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A222471
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Decimal expansion of the negative of the limit of the continued fraction 1/(1-2/(2-2/(3-2/(4-... in terms of Bessel functions.
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3
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1, 4, 3, 9, 7, 4, 9, 3, 2, 1, 8, 7, 0, 2, 3, 2, 8, 0, 5, 8, 9, 5, 7, 0, 6, 9, 5, 7, 4, 1, 1, 2, 2, 7, 4, 2, 5, 1, 5, 2, 7, 3, 3, 7, 6, 2, 2, 3, 8, 1, 1, 6, 1, 7, 5, 2, 8, 1, 4, 5, 3, 0, 7, 8, 8, 7, 7, 2, 3, 6, 1, 6, 8, 1, 6, 4, 3, 4, 5, 9, 6, 3, 8, 5, 0, 1, 9, 5, 1, 3, 1, 8, 5, 9, 7, 7, 0, 4, 8, 7, 6, 3, 4, 1, 7, 8, 7, 4, 0, 2
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OFFSET
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1,2
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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 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).
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- ... =
(1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2))
= -1.4397493218702328058...
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MATHEMATICA
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RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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