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A016628
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Decimal expansion of log(5).
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22
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1, 6, 0, 9, 4, 3, 7, 9, 1, 2, 4, 3, 4, 1, 0, 0, 3, 7, 4, 6, 0, 0, 7, 5, 9, 3, 3, 3, 2, 2, 6, 1, 8, 7, 6, 3, 9, 5, 2, 5, 6, 0, 1, 3, 5, 4, 2, 6, 8, 5, 1, 7, 7, 2, 1, 9, 1, 2, 6, 4, 7, 8, 9, 1, 4, 7, 4, 1, 7, 8, 9, 8, 7, 7, 0, 7, 6, 5, 7, 7, 6, 4, 6, 3, 0, 1, 3, 3, 8, 7, 8, 0, 9, 3, 1, 7, 9, 6, 1
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OFFSET
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1,2
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REFERENCES
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Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
Horace S. Uhler, Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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log(5) = 2*sqrt(2)*Integral_{t = 0..sqrt(2)/2} (1 - t^2)/(1 + t^4) dt.
log(5) = Sum_{n >= 0} (4*n+5)/((4*n+1)*(4*n+3))*(-1/4)^n.
log(5) = (1/4)*Sum_{n >= 0} ( 8/(8*n+1) - 4/(8*n+3) - 2/(8*n+5) + 1/(8*n+7) )*(1/16)^n, a BBP-type formula. (End)
log(5) = 2*Sum_{n >= 0} (-1)^(n*(n+1)/2)*1/((2*n+1)*2^n). - Peter Bala, Oct 29 2020
log(5) = Integral_{x = 0..1} (x^4 - 1)/log(x) dx. - Peter Bala, Nov 14 2020
log(5) = 2*Sum_{n >= 1} 1/(n*P(n, 3/2)*P(n-1, 3/2)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(5) = 1.6094379124341003(29...), correct to 16 decimal places. - Peter Bala, Mar 18 2024
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EXAMPLE
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1.60943791243410037460075933322618763952560135426851772191264789... - Harry J. Smith, May 16 2009
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MATHEMATICA
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PROG
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(PARI) default(realprecision, 20080); x=log(5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016628.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009
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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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