OFFSET
1,2
COMMENTS
The continued fraction (0 + K_{k=1}^{infinity}(-2/k)/(-2) = 1/(1-2/(2-2/(3-2/(4- ... converges, and its negative 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 function. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems for given x. This results in the 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 = 1..5000
FORMULA
(0 + K_{k=1}^{infinity}(-2/k)/(-2) = 1/(1-2/(2-2/(3-2/(4- ... =
(1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2))
= -1.4397493218702328058...
MATHEMATICA
RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)
PROG
(PARI) besselj(1, sqrt(8))/besselj(0, sqrt(8))/sqrt(2) \\ Charles R Greathouse IV, Feb 19 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Mar 23 2013
STATUS
approved