OFFSET
0,1
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).
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.
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)
EXAMPLE
0.231049060186648...
MATHEMATICA
RealDigits[(Log[2]/3), 10, 100][[1]]
PROG
(PARI) log(2)/3 \\ Charles R Greathouse IV, Jul 29 2011
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Alonso del Arte, Jul 29 2011
STATUS
approved