A193535 Decimal expansion of log(2)/3.

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

Offset: 0

Author

Alonso del Arte, Jul 29 2011

Keywords

• nonn
• cons

Comments

This number is involved as an addend or subtrahend in the closed forms of certain series of reciprocals of integers (see for example A113476).

Examples

			0.231049060186648...
		

References

• L. B. W. Jolley, Summation of Series, Dover (1961).
• Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equations 21.16 and 21.18.

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A052502, A113476.

Programs

Mathematica

RealDigits[(Log[2]/3), 10, 100][[1]]

PARI

log(2)/3 \\ Charles R Greathouse IV, Jul 29 2011

Formula

Equals lim_{n->oo} [Sum_{i = 1..n} i^2/(n^3 + i^3)]. [Jolley eq 292, p.52]

Equals Sum_{n>=1} (-1)^(n-1)/(n*2^n*binomial(2*n, n)). - Arkadiusz Wesolowski, Jan 20 2013

From Amiram Eldar, Aug 05 2020: (Start)

Equals Integral_{x=1..oo} 1/(x^4 + x) dx.

Equals Integral_{x=0..oo} 1/(exp(2*x) + 3) dx. (End)

From Peter Bala, Feb 27 2024: (Start)

Equals (1/2)*Sum_{k >= 0} (-1)^k/((3*k + 1)*(3*k + 2)) = (1/2)*(1/(2 + (1*2)^2/(18 + (4*5)^2/(2*18 + (7*8)^2/(3*18 + (10*11)^2/(4*18 + ... )))))) (continued fraction). See A052502.

Equals 7/32 + (3/2)*Sum_{k >= 0} (-1)^k/((3*k + 1)*(3*k + 2)*(3*k + 3)*(3*k + 4)*(3*k + 5)). (End)