OFFSET
0,1
COMMENTS
The continued fraction (0 + K_{k=1}^{infinity}(2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... converges, and its limit is given in the formula section in terms of Bessel functions.
In general, the continued fraction 0 + K_{k=1}^{infinity}(x/k) = x/(1+x/(2+x/(3+... has n-th approximation x*Phat(n,x)/ Q(n,x), with the row polynomials Phat of A221913 and Q of A084950. These polynomials are written in terms of Bessel functions. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems. This results in the given formula 0 + K_{k=1}^{infinity}(x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
FORMULA
(0 + K_{k=1}^{infinity}(2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... =
sqrt(2)*BesselI(1,2*sqrt(2))/(2*BesselI(0,2*sqrt(2)))
= 0.5631786198117113854257529037035635...
MATHEMATICA
RealDigits[BesselI[1, 2*Sqrt[2]]/(Sqrt[2]*BesselI[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)
PROG
(PARI)
default(realprecision, 120);
sqrt(2)*besseli(1, 2*sqrt(2))/(2*besseli(0, 2*sqrt(2))) \\ Rick L. Shepherd, Jan 18 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Mar 07 2013
EXTENSIONS
Offset corrected and terms added by Rick L. Shepherd, Jan 18 2014
STATUS
approved