A222466 - OEIS

A222466

Decimal expansion of the limit of the continued fraction 1/(1+2/(2+2/(3+2/(4+... in terms of Bessel functions.

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

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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)).

For x=1 see for the limit of the continued fraction A052119 and for the n-th approximation A001053(n+1)/A001040(n+1).

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);