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A002391 Decimal expansion of natural logarithm of 3.
(Formerly M4595 N1960)
25
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 [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). 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]

ln(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: (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

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

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Last modified April 19 06:55 EDT 2015. Contains 256803 sequences.