A016578 Decimal expansion of log(3/2).

4, 0, 5, 4, 6, 5, 1, 0, 8, 1, 0, 8, 1, 6, 4, 3, 8, 1, 9, 7, 8, 0, 1, 3, 1, 1, 5, 4, 6, 4, 3, 4, 9, 1, 3, 6, 5, 7, 1, 9, 9, 0, 4, 2, 3, 4, 6, 2, 4, 9, 4, 1, 9, 7, 6, 1, 4, 0, 1, 4, 3, 2, 4, 1, 4, 4, 1, 0, 0, 6, 7, 1, 2, 4, 8, 9, 1, 4, 2, 5, 1, 2, 6, 7, 7, 5, 2, 4, 2, 7, 8, 1, 7, 3, 1, 3, 4, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Examples

			0.4054651081081643819780131154643491365719904234624941976140143...
		

References

• L. B. W. Jolley, Summation of Series, Dover (1961), eq (102), page 20.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Index entries for transcendental numbers

Crossrefs

Cf. A016529, A002391, A002162, A083679.

Programs

Mathematica

RealDigits[Log[3/2],10,111][[1]] (* Robert G. Wilson v, Aug 08 2011 *)

PARI

default(realprecision, 20080); x=10*log(3/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b016578.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009

Formula

Equals Sum {k>=1} 1/(k*3^k). - Robert G. Wilson v, Aug 08 2011

Equals 1/2 - 1/(2*2^2) + 1/(3*2^3) - 1/(4*2^4) + ... [Jolley].

Equals A002391-A002162. - Michel Marcus, Sep 17 2016

From Amiram Eldar, Aug 07 2020: (Start)

Equals 2 * arctanh(1/5).

Equals Integral_{x=0..oo} 1/(2*exp(x) + 1) dx. (End)

log(3/2) = 2*Sum_{n >= 1} 1/(n*P(n, 5)*P(n-1, 5)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3/2) = 0.40546510810816438197(04...), correct to 20 decimal places. - Peter Bala, Mar 16 2024

Equals Sum_{n >= 1} (-1)^(n+1) * 5/(n*binomial(2*n, n)*6^n). The n-th term of the series is O(5*sqrt(Pi/n)*1/24^n). - Peter Bala, Mar 04 2025

Equals Integral_{x=0..1} (sqrt(x) - 1)/log(x) dx. - Kritsada Moomuang, Jun 14 2025