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Natural Logarithm of 2
The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being
converted to decay constants. 
has numerical value
 |
(1)
|
(OEIS A002162).
The irrationality measure of 
is known to be less than 3.8913998 (Rukhadze 1987, Hata
1990).
It is not known if 
is normal (Bailey and Crandall 2002).
The alternating series and BBP-type formula
 |
(2)
|
converges to the natural logarithm of 2, where 
is the Dirichlet
eta function. This identity follows immediately from setting 
in the Mercator series,
yielding
 |
(3)
|
It is also a special case of the identity
 |
(4)
|
where 
is the Lerch transcendent.
This is the simplest in an infinite class of such identities, the first few of which are
 |  |  |
(5)
|
 |  |  |
(6)
|
(E. W. Weisstein, Oct. 7, 2007).
There are many other classes of BBP-type formulas for 
,
including
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
Plouffe (2006) found the beautiful sum
 |
(12)
|
A rapidly converging Zeilberger-type sum due to A. Lupas is given by
 |
(13)
|
(Lupas 2000; typos corrected).
The following integral is given in terms of 
,
 |
(14)
|
The plot above shows the result of truncating the series for 
after 
terms.
Taking the partial series gives the analytic result
 |  | ![ln2+1/2(-1)^N[psi_0(1/2(N+1))-psi_0(1+1/2N)]](https://mathworld.wolfram.com/images/equations/NaturalLogarithmof2/Inline34.svg) |
(15)
|
 |  | ![ln2+1/2(-1)^N[H_((N-1)/2)-H_(N/2)],](https://mathworld.wolfram.com/images/equations/NaturalLogarithmof2/Inline37.svg) |
(16)
|
where 
is the digamma function and 
is a harmonic number.
Rather amazingly, expanding about infinity gives the series
![sum_(k=1)^N((-1)^(k+1))/k=ln2+(-1)^N[1/(2N)+sum_(k=0)^infty((-1)^kT_k)/(4^kN^(2k))]](https://mathworld.wolfram.com/images/equations/NaturalLogarithmof2/NumberedEquation8.svg) |
(17)
|
(Borwein and Bailey 2002, p. 50), where 
is a tangent number. This
means that truncating the series for 
at half a large power of 10 can give a decimal expansion
for 
whose decimal digits are largely correct, but where wrong digits occur with precise
regularity.
For example, taking 
gives a decimal value equal to the second row
of digits above, where the sequence of differences from the decimal digits of 
in the top row is precisely the tangent numbers with alternating signs (Borwein and
Bailey 2002, p. 49).
Beautiful BBP-type formulas for 
are given by
 |  |  |
(18)
|
 |  |  |
(19)
|
(Bailey et al. 2007, p. 31) and
 |
(20)
|
(Borwein and Bailey 2002, p. 129).
A BBP-type formula for 
discovered using the PSLQ
algorithm is
![(ln2)^2=1/(32)sum_(k=0)^infty1/(64^k)[(64)/((6k+1)^2)-(160)/((6k+2)^2)-(56)/((6k+3)^2)-(40)/((6k+4)^2)+4/((6k+5)^2)-1/((6k+6)^2)]](https://mathworld.wolfram.com/images/equations/NaturalLogarithmof2/NumberedEquation10.svg) |
(21)
|
(Bailey and Plouffe 1997; Borwein and Bailey 2002, p. 128).
The sum
 |
(22)
|
has the limit
 |
(23)
|
(Borwein et al. 2004, p. 10).
See also
Alternating Harmonic Series, Dirichlet Eta Function, Mercator Series, Natural Logarithm, Natural Logarithm of 2 Continued Fraction, Natural Logarithm of 2 Digits, q-Harmonic SeriesExplore with Wolfram|Alpha
References
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Integer Relation Detection." §2.2 in Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 29-31, 2007.Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." Organic Mathematics. Proceedings of the Workshop Held in Burnaby, BC, December 12-14, 1995 (Ed. J. Borwein, P. Borwein, L. Jörgenson, and R. Corless). Providence, RI: Amer. Math. Soc., pp. 73-88, 1997. https://www.cecm.sfu.ca/organics/papers/bailey/.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Gourdon, X. and Sebah, P. "The Constant
." http://numbers.computation.free.fr/Constants/Log2/log2.html.Hata,
M. "Legendre Type Polynomials and Irrationality Measures." J. reine
angew. Math. 407, 99-125, 1990.Huylebrouck, D. "Similarities
in Irrationality Proofs for 
, 
, 
, and 
." Amer. Math. Monthly 108, 222-231,
2001.Lupas, A. "Formulae for Some Classical Constants." In
Proceedings of ROGER-2000. 2000. http://www.lacim.uqam.ca/~plouffe/articles/alupas1.pdf.Plouffe,
S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006.
http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.Rukhadze,
E. A. "A Lower Bound for the Rational Approximation of 
by Rational Numbers." Vestnik Moskov Univ. Ser.
I Math. Mekh., No. 6, 25-29 and 97, 1987. [Russian].Sloane,
N. J. A. Sequences A002162/M4074,
A016730, and A059180
in "The On-Line Encyclopedia of Integer Sequences."Sweeney,
D. W. "On the Computation of Euler's Constant." Math. Comput. 17,
170-178, 1963.Uhler, H. 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, 205-212, 1940.Referenced on Wolfram|Alpha
Natural Logarithm of 2Cite this as:
Weisstein, Eric W. "Natural Logarithm of 2." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NaturalLogarithmof2.html