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%I #39 Mar 18 2024 12:08:31
%S 5,4,9,3,0,6,1,4,4,3,3,4,0,5,4,8,4,5,6,9,7,6,2,2,6,1,8,4,6,1,2,6,2,8,
%T 5,2,3,2,3,7,4,5,2,7,8,9,1,1,3,7,4,7,2,5,8,6,7,3,4,7,1,6,6,8,1,8,7,4,
%U 7,1,4,6,6,0,9,3,0,4,4,8,3,4,3,6,8,0,7,8,7,7,4,0,6,8,6,6,0,4,4
%N Decimal expansion of log(3)/2.
%C Culler & Shalen show a bound of log(3)/2 on maximal injectivity under certain circumstances, see links.
%C Equals arctanh(1/2), the rapidity of an object traveling at half the speed of light. - _Sean Stroud_, May 13 2019
%H Marc Culler and Peter B. Shalen, <a href="https://www.jstor.org/stable/40590539">Betti numbers and injectivity radii</a>, Proceedings of the American Mathematical Society, Vol. 137, No. 11 (2009), pp. 3919-3922; <a href="http://arxiv.org/abs/0902.0014">preprint</a>, arXiv:0902.0014 [math.GT], 2009.
%H R. S. Melham and A. G. Shannon, <a href="https://www.fq.math.ca/Scanned/33-1/melham2.pdf">Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers</a>, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=EqrSCOMx37U">This gnarly integral is actually easy??</a>, YouTube video, 2023.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals arctanh(1/2) = arccoth(2) = Integral_{x>2} 1/(x^2-1) dx. - _Jean-François Alcover_, Jun 04 2013
%F From _Amiram Eldar_, Aug 05 2020: (Start)
%F Equals Sum_{k>=0} 1/((2*k+1) * 2^(2*k+1)).
%F Equals Integral_{x=0..oo} 1/(exp(x) + 2) dx. (End)
%F Equals Sum_{k>=1} arctanh(1/Fibonacci(2*k+2)) (Melham and Shannon, 1995). - _Amiram Eldar_, Jan 15 2022
%F log(3)/2 = Sum_{n >= 1} 1/(n*P(n, 2)*P(n-1, 2)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3)/2 = 0.54930614433(10...), correct to 11 decimal places. - _Peter Bala_, Mar 16 2024
%e 0.54930614433405484569762261846...
%t RealDigits[Log[3]/2,10,120][[1]] (* _Harvey P. Dale_, Apr 13 2016 *)
%o (PARI) log(3)/2 \\ _Charles R Greathouse IV_, May 15 2019
%Y Cf. A000045, A002391 (decimal expansion of natural logarithm of 3).
%K nonn,cons,easy
%O 0,1
%A _Jonathan Vos Post_, Feb 03 2009
%E All digits were wrong. Corrected by _N. J. A. Sloane_, Feb 05 2009
%E Offset 0 from _Michel Marcus_, May 13 2019
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