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A002162 Decimal expansion of the natural logarithm of 2.
(Formerly M4074 N1689)
112
6, 9, 3, 1, 4, 7, 1, 8, 0, 5, 5, 9, 9, 4, 5, 3, 0, 9, 4, 1, 7, 2, 3, 2, 1, 2, 1, 4, 5, 8, 1, 7, 6, 5, 6, 8, 0, 7, 5, 5, 0, 0, 1, 3, 4, 3, 6, 0, 2, 5, 5, 2, 5, 4, 1, 2, 0, 6, 8, 0, 0, 0, 9, 4, 9, 3, 3, 9, 3, 6, 2, 1, 9, 6, 9, 6, 9, 4, 7, 1, 5, 6, 0, 5, 8, 6, 3, 3, 2, 6, 9, 9, 6, 4, 1, 8, 6, 8, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Newton calculated the first 16 terms of this sequence.

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, Sections 1.3.3 and 6.2.

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 = 0..20000

D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, May 2005, Volume 52, Issue 5.

P. Bala, New series for old functions

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

Paul Cooijmans, Odds.

X. Gourdon and P. Sebah, The logarithm constant:log(2)

I. Newton, The method of fluxions and infinite series; with its application to the geometry of curve-lines, 1736; see p. 96.

Simon Plouffe, log(2), natural logarithm of 2 to 2000 places

S. Ramanujan, Question 260, J. Ind. Math. Soc.

D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.

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, Natural Logarithm of 2, Masser-Gramain Constant, Logarithmic Integral

Wikipedia, Natural logarithm of 2.

FORMULA

log(2) = Sum_{k>=1} 1/(k*2^k) = Sum_{j>=1} (-1)^(j+1)/j.

log(2) = Integral_{t=0..1} dt/(1+t).

log(2) = 2/3 * (1 + sum(k>=1, 2/[(4k)^3-4k] )) (Ramanujan).

log(2) = 4*sum_{k>=0} [3-2*sqrt(2)]^(2k+1)/(2k+1) (Y. Luke). - R. J. Mathar, Jul 13 2006

log(2) = 1-(1/2)*sum_{k>=1} 1/(k*(2k+1)). - Jaume Oliver Lafont, Jan 06 2009, Jan 08 2009

log(2) = 4*sum_{k>=0} 1/((4k+1)(4k+2)(4k+3)). - Jaume Oliver Lafont, Jan 08 2009

log(2) = 7/12+24*sum_{k>=1} 1/(A052787(k+4)*A000079(k)). - R. J. Mathar, Jan 23 2009

From Alexander R. Povolotsky, Jul 04 2009: (Start)

log(2) = 1/4*(3 - sum(n>=1, 1/(n*(n+1)*(2*n+1)) )).

log(2) = (230166911/9240-sum(k>=1, (1/2)^k* (11/k+10/(k+1)+9/(k+2)+8/(k+3)+7/(k+4)+6/(k+5)-6/(k+7)-7/(k+8)-8/(k+9) -9/(k+10)-10/(k+11)) ))/35917. (End)

log(2) = A052882/A000670. - Mats Granvik, Aug 10 2009

From log(1-x-x^2) at x=1/2, log(2)=(1/2)*Sum_{k>=1}L(k)/(k*2^k), where L(n) is the n-th Lucas number (A000032). - Jaume Oliver Lafont, Oct 24 2009

log(2) = Sum_{k>=1} 1/(cos(k*Pi/3)*k*2^k) (Cf. A176900). - Jaume Oliver Lafont, Apr 29 2010

log(2) = (sum(n>=1, 1/(n^2*(n+1)^2*(2*n+1)) ) +11)/16. - Alexander R. Povolotsky, Jan 13 2011

log(2) = sum(n>=1, (2*n+1)/(sum(k=1..n, k^2 )^2)) +396)/576. - Alexander R. Povolotsky, Jan 14 2011

log(2) = 105*(sum(n>=1, 1/(2*n*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)) ) - 319/44100) log(2) = (319/420 - 3/2*sum(n>=1, 1/(6*n^2+39*n+63) )). - Alexander R. Povolotsky, Dec 16 2008

log(2) = sum(k>=1, A191907(2,k)/k ). - Mats Granvik, Jun 19 2011

log(2) = Integral_{x=0..oo} 1/(1 + e^x) dx. - Jean-Fran├žois Alcover, Mar 21 2013

log(2) = limit of zeta(s)*(1-1/2^(s-1)) as s -> 1. - Mats Granvik, Jun 18 2013

From Peter Bala, Dec 10 2013: (Start)

log(2) = 2*sum {n = 1..inf} 1/( n*A008288(n-1,n-1)*A008288(n,n) ), a result due to Burnside.

log(2) = 1/3*sum {n >= 0} (5*n+4)/( (3*n+1)*(3*n+2)*C(3*n,n) )*(1/2)^n = 1/12*sum {n >= 0} (28*n+17)/( (3*n+1)*(3*n+2)*C(3*n,n) )*(-1/4)^n.

log(2) = 3/16*sum {n >= 0} (14*n+11)/( (4*n+1)*(4*n+3)*C(4*n,2*n) )*(1/4)^n = 1/12*sum {n >= 0} (34*n+25)/( (4*n+1)*(4*n+3)*C(4*n,2*n) )*(-1/18)^n. For more series of this type see the Bala link.

See A142979 for series acceleration formulas for log(2) obtained from the Mercator series log(2) = sum {n >= 1} (-1)^(n+1)/n. See A142992 for series for log(2) related to the root lattice C_n. (End)

log(2) = sum(k=2^n..2^(n+1)-1, 1/k) as n -> Infinity. - Richard R. Forberg, Aug 16 2014

From Peter Bala, Feb 03: (Start)

log(2) = 2/3*Sum {k >= 0} 1/((2*k + 1)*9^k).

Define a pair of integer sequences A(n) = 9^n*(2*n + 1)!/n! and B(n) = A(n)*sum {k = 0..n} 1/((2*k + 1)*9^k). Both satisfy the same second order recurrence equation u(n) = (40*n + 16)*u(n-1) - 36*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion log(2) = 2/3*(1 + 2/(54 - 36*3^2/(96 - 36*5^2/(136 - ... - 36*(2*n - 1)^2/((40*n + 16) - ... ))))). Cf. A002391, A073000 and A105531 for similar expansions. (End)

log(2) = Sum_{n>=1} (Zeta(2*n)-1)/n. - Vaclav Kotesovec, Dec 11 2015

From Peter Bala, Oct 30 2016: (Start)

Asymptotic expansions:

for N even, log(2) - Sum_{k = 1..N/2} (-1)^(k-1)/k ~ (-1)^(N/2)*(1/N - 1/N^2 + 2/N^4 - 16/N^6 + 272/N^8 - ...), where the sequence of unsigned coefficients [1, 1, 2, 16, 272, ...] is A000182 with an extra initial term of 1. See Borwein et al., Theorem 1 (b);

for N odd, log(2) - Sum_{k = 1..(N-1)/2} (-1)^(k-1)/k ~ (-1)^((N-1)/2)*(1/N - 1/N^3 + 5/N^5 - 61/N^7 + 1385/N^9 - ...), by Borwein et al., Lemma 2 with f(x) := 1/(x + 1/2), h := 1/2 and then set x = (N - 1)/2, where the sequence of unsigned coefficients [1, 1, 5, 61, 1385, ...] is A000364. (End)

EXAMPLE

0.693147180559945309417232121458176568075500134360255254120680009493393...

MATHEMATICA

RealDigits[N[Log[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

PROG

(PARI) { default(realprecision, 20080); x=10*log(2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b002162.txt", n, " ", d)); } /* Harry J. Smith, Apr 21 2009 */

CROSSREFS

Cf. A016730 Continued fraction. A008288, A142979, A142992.

Sequence in context: A129938 A022698 A013707 * A257945 A271526 A072365

Adjacent sequences:  A002159 A002160 A002161 * A002163 A002164 A002165

KEYWORD

cons,nonn

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

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Last modified December 9 06:57 EST 2016. Contains 278963 sequences.