OFFSET
1,1
COMMENTS
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
REFERENCES
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.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..20000
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].
O. Espinosa, V. H. Moll, On some integrals involving the Hurwitz Zeta function: Part 1, Raman. J. 6 (2002) 159-188, eq. (5.7)
R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), chap 7.1
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021.
FORMULA
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 log(sqrt(2)) with offset 0. - Michel Marcus, Feb 19 2017
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
From Amiram Eldar, Jun 29 2020: (Start)
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
Equals 10 * Integral_{1..oo} dx/(x*(1+x^2)). [Nahin] - R. J. Mathar, May 22 2024
Equals -10*Integral_{q=0..1} q*log(sin(Pi*q))dq. [Espinosa] - R. J. Mathar, Aug 13 2024
log(2)/2 = Sum_{k>=2} (-1)^(k) * arccoth(k). - Antonio Graciá Llorente, Sep 16 2024
EXAMPLE
3.465735902799726547086160607290882840377500671801276270603400047466968...
MATHEMATICA
RealDigits[5 N [Log[2], 100]] [[1]] (* Vincenzo Librandi, Jan 02 2016 *)
PROG
(PARI) log(32) \\ Charles R Greathouse IV, Jan 24 2012
(Magma) [5*Log(2)]; // Vincenzo Librandi, Jan 02 2016
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved