Another New Sequence Which Converges Faster Towards to the Euler–Mascheroni Constant

Abstract

In this paper, we introduce a new sequence, which approximates the Euler–Mascheroni constant $\gamma$ and converges faster to its limit, with the convergence rate $n^{-5}$. Also, for this constant, new inequalities are established. Our result, compared to other sequences with convergence rates $n^{-2}$, $n^{-3}$, or $n^{-4}$, improves some known results.

1. Introduction

The sequence ${\gamma }_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-logn,n\ge 1,$ converges very slowly to a limit denoted by $\gamma =0.5772\dots$, defined by $\gamma ={\sum }_{n=1}^{\infty }\left[{\displaystyle \frac{1}{n}}-log\left(1+{\displaystyle \frac{1}{n}}\right)\right]$ and known as the Euler–Mascheroni constant. This constant serves as an important bridge between discrete and continuous mathematics, linking sums and integrals. Both the sequence ${\left({\gamma }_n\right)}_{n\ge 1}$ and the Euler–Mascheroni constant $\gamma$ have multiple applications in diverse fields of mathematics, such as number theory ($\gamma$ is involved in the asymptotic distribution of prime numbers, particularly in relation to the harmonic series and the prime number theorem), analysis ($\gamma$ appears in integrals and sums involving logarithmic functions), special function theory ($\gamma$ appears in the definitions and properties of special functions, like the digamma function), applied statistics and probability theory ($\gamma$ appears in the study of distributions, including the normal distribution and in the analysis of algorithms, particularly in expected values), and other sciences (for example, in quantum physics, $\gamma$ shows up in calculations related to quantum field theory and other advanced topics).

Over the years, numerous estimations for ${\gamma }_n-\gamma$ have been obtained. For example, Tims and Tyrrel [1] proved that

$${\displaystyle \frac{1}{2(n+1)}}<{\gamma }_n-\gamma <{\displaystyle \frac{1}{2(n-1)}},n\ge 2,$$

Young [2] obtained that

$${\displaystyle \frac{1}{2(n+1)}}<{\gamma }_n-\gamma <{\displaystyle \frac{1}{2n}},n\ge 1,$$

Vernescu [3] demonstrated that

$${\displaystyle \frac{1}{2n+1}}<{\gamma }_n-\gamma <{\displaystyle \frac{1}{2n}},n\ge 1,$$

and Tóth [4] showed that

$${\displaystyle \frac{1}{2n+{\displaystyle \frac{2}{5}}}}<{\gamma }_n-\gamma <{\displaystyle \frac{1}{2n+{\displaystyle \frac{1}{3}}}},n\ge 1.$$

Anderson et al. [5] proved that

$${\displaystyle \frac{1-\gamma }{n}}<{\gamma }_n-\gamma <{\displaystyle \frac{1}{2n}},n\ge 1$$

and Alzer [6] obtained that

$${\displaystyle \frac{1}{2n+{\displaystyle \frac{2\gamma -1}{1-\gamma }}}}<{\gamma }_n-\gamma <{\displaystyle \frac{1}{2n+{\displaystyle \frac{1}{3}}}},n\ge 1.$$

The convergence of the sequence ${\gamma }_n$ to $\gamma$ is very slow.

Modifying the logarithmic term of ${\gamma }_n$, DeTemple [7] proved that the sequence

$$R_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}\right)$$

converges to $\gamma$ with a rate of convergence of $n^{-2}$, since

$${\displaystyle \frac{1}{24{(n+1)}^2}}<R_n-\gamma <{\displaystyle \frac{1}{24n^2}},n\ge 1.$$

Chen [8] showed that for all integers $n\ge 1$,

$${\displaystyle \frac{1}{24{(n+a)}^2}}\le R_n-\gamma <{\displaystyle \frac{1}{24{(n+b)}^2}},$$

with the best possible constants

$$a={\displaystyle \frac{1}{\sqrt{24\left[-\gamma +1-log\left(\frac{3}{2}\right)\right]}}}-1=0.55106\dots \mathrm{and}b={\displaystyle \frac{1}{2}}.$$

Vernescu [9] provided the sequence

$$V_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n-1}}+{\displaystyle \frac{1}{2n}}-logn,$$

which also converges to $\gamma$ with a rate of convergence of $n^{-2}$, since

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

Similarly, Ivan [10] obtained the convergence to $\gamma$ with a rate of convergence of $n^{-2}$:

$$\underset{n\to \infty }{lim}{n}^2(c_n-\gamma )={\displaystyle \frac{1}{6}},$$

where

$$c_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\sqrt{n(n+1)}.$$

Cristea and Mortici [11] introduced the family of sequences

$$v_n(a,b)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n-2}}+{\displaystyle \frac{an+b}{n(n-1)}}-logn,$$

where $a,b$ are real parameters. They proved that, among these sequences, ${\left(v_n\left({\displaystyle \frac{3}{2}},-{\displaystyle \frac{5}{12}}\right)\right)}_{n\ge 1}$ is the privileged one because it provides the best approximation to $\gamma$, since it has a rate of convergence of $n^{-3}$. More precisely, they obtained the bounds

$${\displaystyle \frac{1}{12n^3}}+{\displaystyle \frac{11}{120n^4}}<v_n\left({\displaystyle \frac{3}{2}},-{\displaystyle \frac{5}{12}}\right)-\gamma <{\displaystyle \frac{1}{12n^3}}+{\displaystyle \frac{13}{120n^4}}(n\ge 9),$$

where

$$v_n\left({\displaystyle \frac{3}{2}},-{\displaystyle \frac{5}{12}}\right)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n-2}}+{\displaystyle \frac{13}{12(n-1)}}+{\displaystyle \frac{5}{12n}}-logn.$$

Using continued fraction approximation, Lu [12] obtained the following faster sequences converging to the Euler–Mascheroni constant:

$$r_n^{\left(2\right)}=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-{\displaystyle \frac{3}{6n+1}}-logn,$$

$$r_n^{\left(3\right)}=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-{\displaystyle \frac{6n-1}{12n^2}}-logn,$$

which satisfy

$${\displaystyle \frac{1}{72{(n+1)}^3}}<\gamma -r_n^{\left(2\right)}<{\displaystyle \frac{1}{72n^3}},$$

$${\displaystyle \frac{1}{120{(n+1)}^4}}<r_n^{\left(3\right)}-\gamma <{\displaystyle \frac{1}{120{(n-1)}^4}}.$$

Modifying the logarithmic term of ${\gamma }_n$, Negoi [13] proved that the sequence

$$T_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}\right),$$

is strictly increasing and convergent to $\gamma$ with the rate of convergence $n^{-3}$. Furthermore, he proved that

$${\displaystyle \frac{1}{48{(n+1)}^3}}<\gamma -T_n<{\displaystyle \frac{1}{48n^3}},n\ge 1.$$

Chen and Mortici [14] showed that for all integers $n\ge 1$,

$${\displaystyle \frac{1}{48{(n+a)}^3}}\le \gamma -T_n<{\displaystyle \frac{1}{48{(n+b)}^3}},$$

with the best possible constants

$$a={\displaystyle \frac{1}{\sqrt[3]{48\left[1-\gamma +log\left({\displaystyle \frac{37}{24}}\right)\right]}}}-1=0.27380525\dots$$

and

$$b={\displaystyle \frac{83}{360}}=0.23055555\dots$$

Recently, You and Chen [15] modified the logarithmic term of ${\gamma }_n$ and demonstrated that the sequences

$$r_2\left(n\right)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}+{\displaystyle \frac{1}{48n^2}}\right),$$

$$r_3\left(n\right)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}-{\displaystyle \frac{1}{48n^3}}-{\displaystyle \frac{1}{576n^4}}-{\displaystyle \frac{1}{1152n^5}}\right),$$

converge to $\gamma$ with rates of convergence $n^{-3}$ and $n^{-4}$, respectively, since

$${\displaystyle \frac{1}{24{(n+1)}^3}}<\gamma -r_2\left(n\right)<{\displaystyle \frac{1}{24n^3}},$$

$${\displaystyle \frac{143}{5760{(n+1)}^4}}<r_3\left(n\right)-\gamma <{\displaystyle \frac{143}{5760n^4}}.$$

Quite recently, also by modifying the logarithmic term of ${\gamma }_n$, Crînganu [16] defined a new sequence

$$S_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}\right)$$

and showed that it converges to $\gamma$, with the rate of convergence $n^{-4}$, since

$${\displaystyle \frac{23}{5760{(n+a)}^4}}\le S_n-\gamma <{\displaystyle \frac{23}{5760{(n+b)}^4}},$$

with the best possible constants

$$a=\sqrt[4]{{\displaystyle \frac{23}{5760\left[1-\gamma -log\left({\displaystyle \frac{73}{48}}\right)\right]}}}-1=0.0315\dots$$

and

$$b=-{\displaystyle \frac{7}{46}}=-0.1521\dots .$$

Our aim is to obtain a new sequence that converges to the Euler–Mascheroni constant at a faster rate of convergence.

2. Main Results

Inspired by Crînganu [16], we consider, for $a,b,c,d\in \mathbb{R},n\ge 1$, the family of sequences

$$L_n(a,b,c,d)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+a+{\displaystyle \frac{b}{n}}+{\displaystyle \frac{c}{n^2}}+{\displaystyle \frac{d}{n^3}}\right),$$

and

$$K_n(a,b,c,d)=L_n(a,b,c,d)-\gamma =1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+a+{\displaystyle \frac{b}{n}}+{\displaystyle \frac{c}{n^2}}+{\displaystyle \frac{d}{n^3}}\right)-\gamma ,$$

which converges to zero.

Using a Maclaurin growth series, we get

$$\begin{array}{ccc}\hfill K_{n+1}(a,b,c,d)-K_n(a,b,c,d)& =& {\displaystyle \frac{\tilde{A}}{n^2}}+{\displaystyle \frac{\tilde{B}}{n^3}}+{\displaystyle \frac{\tilde{C}}{n^4}}+{\displaystyle \frac{\tilde{D}}{n^5}}+{\displaystyle \frac{\tilde{E}}{n^6}}+\mathcal{O}\left({\displaystyle \frac{1}{n^7}}\right),\hfill \end{array}$$

where

$$\begin{array}{c}\tilde{A}=a-{\displaystyle \frac{1}{2}},\\ \tilde{B}=-a^2-a+2b+{\displaystyle \frac{2}{3}},\\ \tilde{C}=a^3-3ab+{\displaystyle \frac{3a^2}{2}}+a-3b+3c-{\displaystyle \frac{3}{4}},\\ \tilde{D}=-a^4-2a^3+4a^{2}b-2a^2-2b^2+6ab-4ac-a+4b-6c+4d+{\displaystyle \frac{4}{5}},\\ \tilde{E}=a^5+{\displaystyle \frac{5a^4}{2}}-5a^{3}b-10a^{2}b+5a{b}^2+5a^{2}c+{\displaystyle \frac{10a^3}{3}}+{\displaystyle \frac{5a^2}{2}}+\\ +5b^2-10ab+10ac-5ad-5bc+a-5b+10c-10d-{\displaystyle \frac{5}{6}}.\end{array}$$

If

$$\tilde{A}=0,\tilde{B}=0,\tilde{C}=0,\tilde{D}=0,$$

then

$$a={\displaystyle \frac{1}{2}},b={\displaystyle \frac{1}{24}},c=-{\displaystyle \frac{1}{48}},d={\displaystyle \frac{23}{5760}}$$

and so

$$\tilde{E}=-{\displaystyle \frac{17}{768}}.$$

Thus

$$\begin{array}{cc}\hfill L_n& =L_n\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{24}},-{\displaystyle \frac{1}{48}},{\displaystyle \frac{23}{5760}}\right)\hfill \\ & =1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}+{\displaystyle \frac{23}{5760n^3}}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill K_n& =K_n\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{24}},-{\displaystyle \frac{1}{48}},{\displaystyle \frac{23}{5760}}\right)\hfill \\ & =1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}+{\displaystyle \frac{23}{5760n^3}}\right)-\gamma .\hfill \end{array}$$

Thus, we get

$$K_{n+1}-K_n=-{\displaystyle \frac{17}{768n^6}}+\mathcal{O}\left({\displaystyle \frac{1}{n^7}}\right).$$

By a variant of the Stolz lemma, we know that if a sequence ${\left(x_n\right)}_{n\ge 1}$ converges to zero and there exists the limit

$$\underset{n\to \infty }{lim}{n}^k(x_n-x_{n+1})=\mathcal{l}\in \mathbb{R},k>1,$$

then

$$\underset{n\to \infty }{lim}{n}^{k-1}{x}_n={\displaystyle \frac{\mathcal{l}}{k-1}}.$$

Therefore, for $K_n$, we have

$$\underset{n\to \infty }{lim}{n}^6(K_n-K_{n+1})={\displaystyle \frac{17}{768}}$$

and so

$$\underset{n\to \infty }{lim}{n}^5{K}_n={\displaystyle \frac{17}{3840}}.$$

Starting from this result, using an elementary sequence method and MATLAB 24.1 software for calculations, we obtain the following:

Theorem 1.

For every integer $n\ge 1$ we have

$${\displaystyle \frac{17}{3840{(n+a)}^5}}\le W_n-\gamma <{\displaystyle \frac{17}{3840{(n+b)}^5}},$$

with the best possible constants

$$a=\sqrt[5]{{\displaystyle \frac{17}{3840\left[1-\gamma -log\left({\displaystyle \frac{8783}{5760}}\right)\right]}}}-1=0.37407\dots$$

and

$$b={\displaystyle \frac{3305}{12852}}=0.25715\dots .$$

Proof. 

We define the sequence

$$\begin{array}{ccc}\hfill a_n& =& L_n-\gamma -{\displaystyle \frac{17}{3840{(n+a)}^5}}\hfill \\ & & =1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}+{\displaystyle \frac{23}{5760n^3}}\right)-\gamma -{\displaystyle \frac{17}{3840{(n+a)}^5}},\hfill \end{array}$$

for $a>0$ and so

$$a_{n+1}-a_n=f\left(n\right),$$

where

$$\begin{array}{ccc}\hfill f\left(n\right)& =& {\displaystyle \frac{1}{n+1}}-log\left(n+{\displaystyle \frac{3}{2}}+{\displaystyle \frac{1}{24(n+1)}}-{\displaystyle \frac{1}{48{(n+1)}^2}}+{\displaystyle \frac{23}{5760{(n+1)}^3}}\right)+\hfill \\ & & +log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}+{\displaystyle \frac{23}{5760n^3}}\right)-\hfill \\ & & -{\displaystyle \frac{17}{3840{(n+a+1)}^5}}+{\displaystyle \frac{17}{3840{(n+a)}^5}}.\hfill \end{array}$$

The derivative of function f is equal to

$$\begin{array}{ccc}\hfill f^{\prime }\left(n\right)& =& -{\displaystyle \frac{1}{{(n+1)}^2}}-{\displaystyle \frac{3(1920n^4+7680n^3+11440n^2+7600n+1897)}{(n+1)(5760n^4+25920n^3+43440n^2+32040n+8783)}}+\hfill \\ & & +{\displaystyle \frac{3(1920n^4-80n^2+80n-23)}{n(5760n^4+2880n^3+240n^2-120n+23)}}-{\displaystyle \frac{17}{768}}{\displaystyle \frac{{(n+a+1)}^6-{(n+a)}^6}{{(n+a)}^6{(n+a+1)}^6}}\hfill \\ & =& {\displaystyle \frac{P\left(n\right)}{Q\left(n\right)}},\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill P\left(n\right)& =& 768(4406400n^4+7490400n^3+2117280n^2-1647796n-606027)·\hfill \\ & & ·{(n+a)}^6·{(n+a+1)}^6-17n{(n+1)}^2(5760n^4+2880n^3+240n^2-120n+23)·\hfill \\ & & ·(5760n^4+25920n^3+43440n^2+32040n+8783)·\left[{(n+a+1)}^6-{(n+a)}^6\right],\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Q\left(n\right)& =& 768n{(n+1)}^2(5760n^4+2880n^3+240n^2-120n+23)·\hfill \\ & & ·(5760n^4+25920n^3+43440n^2+32040n+8783)·{(n+a)}^6{(n+a+1)}^6.\hfill \end{array}$$

Using MATLAB 24.1 software, we obtain that

$$\begin{array}{c}P\left(n\right)=p_0·n^{15}+p_1·n^{14}+p_2·n^{13}+p_3·n^{12}+p_4·n^{11}+p_5·n^{10}+p_6·n^9+p_7·n^8+\\ +p_8·n^7+p_9·n^6+p_{10}·n^5+p_{11}·n^4+p_{12}·n^3+p_{13}·n^2-p_{14}·n-p_{15},\end{array}$$

where

$$\begin{array}{c}{p}_0=23688806400a-6091776000,\\ p_1=189510451200a^2+140097945600a-54101237760,\\ p_2=710664192000a^3+1208781619200a^2+283562311680a-216840281088,\\ p_3=1658216448000a^4+4345122816000a^3+3195768176640a^2+27217096704a-517434349824,\\ p_4=2676835123200a^5+9421194240000a^4+11191051468800a^3+\\ +4313542551552a^2-951258977280a-816556190976,\\ p_5=3126922444800a^6+13913159270400a^5+22443335884800a^4+\\ +15431215165440a^3+2584804713984a^2-2085448478976a-895178414400,\\ p_6=2680219238400a^7+14696194867200a^6+29955583918080a^5+28487564835840a^4+\\ +11558431964160a^3-677704596480a^2-2380540040640a-696766207680,\\ p_7=1675137024000a^8+11256628838400a^7+28108735119360a^6+33662464303104a^5+\\ +19582377488640a^4+3364432450560a^3-2380953893760a^2-1694043099360a-386195711526,\\ p_8=744505344000a^9+6197824512000a^8+18769457479680a^7+27144229367808a^6+\\ +19496312727552a^5+5329924312320a^4-1566049991040a^3-1857543344640a^2-\\ -777633703566a-149937998355,\\ p_9=223351603200a^{10}+2382336000000a^9+87842838528008a^8+15008630759424a^7+\\ +12269823390720a^6+3216811753728a^5-1958475850560a^4-1844984923200a^3-\\ -744713732028a^2-224024317080a-38882029144,\\ p_{10}=40609382400a^{11}+603024998400a^{10}+2763719884800a^9+5475548298240a^8+\\ +4600589211648a^7+13519070208a^6-2858407439040a^5-2140061613120a^4-\\ -720254785500a^3-155641811586a^2-36388303746a-6000763190,\\ p_{11}=3384115200a^{12}+89336217600a^{11}+537755811840a^{10}+1188768215040a^9+\\ +700593096960a^8-1318070983680a^7-2664208535040a^6-1999216550880a^5-\\ -736245769470a^4-1130981320660a^3-12904066140a^2-2408847810a-423322168,\\ p_{12}=5752627200a^{12}+54028615680a^{11}+110086612992a^{10}-161048540160a^9-\\ -936009699840a^8-1519292712960a^7-1239385743360a^6-540538312566a^5-\\ -116468696355a^4-9205598920a^3-99484425a^2-131085198a-40496924,\\ p_{13}=1626071040a^{12}-5429661696a^{11}-89850714624a^{10}-310896161280a^9-\\ -517564650240a^8-477213143040a^7-247005305856a^6-66245936460a^5-\\ -7106635020a^4-235621700a^3-228228570a^2-111896346a-22083544)n^2\\ p_{14}=1265507328a^{12}+13178188800a^{11}+49700906496a^{10}+95124456960a^9+\\ +102759782400a^8+63444492288a^7+20813514240a^6+2813177334a^5+\\ +51512295a^4+68683060a^3+51512295a^2+20604918a+3434153,\\ p_{15}=465428736a^{12}+2792572416a^{11}+6981431040a^{10}+9308574720a^9+\\ +6981431040a^8+2792572416a^7+465428736a^6.\end{array}$$

If $p_0=0$, i.e., $a={\displaystyle \frac{3305}{12852}},$ then

$$\begin{array}{cc}\hfill P\left(n\right)& =-{\displaystyle \frac{1978301112320}{357}}{n}^{14}-{\displaystyle \frac{6614222397555712}{127449}}{n}^{13}-{\displaystyle \frac{267750738778833992704}{1228480911}}{n}^{12}-\hfill \\ \hfill & -{\displaystyle \frac{6410863378097015826387904}{11841327501129}}{n}^{11}-{\displaystyle \frac{11203462396425132464117874560}{12682061753709159}}{n}^{10}-\hfill \\ \hfill & -{\displaystyle \frac{283656463491678607733321331433820}{285232250902672695069}}{n}^9-\hfill \\ & -{\displaystyle \frac{9458940152242502727508996226916173}{11979754537912253192898}}{n}^8-\hfill \\ \hfill & -{\displaystyle \frac{31432492089149854440508935518270759137255}{70669386642452959618122419064}}{n}^7-\hfill \\ \hfill & -{\displaystyle \frac{80478660328007330472740977151857448170459391}{454121478564402718506054664905264}}{n}^6-\hfill \\ \hfill & -{\displaystyle \frac{16214760589885408521174449082511424923648820905}{324242735694983541013323030742358496}}{n}^5-\hfill \\ \hfill & -{\displaystyle \frac{1566232538811809753079292526805082299432018352507841}{150018035009469424887716193279628490061312}}{n}^4-\hfill \\ \hfill & -{\displaystyle \frac{508047894224609605857698089286221126579528398762921383}{275433112277385864093846930861397907752568832}}{n}^3-\hfill \\ \hfill & -{\displaystyle \frac{20244841378617164823764709930798913021994554417695397}{68858278069346466023461732715349476938142208}}{n}^2-\hfill \\ \hfill & -{\displaystyle \frac{197879846675513964816443737481433155667954876842757889}{6610394694657260738252326340673549786061651968}}n-\hfill \\ \hfill & -{\displaystyle \frac{4683428037913070737204329712156882479692563209390625}{8813859592876347651003101787564733048082202624}}<0,\hfill \end{array}$$

for all $n\ge 1,$ and then f is strictly decreasing.

We have ${lim}_{n\to \infty }f\left(n\right)=0$, and then it follows that $f\left(n\right)>0$ for all $n\ge 1,$ such that ${\left(a_n\right)}_{n\ge 1}$ is strictly increasing.

Since $\left(a_n\right)$ converges to zero, it results that $a_n<0$ for all $n\ge 1,$ such that

$$L_n-\gamma <{\displaystyle \frac{17}{3840{\left(n+{\displaystyle \frac{3305}{12852}}\right)}^5}},\mathrm{for}\mathrm{all}n\ge 1.$$

If

$$a=\sqrt[5]{{\displaystyle \frac{17}{3840\left[1-\gamma -log\left({\displaystyle \frac{8783}{5760}}\right)\right]}}}-1=0.37407\dots ,$$

then

$$a_1=L_1-\gamma -{\displaystyle \frac{17}{3840{(1+a)}^5}}=0,$$

$$P\left(n\right)>0\mathrm{for}\mathrm{all}n\ge 2$$

and, consequently, f is strictly increasing on $[2,\infty ).$

Since ${lim}_{n\to \infty }f\left(n\right)=0$, it results that $f\left(n\right)<0$ for all $n\ge 2,$ such that ${\left(a_n\right)}_{n\ge 2}$ is strictly decreasing.

The sequence $\left(a_n\right)$ converges to zero and then it results that $a_n>0$ for all $n\ge 2,$ such that

$${\displaystyle \frac{17}{3840{(n+a)}^5}}\le L_n-\gamma ,\mathrm{for}\mathrm{all}n\ge 1.$$

□

Remark 1.

Let us remark that, if $a>{\displaystyle \frac{3305}{12852}},$ then $23688806400a-6091776000>0,$ so there exists $n_a\ge 1$ such that $P\left(n\right)>0$ for all $n\ge n_a$ and we have

$${\displaystyle \frac{17}{3840{(n+a)}^5}}<L_n-\gamma <{\displaystyle \frac{17}{3840{\left(n+{\displaystyle \frac{3305}{12852}}\right)}^5}},$$

for all $n\ge n_a.$

Remark 2.

After a few iterations, we can observe that, compared to sequences

$$R_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}\right),$$

$$T_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}\right),$$

and

$$S_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}\right),$$

with rates of convergence $n^{-2}$, $n^{-3}$, and $n^{-4}$, respectively, the sequence

$$L_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}+{\displaystyle \frac{23}{5760n^3}}\right)$$

with rate of convergence $n^{-5}$ convergences faster to γ:

n	$R_n$	$T_n$	$S_n$	$L_n$
10	$0.5775929968\dots$	$0.5771962501\dots$	$0.5772160837\dots$	$0.5772157035\dots$
11	$0.5775303095\dots$	$0.5772009829\dots$	$0.5772159500\dots$	$0.5772156892\dots$
12	$0.5774820339\dots$	$0.5772042946\dots$	$0.5772158656\dots$	$0.5772156808\dots$
13	$0.5774440696\dots$	$0.5772066809\dots$	$0.5772158102\dots$	$0.5772156756\dots$
14	$0.5774136771\dots$	$0.5772084436\dots$	$0.5772157727\dots$	$0.5772156723\dots$
15	$0.5773889693\dots$	$0.5772097738\dots$	$0.5772157465\dots$	$0.5772156702\dots$
16	$0.5773686123\dots$	$0.5772107964\dots$	$0.5772157278\dots$	$0.5772156687\dots$
17	$0.5773516417\dots$	$0.5772115954\dots$	$0.5772157142\dots$	$0.5772156677\dots$
18	$0.5773373461\dots$	$0.5772122288\dots$	$0.5772157040\dots$	$0.5772156670\dots$
19	$0.5773251915\dots$	$0.5772127372\dots$	$0.5772156964\dots$	$0.5772156665\dots$
20	$0.5773147709\dots$	$0.5772131501\dots$	$0.5772156905\dots$	$0.5772156661\dots$
21	$0.5773057696\dots$	$0.5772134889\dots$	$0.5772156859\dots$	$0.5772156659\dots$
22	$0.5772979410\dots$	$0.5772137694\dots$	$0.5772156823\dots$	$0.5772156657\dots$
23	$0.5772910899\dots$	$0.5772140037\dots$	$0.5772156795\dots$	$0.5772156655\dots$
24	$0.5772850602\dots$	$0.5772142010\dots$	$0.5772156772\dots$	$0.5772156654\dots$
25	$0.5772797255\dots$	$0.5772143682\dots$	$0.5772156753\dots$	$0.5772156653\dots$
26	$0.5772749832\dots$	$0.5772145109\dots$	$0.5772156738\dots$	$0.5772156652\dots$
27	$0.5772707485\dots$	$0.5772146334\dots$	$0.5772156725\dots$	$0.5772156651\dots$
28	$0.5772669516\dots$	$0.5772147391\dots$	$0.5772156715\dots$	$0.5772156651\dots$
29	$0.5772635342\dots$	$0.5772148309\dots$	$0.5772156706\dots$	$0.5772156651\dots$
30	$0.5772604473\dots$	$0.5772149110\dots$	$0.5772156699\dots$	$0.5772156650\dots$
40	$0.5772410648\dots$	$0.5772153449\dots$	$0.5772156664\dots$	$0.5772156649\dots$
50	$0.5772320020\dots$	$0.5772155005\dots$	$0.5772156654\dots$	$0.5772156649\dots$

3. Conclusions

By modifying the logarithmic term of ${\gamma }_n$, we have constructed the sequence

$$L_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{24n}}-{\displaystyle \frac{1}{48n^2}}+{\displaystyle \frac{23}{5760n^3}}\right),\mathrm{for}n\ge 1,$$

that converges faster to the Euler–Mascheroni constant, with a rate of convergence of $n^{-5}$, compared to the sequences from [7,8,9,10] which have a rate of convergence of $n^{-2}$, those from [11,12,13,14,15] which have a rate of convergence of $n^{-3}$, or those from [12,15,16] which have a convergence rate of $n^{-4}$.

Furthermore, the idea of constructing the sequence $L_n$ allows the construction of a new sequence with a rate of convergence of $n^{-6}$, starting from the family of sequences

$$L_n^{\left(5\right)}(a,b,c,d,e)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+a+{\displaystyle \frac{b}{n}}+{\displaystyle \frac{c}{n^2}}+{\displaystyle \frac{d}{n^3}}+{\displaystyle \frac{e}{n^4}}\right),$$

for $a,b,c,d,e\in \mathbb{R}$,$n\ge 1.$

More generally, it can be proved that starting from the family of sequences

$$L_n^{\left(k\right)}(a_1,a_2,\dots ,a_k)=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}-log\left(n+a_1+{\displaystyle \frac{a_2}{n}}+\cdots +{\displaystyle \frac{a_k}{n^{k-1}}}\right),$$

for $a_1,a_2,\dots ,a_k\in \mathbb{R}$,$k\ge 5,n\ge 1,$ we obtain a sequence with a rate of convergence of $n^{-k-1}$.

Focusing on faster convergence for sequences related to the Euler–Mascheroni constant $\gamma$ is crucial for advancing both theoretical understanding and practical applications of this important mathematical constant.