A083679 Decimal expansion of log(4/3).
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, 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, 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
Offset: 0
Examples
log(4/3) = 0.2876820724517809274392190059938274315035097108977610565....
Programs
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Mathematica
RealDigits[Log[4/3],10,120][[1]] (* Harvey P. Dale, Feb 04 2015 *)
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PARI
log(4/3) \\ Charles R Greathouse IV, May 15 2019
Formula
Limit of a special sum: log(4/3) = Sum_{k>=1} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1).
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).
From Amiram Eldar, Aug 07 2020: (Start)
Equals 2 * arctanh(1/7).
Equals Sum_{n>=1} 1/(n * 4^n) = Sum_{n>=1} 1/A018215(n).
Equals Sum_{n>=1} (-1)^(n+1)/(n * 3^n) = Sum_{n>=1} (-1)^(n+1)/A036290(n).
Equals Integral_{x=0..oo} 1/(3*exp(x) + 1) dx. (End)
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
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
Equals Integral_{x=0..1} (x^(1/3) - 1)/log(x) dx. - Kritsada Moomuang, May 27 2025