A229495 - OEIS

%I #32 Jul 08 2023 04:12:25

%S 1,0,8,4,4,3,7,5,5,1,4,1,9,2,2,7,5,4,6,6,1,1,5,7,7,3,1,3,4,2,2,9,4,7,

%T 9,8,5,8,3,9,5,9,6,9,3,1,9,6,4,7,2,6,2,6,8,2,2,5,1,3,4,3,4,7,1,2,2,8,

%U 7,5,1,4,7,9,6,2,6,9,0,0,2,4,9,9,0,3,4,7,1,6,8,2,8,8,4,8,4,7,5,3,1,5,2,3,6,6,7,9,3,9,1,9,7,3,4,9,3,6,4,3,5,3,4,7,6,8,3,8,1,5,4,1,3,1,9,5,6,3,3,6,6,3,3,4,2,9,5,1,9,7

%N Stirling's approximation constant e / sqrt(2*Pi).

%D Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 1.5, pages 2 and 27-28.

%H G. C. Greubel, <a href="/A229495/b229495.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation">Stirling's approximation</a>.

%F Equals exp(1)/sqrt(2*Pi).

%F Equals lim_{n->oo} (A001142(n)^(1/n)*sqrt(n)/(exp(n/2))) (Furdui, 2013). - _Amiram Eldar_, Mar 26 2022

%F Equals Product_{n>=1} (1 + 1/n)^(n+1/2)/e. - _Amiram Eldar_, Jul 08 2023

%e 1.0844375514192275466115773134229479858...

%p evalf(exp(1)/sqrt(2*Pi),120); # _Muniru A Asiru_, Oct 07 2018

%t RealDigits[E/Sqrt[2Pi],10,120][[1]] (* _Harvey P. Dale_, Jan 21 2017 *)

%o (PARI) exp(1)/sqrt(2*Pi) \\ _Ralf Stephan_, Sep 26 2013

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Exp(1)/Sqrt(2*Pi(R)); // _G. C. Greubel_, Oct 06 2018

%Y Cf. A001113 (e), A019727 (sqrt(2*Pi)), A001142.

%K nonn,cons

%O 1,3

%A _John W. Nicholson_, Sep 24 2013

%E More terms from _Ralf Stephan_, Sep 26 2013

%E Corrected and extended by _Harvey P. Dale_, Jan 21 2017