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A222466
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Decimal expansion of the limit of the continued fraction 1/(1+2/(2+2/(3+2/(4+... in terms of Bessel functions.
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4
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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
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0,1
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COMMENTS
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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).
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LINKS
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FORMULA
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(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...
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MATHEMATICA
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RealDigits[BesselI[1, 2*Sqrt[2]]/(Sqrt[2]*BesselI[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)
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PROG
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(PARI)
default(realprecision, 120);
sqrt(2)*besseli(1, 2*sqrt(2))/(2*besseli(0, 2*sqrt(2))) \\ Rick L. Shepherd, Jan 18 2014
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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