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%I #103 Apr 21 2023 03:29:04
%S 3,4,6,5,7,3,5,9,0,2,7,9,9,7,2,6,5,4,7,0,8,6,1,6,0,6,0,7,2,9,0,8,8,2,
%T 8,4,0,3,7,7,5,0,0,6,7,1,8,0,1,2,7,6,2,7,0,6,0,3,4,0,0,0,4,7,4,6,6,9,
%U 6,8,1,0,9,8,4,8,4,7,3,5,7,8,0,2,9,3,1,6,6,3,4,9,8,2,0,9,3,4,3
%N Decimal expansion of log(32) = 5*log(2).
%C The function exp(x) has its maximum curvature where x = -(1/10)*5*log(2) = -log(2)/2 = 0.34657... - _Dimitri Papadopoulos_, Oct 27 2022
%C This maximum curvature occurs at the point with coordinates (x_M = -log(2)/2 = -(this constant)/10; y_M = sqrt(2)/2 = A010503) and is equal to 2*sqrt(3)/9 = A212886. - _Bernard Schott_, Dec 23 2022
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
%H Harry J. Smith, <a href="/A016655/b016655.txt">Table of n, a(n) for n = 1..20000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%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 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2021.
%H <a href="/index/Cu#curves">Index to sequences related to curves</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F log(2)/2 = (1 - 1/2 - 1/4) + (1/3 - 1/6 - 1/8) + (1/5 - 1/10 - 1/12) + ... [Jolley, Summation of Series, Dover (1961) eq (73)]
%F Equals 10*log(2)/2 = 5*log(2) = 5*A002162, so 10*(1/2 - 1/4 + 1/6 - 1/8 + 1/10 - ... + (-1)^(k+1)/(2*k) + ...) = log(32). - _Eric Desbiaux_, Nov 26 2008
%F -log(2)/2 = lim_{n->oo} ((Sum_{k=2..n} arctanh(1/k)) - log(n)). - _Jean-François Alcover_, Aug 07 2014, after Steven Finch
%F Equals log(sqrt(2)) with offset 0. - _Michel Marcus_, Feb 19 2017
%F Equals (5/4)*Sum_{k=1..4} (-1)^(k+1) gamma(0, k/4) where gamma(n,x) denotes the generalized Stieltjes constants. - _Peter Luschny_, May 16 2018
%F From _Amiram Eldar_, Jun 29 2020: (Start)
%F log(2)/2 = arctanh(1/3) = arcsinh(1/sqrt(8)).
%F log(2)/2 = Integral_{x=0..Pi/4} tan(x) dx.
%F log(2)/2 = Sum_{k>=0} (-1)^k/(2*k+2).
%F log(2)/2 = Sum_{k>=1} 1/A060851(k). (End)
%F log(2)/2 = Sum_{k>=1} (-1)^(k+1) * arctanh(Lucas(2*k+3)/Fibonacci(2*k+3)^2) (Melham and Shannon, 1995). - _Amiram Eldar_, Jan 15 2022
%e 3.465735902799726547086160607290882840377500671801276270603400047466968...
%t RealDigits[5 N [Log[2], 100]] [[1]] (* _Vincenzo Librandi_, Jan 02 2016 *)
%o (PARI) log(32) \\ _Charles R Greathouse IV_, Jan 24 2012
%o (Magma) [5*Log(2)]; // _Vincenzo Librandi_, Jan 02 2016
%Y Cf. A000045, A000032, A060851, A195909, A195913, A195697, A016460 (continued fraction).
%Y Cf. A010503, A212886.
%K nonn,cons
%O 1,1
%A _N. J. A. Sloane_
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