Editing Euler's constant (section)

=== Series expansions ===
In general,

<math display="block">
\gamma = \lim_{n \to \infty}\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3} + \ldots + \frac{1}{n} - \log(n+\alpha) \right) \equiv \lim_{n \to \infty} \gamma_n(\alpha)
</math>

for any {{math|{{var|α}} > −{{var|n}}}}. However, the rate of convergence of this expansion depends significantly on {{mvar|α}}. In particular, {{math|{{var|γ}}{{sub|{{var|n}}}}(1/2)}} exhibits much more rapid convergence than the conventional expansion {{math|{{var|γ}}{{sub|{{var|n}}}}(0)}}.{{r|DeTemple1993}}{{sfn|Havil|2003|pp=75–8}} This is because

<math display="block">
\frac{1}{2(n+1)} < \gamma_n(0) - \gamma < \frac{1}{2n},
</math>

while

<math display="block">
\frac{1}{24(n+1)^2} < \gamma_n(1/2) - \gamma < \frac{1}{24n^2}.
</math>

Even so, there exist other series expansions which converge more rapidly than this; some of these are discussed below.

Euler showed that the following [[infinite series]] approaches {{mvar|γ}}:
<math display="block">\gamma = \sum_{k=1}^\infty \left(\frac 1 k - \log\left(1+\frac 1 k \right)\right).</math>

The series for {{mvar|γ}} is equivalent to a series [[Niels Nielsen (mathematician)|Nielsen]] found in 1897:{{r|Krämer2005}}{{sfn|Blagouchine|2016}}

<math display="block">\gamma = 1 - \sum_{k=2}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor}{k+1}.</math>

In 1910, [[Giovanni Vacca (mathematician)|Vacca]] found the closely related series{{r|Vacca1910|Glaisher1910|Hardy1912|Vacca1926|Kluyver1927|Krämer2005|Blagouchine2016}}

<math display="block">\begin{align}
\gamma & = \sum_{k=1}^\infty (-1)^k\frac{\left\lfloor\log_2 k\right\rfloor} k \\[5pt]
& = \tfrac12-\tfrac13 + 2\left(\tfrac14 - \tfrac15 + \tfrac16 - \tfrac17\right) + 3\left(\tfrac18 - \tfrac19 + \tfrac1{10} - \tfrac1{11} + \cdots - \tfrac1{15}\right) + \cdots,
\end{align}</math>

where {{math|log{{sub|2}}}} is the [[binary logarithm|logarithm to base 2]] and {{math|{{floor|&nbsp;}}}} is the [[Floor and ceiling functions|floor function]].

This can be generalized to:<ref>{{Citation |last1=Pilehrood |first1=Khodabakhsh Hessami |title=Vacca-type series for values of the generalized-Euler-constant function and its derivative |date=2008-08-04 |last2=Pilehrood |first2=Tatiana Hessami|arxiv=0808.0410 }}</ref>

<math display="block">\gamma= \sum_{k=1}^\infty \frac{\left\lfloor\log_B k\right\rfloor}{k} \varepsilon(k)</math>where:<math display="block">\varepsilon(k)= \begin{cases} B-1, &\text{if } B\mid n \\ -1, &\text{if }B\nmid n \end{cases}</math>

In 1926 Vacca found a second series:

<math display="block">\begin{align}
\gamma + \zeta(2) & = \sum_{k=2}^\infty \left( \frac1{\left\lfloor\sqrt{k}\right\rfloor^2} - \frac1{k}\right) \\[5pt]
& = \sum_{k=2}^\infty \frac{k - \left\lfloor\sqrt{k}\right\rfloor^2}{k \left\lfloor \sqrt{k} \right\rfloor^2} \\[5pt]
&= \frac12 + \frac23 + \frac1{2^2}\sum_{k=1}^{2\cdot 2} \frac{k}{k+2^2} + \frac1{3^2}\sum_{k=1}^{3\cdot 2} \frac{k}{k+3^2} + \cdots
\end{align}</math>

From the [[Carl Johan Malmsten|Malmsten]]–[[Ernst Kummer|Kummer]] expansion for the logarithm of the gamma function<ref name=":1" /> we get:

<math display="block">\gamma = \log\pi - 4\log\left(\Gamma(\tfrac34)\right) + \frac 4 \pi \sum_{k=1}^\infty (-1)^{k+1}\frac{\log(2k+1)}{2k+1}.</math>

Ramanujan, in his [[Ramanujan's lost notebook|lost notebook]] gave a series that approaches {{mvar|γ}}{{r|Berndt2008}}:

<math display="block">\gamma = \log 2 - \sum_{n=1}^{\infty} \sum_{k=\frac{3^{n-1}+1}{2}}^{\frac{3^{n}-1}{2}} \frac{2n}{(3k)^3-3k}</math>

An important expansion for Euler's constant is due to [[Gregorio Fontana|Fontana]] and [[Lorenzo Mascheroni|Mascheroni]]

<math display="block">\gamma = \sum_{n=1}^\infty \frac{|G_n|}{n} = \frac12 + \frac1{24} + \frac1{72} + \frac{19}{2880} + \frac3{800} + \cdots,</math>
where {{math|{{var|G}}{{sub|{{var|n}}}}}} are [[Gregory coefficients]].{{r|Krämer2005|Blagouchine2016|Blagouchine2018}} This series is the special case {{math|1={{var|k}} = 1}} of the expansions

<math display="block">\begin{align}
 \gamma &= H_{k-1}  - \log k + \sum_{n=1}^{\infty}\frac{(n-1)!|G_n|}{k(k+1) \cdots (k+n-1)} && \\
     &= H_{k-1} - \log k + \frac1{2k} + \frac1{12k(k+1)} + \frac1{12k(k+1)(k+2)} + \frac{19}{120k(k+1)(k+2)(k+3)} + \cdots &&
\end{align}</math>

convergent for {{math|1={{var|k}} = 1, 2, ...}}

A similar series with the Cauchy numbers of the second kind {{math|{{var|C}}{{sub|{{var|n}}}}}} is{{r|Blagouchine2016|Alabdulmohsin2018_1478}} 

<math display="block">\gamma = 1 -  \sum_{n=1}^\infty \frac{C_{n}}{n \, (n+1)!} =1- \frac{1}{4} -\frac{5}{72} - \frac{1}{32} - \frac{251}{14400} - \frac{19}{1728} - \ldots</math>

Blagouchine (2018) found an interesting generalisation of the Fontana–Mascheroni series

<math display="block">\gamma=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n}\Big\{\psi_{n}(a)+ \psi_{n}\Big(-\frac{a}{1+a}\Big)\Big\},
\quad a>-1</math>

where {{math|{{var|ψ}}{{sub|{{var|n}}}}({{var|a}})}} are the [[Bernoulli polynomials of the second kind]], which are defined by the generating function

<math display="block">
\frac{z(1+z)^s}{\log(1+z)}= \sum_{n=0}^\infty z^n \psi_n(s) ,\qquad |z|<1.
</math>

For any rational {{mvar|a}} this series contains rational terms only. For example, at {{math|1={{var|a}} = 1}}, it becomes{{r|OEIS_A302120|OEISA302121}}

<math display="block">\gamma=\frac{3}{4} - \frac{11}{96} - \frac{1}{72} - \frac{311}{46080} - \frac{5}{1152} - \frac{7291}{2322432}  - \frac{243}{100352} - \ldots</math>
Other series with the same polynomials include these examples:

<math display="block">\gamma= -\log(a+1) - \sum_{n=1}^\infty\frac{(-1)^n \psi_{n}(a)}{n},\qquad \Re(a)>-1 </math>

and

<math display="block">\gamma= -\frac{2}{1+2a} \left\{\log\Gamma(a+1) -\frac{1}{2}\log(2\pi) + \frac{1}{2} + \sum_{n=1}^\infty\frac{(-1)^n \psi_{n+1}(a)}{n}\right\},\qquad \Re(a)>-1 </math>

where {{math|Γ({{var|a}})}} is the [[gamma function]].{{r|Blagouchine2018}}

A series related to the Akiyama–Tanigawa algorithm is

<math display="block">\gamma= \log(2\pi) - 2 - 2 \sum_{n=1}^\infty\frac{(-1)^n G_{n}(2)}{n}=
\log(2\pi) - 2 + \frac{2}{3} + \frac{1}{24}+ \frac{7}{540} + \frac{17}{2880}+ \frac{41}{12600} + \ldots  </math>

where {{math|{{var|G}}{{sub|{{var|n}}}}(2)}} are the [[Gregory coefficients]] of the second order.{{r|Blagouchine2018}}
  
As a series of [[prime number]]s:

<math display="block">\gamma = \lim_{n\to\infty}\left(\log n - \sum_{p\le n}\frac{\log p}{p-1}\right).</math>