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

0,1

Links

Table of n, a(n) for n=0..103.

Index entries for transcendental numbers

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

Cf. A018215, A036290.

Sequence in context: A027606 A202693 A174552 * A213930 A319463 A079031

Adjacent sequences: A083676 A083677 A083678 * A083680 A083681 A083682

Keyword

cons,nonn

Author

Benoit Cloitre, Jun 15 2003

Status

approved