

A019727


Decimal expansion of sqrt(2*Pi).


40



2, 5, 0, 6, 6, 2, 8, 2, 7, 4, 6, 3, 1, 0, 0, 0, 5, 0, 2, 4, 1, 5, 7, 6, 5, 2, 8, 4, 8, 1, 1, 0, 4, 5, 2, 5, 3, 0, 0, 6, 9, 8, 6, 7, 4, 0, 6, 0, 9, 9, 3, 8, 3, 1, 6, 6, 2, 9, 9, 2, 3, 5, 7, 6, 3, 4, 2, 2, 9, 3, 6, 5, 4, 6, 0, 7, 8, 4, 1, 9, 7, 4, 9, 4, 6, 5, 9, 5, 8, 3, 8, 3, 7, 8, 0, 5, 7, 2, 6
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OFFSET

1,1


COMMENTS

Pickover says that the expression: lim_{n>oo} e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits.  Jason Earls, Mar 16 2001
sqrt(2*Pi)*sqrt(n) is the expected height of a labeled random tree of order n (see Rényi, Szekeres, 1967, formula (4.6)).  Hugo Pfoertner, May 18 2023
The constant in the formula known as "Stirling's approximation" (or "Stirling's formula"). It is sometimes called Stirling constant. The formula without the exact value of the constant was discovered by the French mathematician Abraham de Moivre (16671754), and was published in his book (1730). The exact value of the constant was found by the Scottish mathematician James Stirling (16921770) and was published in his book "Methodus differentialis" (1730).  Amiram Eldar, Jul 08 2023


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 GlaisherKinkelin constant, p. 137.
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.


LINKS

Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 6264.
A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497507.


FORMULA

Equals lim_{n>oo} e^n*(n!)/n^n*sqrt(n).
Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog.  JeanFrançois Alcover, May 27 2013
Also equals the generalized GlaisherKinkelin constant A_0, see the Finch reference.  JeanFrançois Alcover, Dec 23 2014
Equals exp(zeta'(0)). See Kimoto et al.  Michel Marcus, Jun 27 2019


EXAMPLE

2.506628274631000502415765284811045253006986740609938316629923576342293....


MATHEMATICA



PROG

(PARI) default(realprecision, 20080); x=sqrt(2*Pi); for (n=1, 20000, d=floor(x); x=(xd)*10; write("b019727.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009
(Maxima) fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* Martin Ettl, Oct 11 2012 */
(Magma) R:= RealField(100); Sqrt(2*Pi(R)); // G. C. Greubel, Mar 08 2018


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



