|
Search: id:A002162
|
|
|
| A002162 |
|
Decimal expansion of natural logarithm of 2.
(Formerly M4074 N1689)
|
|
+0
19
|
|
| 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; cons; graph; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
ln(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)),n=1...infinity)) - from and by Alexander R. Povolotsky [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
ln(2) = 105*(sum(1/(2*n*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)),n=1...infinity) - 319/44100) ln(2) = (319/420 - 3/2*sum(1/(6*n^2+39*n+63),n=1...infinity)) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
i^2*ln(1/2) [From Eric Desbiaux (moongerms(AT)wanadoo.fr), Jun 27 2009]
|
|
REFERENCES
|
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.3.
D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.
Uhler, Horace S.; 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.
|
|
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
Paul Cooijmans, Odds.
X. Gourdon and P. Sebah, The logarithm constant:log(2)
S. Plouffe, log(2), natural logarithm of 2 to 2000 places
S. Ramanujan, Question 260, J. Ind. Math. Soc.
Eric Weisstein's World of Mathematics, Natural Logarithm of 2
Eric Weisstein's World of Mathematics, Masser-Gramain Constant
Eric Weisstein's World of Mathematics, Logarithmic Integral
|
|
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..inf, 2/[(4k)^3-4k]}) (Ramanujan).
log(2)=4*sum_{k=0..inf} [3-2*sqrt(2)]^(2k+1)/(2k+1) (Y. Luke) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 13 2006
log(2) = 1-(1/2)Sum_{ k >= 1 } 1/(k*(2k+1)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 06 2009, Jan 08 2009]
log(2) = 4*Sum_{ k >= 0 } 1/((4k+1)(4k+2)(4k+3)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 08 2009]
Equals 7/12+24*sum_{k=1..infinity} 1/(A052787(k+4)*A000079(k)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2009]
|
|
EXAMPLE
|
.6931471805599453...
0.693147180559945309417232121458176568075500134360255254120680009493393... [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
|
|
MAPLE
|
ln(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)), n=1...infinity)) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
|
|
PROGRAM
|
(PARI) { default(realprecision, 20080); x=10*log(2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b002162.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
|
|
CROSSREFS
|
Cf. A016730 Continued fraction. [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 16 2009]
Adjacent sequences: A002159 A002160 A002161 this_sequence A002163 A002164 A002165
Sequence in context: A129938 A022698 A013707 this_sequence A072365 A085138 A143735
|
|
KEYWORD
|
cons,nonn,new
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
|
|
|
Search completed in 0.003 seconds
|
