}+ \bigl( 72\text{,}450p^{6}+21\text{,}410p ^{5}+1559p^{4} \\ & \quad {} +4p^{3}+p^{2} \bigr)x+ \bigl(3675p^{6}+1050p^{5}+75p^{4} \bigr) . \end{aligned}$$

Since \(p>\frac{1}{9}\), we conclude that \({{Q}}(x)<0\). Thus we have \(g{{^{\prime }}}\)\((x)<0\) for \({{x\in {}[ 1,+\infty ).}}\) It follows that g\((x)\) is strictly decreasing on \([1,+\infty )\). Since \(\lim_{x\rightarrow +\infty }g(x)=0\), we have \(g(x)>0\) for \({{x\in {}[ 1,+\infty ).}}\) This yields \(u_{n+1}-u_{n}=g(n)>0\), which implies that \((u_{n})_{n\geq 1}\) is strictly increasing. We obtain \(u_{n}<0\) since \(\lim_{n\rightarrow \infty }u_{n}=0\). The right-hand inequality of (2.3) is proved.

This completes the proof of Theorem 2.1. □

Some remarks on Theorem 2.1

Remark 3.1

Lu [16] constructed the sequence

$$ r_{n}^{(3)}=1+\frac{1}{2}+\cdots +\frac{1}{n}- \log n-\frac{a_{1}}{n+\frac{a _{2}\cdot n}{n+a_{3}}}, $$
(3.1)

where \(a_{1}=\frac{1}{2}\), \(a_{2}=\frac{1}{6}\), and \(a_{3}=- \frac{1}{6}\) and proved the inequality

$$ \frac{1}{120(n+1)^{4}}< r_{n}^{(3)}-\gamma < \frac{1}{120(n-1)^{4}}. $$
(3.2)

In Theorem 2.1, if we take \(p=\frac{1}{5}\) in inequality (2.3), then we get

$$ \frac{1}{120n^{4}}-\frac{1}{50n^{5}}< \gamma -\Gamma_{n}^{(2)}< \frac{1}{120n ^{4}}. $$
(3.3)

Since

$$ \frac{1}{120(n+1)^{4}}< \frac{1}{120n^{4}}-\frac{1}{50n^{5}} $$

and

$$ \frac{1}{120n^{4}}< \frac{1}{120(n-1)^{4}} $$

for all natural numbers \(n\geq 5\), the sequence \(( \Gamma_{n} ^{(2)} ) _{n\geq 1}\) provides a more accurate double inequality for the difference between the sequence and the Euler–Mascheroni constant than the sequence \(( r_{n}^{(3)} ) _{n\geq 1}\) from [16].

Remark 3.2

Lu et al. [18] considered the following sequence converging to the Euler–Mascheroni constant:

$$ r_{n,2}^{(3)}=1+\frac{1}{2}+\cdots +\frac{1}{n}- \log n-\frac{1}{2} \log \biggl( 1+\frac{a_{1}}{n+\frac{a_{2}\cdot n}{n+a_{3}}} \biggr) , $$
(3.4)

where \(a_{1}=1\), \(a_{2}=-\frac{1}{3}\), and \(a_{3}=\frac{1}{3}\), and they proved that

$$ \frac{1}{180(n+1)^{4}}< \gamma -r_{n,2}^{(3)}< \frac{1}{180n^{4}}. $$
(3.5)

In Theorem 2.1, if we choose \(p=\frac{1}{6}\) in inequality (2.3), then we obtain

$$ \frac{1}{180n^{4}}-\frac{1}{72n^{5}}< \gamma -\Gamma_{n}^{(2)}< \frac{1}{180n ^{4}}. $$
(3.6)

It is easy to find that

$$ \frac{1}{180(n+1)^{4}}< \frac{1}{180n^{4}}-\frac{1}{72n^{5}} $$

for all natural numbers \(n\geq 5\), so the sequence \(( \Gamma_{n} ^{(2)} ) _{n\geq 1}\) improves inequality (3.5) from [18].

Remark 3.3

For more results relating to the Euler constant, sequences, and some estimates, we refer the interested reader to Sîntǎmǎrian [2326] and the references therein.

Conclusion

To provide a sequence converging faster to the Euler–Mascheroni constant, we construct a sequence \(\Gamma_{n}^{(2)}\) by reference to the Padé approximant method, which improves the rate of convergence of the sequences introduced by Lu [16, 18]. Our sequence depends on a real parameter and has a relatively simple form. It is worth noting that the method mentioned is also applicable to establishing estimates of bounds for some special means. For example, the method can be used for further study on the results obtained previously by Chu et al. [69], Qian and Chu [22], Yang et al. [3134], and Zhao et al. [35].

References

  1. 1.

    Alzer, H., Koumandos, S.: Series representations for γ and other mathematical constants. Anal. Math. 34(1), 1–8 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bailey, D.H.: Numerical results on the transcendence of constants involving π, e and Euler’s constant. Math. Comput. 50, 275–281 (1988)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bercu, G., Wu, S.: Refinements of certain hyperbolic inequalities via the Padé approximation method. J. Nonlinear Sci. Appl. 9(7), 5011–5020 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bernardo, J.M.: Algorithm AS 103: Psi (digamma) function. Appl. Stat. 25(3), 315–317 (1976)

    Article  Google Scholar 

  5. 5.

    Boas, R.P.: Estimating remainders. Math. Mag. 51(2), 83–89 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Chu, Y.M., Wang, M.K.: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3–4), 223–229 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Chu, Y.M., Wang, M.K., Gong, W.M.: Two sharp double inequalities for Seiffert mean. J. Inequal. Appl. 2011, Article ID 44 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Chu, Y.M., Wang, M.K., Qiu, S.L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Chu, Y.M., Xia, W.F.: Two sharp inequalities for power mean, geometric mean, and harmonic mean. J. Inequal. Appl. 2009, Article 741923, 6 pages (2009)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Cringanu, J.: Better bounds in Chen’s inequalities for the Euler’s constant. Bull. Aust. Math. Soc. 92(1), 94–97 (2015)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cristea, V.G., Mortici, C.: Latter research on Euler–Mascheroni constant. arXiv:1312.4397 [math.CA] (2013)

  12. 12.

    DeTemple, D.W.: A quicker covergences to Euler’s constant. Am. Math. Mon. 100(5), 468–470 (1993)

    Article  MATH  Google Scholar 

  13. 13.

    Hu, Y., Mortici, C.: Sharp inequalities related to the constant e. J. Inequal. Appl. 2014, Article ID 382 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Hu, Y., Mortici, C.: On the Keller limit and generalization. J. Inequal. Appl. 2016, Article ID 97 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Am. Math. Soc. 50(4), 527–628 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lu, D.: A new quicker sequence convergent to Euler’s constant. J. Number Theory 136, 320–329 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Lu, D.: Some new improved classes of convergence towards Euler’s constant. Appl. Math. Comput. 243, 24–32 (2014)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Lu, D., Song, L., Yu, Y.: Some new continued fraction approximation of Euler’s constant. J. Number Theory 147, 69–80 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Mortici, C., Hu, Y.: On some convergences to the constant e and improvements of Carleman’s inequality. Carpath. J. Math. 31(2), 249–254 (2015)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mortici, C., Vernescu, A.: An improvement of the convergence speed of the sequence \((\gamma_{n})_{n\geq 1}\) converging to Euler’s constant. An. Ştiinţ. Univ. ‘Ovidius’ Constanţa 13, 97–100 (2005)

    MATH  Google Scholar 

  21. 21.

    Mortici, C., Vernescu, A.: Some new facts in discrete asymptotic analysis. Math. Balk. 21, 301–308 (2007)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Qian, W.M., Chu, Y.M.: Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters. J. Inequal. Appl. 2017, Article ID 274 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Sîntǎmǎrian, A.: A generalization of Euler’s constant. Numer. Algorithms 46(2), 141–151 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Sîntǎmǎrian, A.: Some new sequences that converge to a generalization of Euler’s constant. Creative Math. Inform. 20(2), 191–196 (2011)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Sîntǎmǎrian, A.: New sequences that converge to a generalization of Euler’s constant. Integers 11(2), 127–138 (2011)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Sîntǎmǎrian, A.: Euler’s constant, sequences and some estimates. Surv. Math. Appl. 8, 103–114 (2013)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Sweeney, D.W.: On the computation of Euler’s constant. Math. Comput. 17, 170–178 (1963)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Vernescu, A.: A new accelerate convergence to the constant of Euler. Gaz. Mat., Ser. A 104(4), 273–278 (1999) (in Romanian)

    Google Scholar 

  29. 29.

    Wu, S., Bercu, G.: Fast convergence of generalized DeTemple sequences and the relation to the Riemann zeta function. J. Inequal. Appl. 2017, Article ID 110 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Wu, S., Bercu, G.: Padé approximants for inverse trigonometric functions and their applications. J. Inequal. Appl. 2017, Article ID 31 (2017)

    Article  MATH  Google Scholar 

  31. 31.

    Yang, Z.H., Chu, Y.M.: A monotonicity property involving the generalized elliptic integral of the first kind. Math. Inequal. Appl. 20(3), 729–735 (2017)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Yang, Z.H., Qian, W.M., Chu, Y.M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Yang, Z.H., Qian, W.M., Chu, Y.M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Yang, Z.H., Zhang, W., Chu, Y.M.: Sharp Gautschi inequality for parameter \(0< p<1\) with applications. Math. Inequal. Appl. 20(4), 1107–1120 (2017)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Zhao, T.H., Chu, Y.M., Jiang, Y.P.: Monotonic and logarithmically convex properties of a function involving gamma functions. J. Inequal. Appl. 2009, Article ID 728612 (2009)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Funding

The work of the first author is supported by the Natural Science Foundation of Fujian Province of China under Grant 2016J01023.

Author information

Affiliations

Authors

Contributions

Both authors read and approved the final manuscript.

Corresponding author

Correspondence to Shanhe Wu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wu, S., Bercu, G. A new sequence related to the Euler–Mascheroni constant. J Inequal Appl 2018, 151 (2018). https://doi.org/10.1186/s13660-018-1746-3

Download citation

MSC

  • 11Y60
  • 41A60
  • 41A25

Keywords

  • Sequences
  • Euler–Mascheroni constant
  • Rate of convergence
  • Lower and upper bounds