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A083679 Decimal expansion of log(4/3).

%I A083679 #27 May 27 2025 09:01:25
%S A083679 2,8,7,6,8,2,0,7,2,4,5,1,7,8,0,9,2,7,4,3,9,2,1,9,0,0,5,9,9,3,8,2,7,4,
%T A083679 3,1,5,0,3,5,0,9,7,1,0,8,9,7,7,6,1,0,5,6,5,0,6,6,6,5,6,8,5,3,4,9,2,9,
%U A083679 2,9,5,0,7,2,0,7,8,0,4,6,4,3,3,8,1,1,0,8,9,9,1,7,9,1,0,5,2,8,6,2,9,6,0,3
%N A083679 Decimal expansion of log(4/3).
%H A083679 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A083679 Limit of a special sum: log(4/3) = Sum_{k>=1} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1).
%F A083679 Asymptotically: log(4/3) = Sum_{k=1..n} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1) + log(2)/2^(n+1) + o(1/2^n).
%F A083679 From _Amiram Eldar_, Aug 07 2020: (Start)
%F A083679 Equals 2 * arctanh(1/7).
%F A083679 Equals Sum_{n>=1} 1/(n * 4^n) = Sum_{n>=1} 1/A018215(n).
%F A083679 Equals Sum_{n>=1} (-1)^(n+1)/(n * 3^n) = Sum_{n>=1} (-1)^(n+1)/A036290(n).
%F A083679 Equals Integral_{x=0..oo} 1/(3*exp(x) + 1) dx. (End)
%F A083679 log(4/3) = 2*Sum_{n >= 1} 1/(n*P(n, 7)*P(n-1, 7)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(4/3) = 0.28768207245178092743921(31...), correct to 23 decimal places. - _Peter Bala_, Mar 18 2024
%F A083679 Equals Sum_{n >= 1} (-1)^(n+1) * 7/(n*binomial(2*n, n)*12^n). The n-th term of the series is O(7*sqrt(Pi/n)*1/48^n). - _Peter Bala_, Mar 04 2025
%F A083679 Equals Integral_{x=0..1} (x^(1/3) - 1)/log(x) dx. - _Kritsada Moomuang_, May 27 2025
%e A083679 log(4/3) = 0.2876820724517809274392190059938274315035097108977610565....
%t A083679 RealDigits[Log[4/3],10,120][[1]] (* _Harvey P. Dale_, Feb 04 2015 *)
%o A083679 (PARI) log(4/3) \\ _Charles R Greathouse IV_, May 15 2019
%Y A083679 Cf. A016578, A018215, A036290.
%K A083679 cons,nonn
%O A083679 0,1
%A A083679 _Benoit Cloitre_, Jun 15 2003