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

Offset: 0

Author

Wolfdieter Lang, Mar 07 2013

Keywords

• nonn
• cons

Comments

The continued fraction (0 + K_{k=1..oo} (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..oo} (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..oo} (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).

Examples

			0.5631786198117113854257529037035635...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

A052119 (x=1), 2*A221913/A084950.

Programs

Mathematica

RealDigits[BesselI[1, 2*Sqrt[2]]/(Sqrt[2]*BesselI[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)

PARI

default(realprecision, 120);
sqrt(2)*besseli(1,2*sqrt(2))/(2*besseli(0,2*sqrt(2))) \\ Rick L. Shepherd, Jan 18 2014

Formula

(0 + K_{k=1..oo} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+ ... =

sqrt(2)*BesselI(1,2*sqrt(2))/(2*BesselI(0,2*sqrt(2)))

Extensions

Offset corrected and terms added by Rick L. Shepherd, Jan 18 2014