OFFSET
0,1
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
EXAMPLE
log(4/3) = 0.2876820724517809274392190059938274315035097108977610565....
MATHEMATICA
RealDigits[Log[4/3], 10, 120][[1]] (* Harvey P. Dale, Feb 04 2015 *)
PROG
(PARI) log(4/3) \\ Charles R Greathouse IV, May 15 2019
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Jun 15 2003
STATUS
approved