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A016655
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Decimal expansion of log(32) = 5*log(2).
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17
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3, 4, 6, 5, 7, 3, 5, 9, 0, 2, 7, 9, 9, 7, 2, 6, 5, 4, 7, 0, 8, 6, 1, 6, 0, 6, 0, 7, 2, 9, 0, 8, 8, 2, 8, 4, 0, 3, 7, 7, 5, 0, 0, 6, 7, 1, 8, 0, 1, 2, 7, 6, 2, 7, 0, 6, 0, 3, 4, 0, 0, 0, 4, 7, 4, 6, 6, 9, 6, 8, 1, 0, 9, 8, 4, 8, 4, 7, 3, 5, 7, 8, 0, 2, 9, 3, 1, 6, 6, 3, 4, 9, 8, 2, 0, 9, 3, 4, 3
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OFFSET
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1,1
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COMMENTS
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The function exp(x) has its maximum curvature where x = -(1/10)*5*log(2) = -log(2)/2 = 0.34657... - Dimitri Papadopoulos, Oct 27 2022
This maximum curvature occurs at the point with coordinates (x_M = -log(2)/2 = -(this constant)/10; y_M = sqrt(2)/2 = A010503) and is equal to 2*sqrt(3)/9 = A212886. - Bernard Schott, Dec 23 2022
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
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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(2)/2 = (1 - 1/2 - 1/4) + (1/3 - 1/6 - 1/8) + (1/5 - 1/10 - 1/12) + ... [Jolley, Summation of Series, Dover (1961) eq (73)]
Equals 10*log(2)/2 = 5*log(2) = 5*A002162, so 10*(1/2 - 1/4 + 1/6 - 1/8 + 1/10 - ... + (-1)^(k+1)/(2*k) + ...) = log(32). - Eric Desbiaux, Nov 26 2008
-log(2)/2 = lim_{n->oo} ((Sum_{k=2..n} arctanh(1/k)) - log(n)). - Jean-François Alcover, Aug 07 2014, after Steven Finch
Equals (5/4)*Sum_{k=1..4} (-1)^(k+1) gamma(0, k/4) where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018
log(2)/2 = arctanh(1/3) = arcsinh(1/sqrt(8)).
log(2)/2 = Integral_{x=0..Pi/4} tan(x) dx.
log(2)/2 = Sum_{k>=0} (-1)^k/(2*k+2).
log(2)/2 = Sum_{k>=1} 1/A060851(k). (End)
log(2)/2 = Sum_{k>=1} (-1)^(k+1) * arctanh(Lucas(2*k+3)/Fibonacci(2*k+3)^2) (Melham and Shannon, 1995). - Amiram Eldar, Jan 15 2022
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EXAMPLE
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3.465735902799726547086160607290882840377500671801276270603400047466968...
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MATHEMATICA
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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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