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 A002391 Decimal expansion of natural logarithm of 3. (Formerly M4595 N1960) 39
 1, 0, 9, 8, 6, 1, 2, 2, 8, 8, 6, 6, 8, 1, 0, 9, 6, 9, 1, 3, 9, 5, 2, 4, 5, 2, 3, 6, 9, 2, 2, 5, 2, 5, 7, 0, 4, 6, 4, 7, 4, 9, 0, 5, 5, 7, 8, 2, 2, 7, 4, 9, 4, 5, 1, 7, 3, 4, 6, 9, 4, 3, 3, 3, 6, 3, 7, 4, 9, 4, 2, 9, 3, 2, 1, 8, 6, 0, 8, 9, 6, 6, 8, 7, 3, 6, 1, 5, 7, 5, 4, 8, 1, 3, 7, 3, 2, 0, 8, 8, 7, 8, 7, 9, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 REFERENCES W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. 2. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Harry J. Smith, Table of n, a(n) for n = 1..20000 D. H. Bailey, Compendium to BBP formulas. P. Bala, New series for old functions. G. Huvent, Formules BBP en base 3. - Jaume Oliver Lafont, Oct 12 2009 Simon Plouffe, Plouffe's Inverter, The natural logarithm of 3 to 10000 digits. Simon Plouffe, log(3), natural logarithm of 3 to 2000 places. S. Ramanujan, Notebook entry. 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. Eric Weisstein's World of Mathematics, BBP-Type Formula. FORMULA log(3) = Sum_{n>=1} (9*n-4)/((3*n-2)*(3*n-1)*3*n). [Jolley, Summation of Series, Dover (1961) eq 74] log(3) = 1/4*(1+ Sum((1/(9)^(k+1))*(27/(2*k+1) + 4/(2*k+2) + 1/(2*k+3)), k = 0 .. infinity) ) (a BBP-type formula). - Alexander R. Povolotsky, Dec 01 2008 log(3) = 4/5 +2/10*sum((1/4)^n*(1/(2*n+1)+1/(2*n+3)),n=0...infinity). - Alexander R. Povolotsky, Dec 18 2008 log(3) = Sum((1/9)^(k+1)(9/(2k+1)+1/(2k+2)),k=0..infinity). - Jaume Oliver Lafont, Dec 22 2008 Sum_{i>=1} 1/(9^i*i) + Sum_{i>=0} 1/(9^i*(i+1/2)) = 2*log(3) (Huvent 2001). - Jaume Oliver Lafont, Oct 12 2009 log(3) = Sum(k>=1, A191907(3,k)/k ) (conjecture). - Mats Granvik, Jun 19 2011 log(3) = Sum_{k=3^n..3^(n+1)-1} 1/k as n -> Infinity. Also see A002162. By analogy to the integral of 1/x, log(m) = Sum_{k=m^n..m^(n+1)-1} 1/k as n -> Infinity, for any value of m > 1. - Richard R. Forberg, Aug 16 2014 From Peter Bala, Feb 04 2015: (Start) log(3) = Sum {k >= 0} 1/((2*k + 1)*4^k). Define a pair of integer sequences A(n) = 4^n*(2*n + 1)!/n! and B(n) = A(n)*sum {k = 0..n} (1/((2*k + 1)*4^k). Both sequences satisfy the same second order recurrence equation u(n) = (20*n + 6)*u(n-1) - 16*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion log(3) = 1 + 2/(24 - 16*3^2/(46 - 16*5^2/(66 - ... - 16*(2*n - 1)^2/((20*n + 6) - ... )))). Cf. A002162, A073000 and A105531 for similar expansions. log(3) = 2 * Sum {k >= 1} (-1)^(k+1)*(4/3)^k/(k*binomial(2*k,k)). log(3) = (1/4) * Sum {k >= 1} (-1)^(k+1) (55*k - 23)*(8/9)^k/( 2*k*(2*k - 1)*binomial(3*k,k) ). log(3) = (1/4) * Sum {k >= 1} (7*k + 1)*(8/3)^k/( 2*k*(2*k - 1)*binomial(3*k,k) ). (End) log(3) = -lim_{n->Infinity} (n+1)th derivative of zeta(n) / n-th derivative of zeta(n). By n = 1000 there is convergence to 25 digits. A related expression: lim_{n->Infinity} n-th derivative of zeta(n-1) / n-th derivative of zeta(n) = 3. Also see A002581. - Richard R. Forberg, Feb 24 2015 From Peter Bala, Nov 02 2019: (Start) log(3) = 2*Integral_{x = 0..1} (1 - x^2)/(1 + x^2 + x^4) dx = 2*( 1 - (2/3) + 1/5 + 1/7 - (2/9) + 1/11 + 1/13 - (2/15) + ... ). log(3) = 16*Sum_{n >= 0} 1/( (6*n + 1)*(6*n + 3)*(6*n + 5) ). log(3) = 4/5 + 64*Sum_{n >= 0} (18*n + 1)/((6*n - 5)*(6*n - 3)*(6*n - 1)*(6*n + 1)*(6*n + 7)). (End) From Amiram Eldar, Jul 05 2020: (Start) Equals 2*arctanh(1/2). Equals Sum_{k>=1} (2/3)^k/k. Equals Integral_{x=0..Pi} sin(x)dx/(2 + cos(x)). (End) log(3) = Integral_{x = 0..1} (x^2 - 1)/log(x) dx. - Peter Bala, Nov 14 2020 EXAMPLE 1.098612288668109691395245236922525704647490557822749451734694333637494... MATHEMATICA RealDigits[Log[3], 10, 120][[1]]  (* Harvey P. Dale, Apr 23 2011 *) PROG (PARI) log(3) \\ Charles R Greathouse IV, Jan 24 2012 CROSSREFS Cf. A058962, A154920, A002162, A016731 (continued fraction), A073000, A105531, A254619. Sequence in context: A059068 A059069 A084660 * A193626 A316600 A087044 Adjacent sequences:  A002388 A002389 A002390 * A002392 A002393 A002394 KEYWORD nonn,cons AUTHOR EXTENSIONS Editing and more terms from Charles R Greathouse IV, Apr 20 2010 STATUS approved

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Last modified December 7 04:21 EST 2021. Contains 349567 sequences. (Running on oeis4.)