

A002391


Decimal expansion of natural logarithm of 3.
(Formerly M4595 N1960)


26



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
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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 [From 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). 205212.
Eric Weisstein's World of Mathematics, BBPType Formula


FORMULA

log(3) = sum_{n>=1} (9*n4)/((3*n2)*(3*n1)*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 BBPtype 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: (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(n1)  16*(2*n  1)^2*u(n2). 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) / nth derivative of zeta(n). By n = 1000 there is convergence to 25 digits. A related expression: lim_{n>Infinity} nth derivative of zeta(n1) / nth derivative of zeta(n) = 3. Also see A002581.  Richard R. Forberg, Feb 24 2015


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 A087044 A246168
Adjacent sequences: A002388 A002389 A002390 * A002392 A002393 A002394


KEYWORD

nonn,cons


AUTHOR

N. J. A. Sloane


EXTENSIONS

Editing and more terms from Charles R Greathouse IV, Apr 20 2010


STATUS

approved



