Editing Euler's constant (section)

=== Continued fraction ===
The simple [[continued fraction]] expansion of Euler's constant is given by:{{r|OEIS_A002852}}

:<math>\gamma=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{4+\dots}}}}}}}</math>

which has no ''apparent'' pattern. It is known to have at least 16,695,000,000 terms,{{r|OEIS_A002852}} and it has infinitely many terms [[if and only if]] {{mvar|γ}} is irrational.

[[File:KhinchinBeispiele.svg|thumb|The Khinchin limits for <math>\pi</math> (red), <math>\gamma</math> (blue) and <math>\sqrt[3]{2}</math> (green).|350x350px]]

Numerical evidence suggests that both Euler's constant {{math|{{var|γ}}}} as well as the constant {{math|{{var|e}}{{i sup|1px|{{var|γ}}}}}} are among the numbers for which the [[geometric mean]] of their simple continued fraction terms converges to [[Khinchin's constant]]. Similarly, when <math>p_n/q_n</math> are the convergents of their respective continued fractions, the limit <math>\lim_{n\to\infty}q_n^{1/n}</math> appears to converge to [[Lévy's constant]] in both cases.<ref name=":4">{{Cite journal |last=Brent |first=Richard P. |date=1977 |title=Computation of the Regular Continued Fraction for Euler's Constant |url=https://www.jstor.org/stable/2006010 |journal=Mathematics of Computation |volume=31 |issue=139 |pages=771–777 |doi=10.2307/2006010 |jstor=2006010 |issn=0025-5718}}</ref> However neither of these limits has been proven.<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Continued Fraction |url=https://mathworld.wolfram.com/Euler-MascheroniConstantContinuedFraction.html |access-date=2024-09-23 |website=mathworld.wolfram.com |language=en}}</ref>

There also exists a generalized continued fraction for Euler's constant.<ref>{{Citation |last1=Pilehrood |first1=Khodabakhsh Hessami |title=On a continued fraction expansion for Euler's constant |date=2013-12-29 |last2=Pilehrood |first2=Tatiana Hessami|arxiv=1010.1420 }}</ref>

A good simple [[approximation]] of {{math|{{var|γ}}}} is given by the [[Multiplicative inverse|reciprocal]] of the [[square root of 3]] or about 0.57735:<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Euler-Mascheroni Constant Approximations |url=https://mathworld.wolfram.com/Euler-MascheroniConstantApproximations.html |access-date=2024-10-19 |website=mathworld.wolfram.com |language=en}}</ref>

:<math>\frac1\sqrt {3}=0+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\dots}}}}}}}</math>

with the difference being about 1 in 7,429.