OFFSET
0,1
COMMENTS
This constant is a transcendental number (Adhikari et al., 2001).
A similar series is Sum_{k>=1} F(k)/2^k = 2.
The corresponding series with Lucas numbers (A000032) is Sum_{k>=1} L(k)/(k*2^k) = 2*log(2) (A016627).
In general, for m>=2, Sum_{k>=1} F(k)/(k*m^k) = log(1 - 2*sqrt(5)/(1 + sqrt(5) - 2*m)) / sqrt(5) and Sum_{k>=1} L(k)/(k*m^k) = log(m^2 / (m^2 - m - 1)). - Vaclav Kotesovec, May 08 2021
LINKS
S. D. Adhikari, N. Saradha, T. N. Shorey and R. Tijdeman, Transcendental infinite sums, Indagationes Mathematicae, Vol. 12, No. 1 (2001), pp. 1-14.
István Mező, Several Generating Functions for Second-Order Recurrence Sequences, Journal of Integer Sequences, Vol. 12 (2009), Article 09.3.7.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.
FORMULA
EXAMPLE
0.86081788192800807777886646590121085084914136508057...
MATHEMATICA
RealDigits[Sum[Fibonacci[n]/n/2^n, {n, 1, Infinity}], 10, 100][[1]]
PROG
(PARI) suminf(k=1, fibonacci(k)/(k*2^k)) \\ Michel Marcus, May 07 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 07 2021
STATUS
approved