A072334 - OEIS

%I #53 Mar 26 2022 10:16:43

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

%T 8,1,3,1,8,0,3,1,5,5,7,0,5,5,1,8,4,7,3,2,4,0,8,7,1,2,7,8,2,2,5,2,2,5,

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

%N Decimal expansion of e^2.

%C Also where x^(1/sqrt(x)) is a maximum. - _Robert G. Wilson v_, Oct 22 2014

%D Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 1.4, pages 2 and 28-29.

%H Harry J. Smith, <a href="/A072334/b072334.txt">Table of n, a(n) for n = 1..20000</a>

%H John Cosgrave, <a href="https://www.johnbcosgrave.com/archive/esquared.htm">e^2 is irrational</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals Sum_{n>=0} Sum_{k>=0} 1/(n!*k!). - _Fredrik Johansson_, Apr 21 2006

%F Equals Sum_{n>=0} 2^n/n!. - _Daniel Hoyt_ Nov 20 2020

%F From _Peter Bala_, Jan 13 2022: (Start)

%F e^2 = Sum_{n >= 0} 2^n/n!. Faster converging series include

%F e^2 = 8*Sum_{n >= 0} 2^n/(p(n-1)*p(n)*n!), where p(n) = n^2 - n + 2 and

%F e^2 = -48*Sum_{n >= 0} 2^n/(q(n-1)*q(n)*n!), where q(n) = n^3 + 5*n - 2.

%F e^2 = 7 + Sum_{n >= 0} 2^(n+3)/((n+2)^2*(n+3)^2*n!) and

%F 7/e^2 = 1 - Sum_{n >= 0} (-2)^(n+1)*n^2/(n+2)!.

%F e^2 = 7 + 2/(5  + 1/(7 + 1/(9  + 1/(11 + ...)))) (follows from the fact that A004273 is the continued fraction expansion of tanh(1) = (e^2 - 1)/ (e^2 + 1)). Cf. A001204. (End)

%F Equals lim_{n->oo} (Sum_{k=1..n} 1/binomial(n,k)^x)^(n^x), for all real x > 1/2 (Furdui, 2013). - _Amiram Eldar_, Mar 26 2022

%e 7.389056098930650...

%t RealDigits[E^2, 10, 100][[1]] (* _Vincenzo Librandi_, Apr 05 2020 *)

%o (PARI) default(realprecision, 20080); x=exp(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b072334.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 30 2009

%o (Magma) SetDefaultRealField(RealField(100)); Exp(1)^2; // _Vincenzo Librandi_, Apr 05 2020

%Y Cf. A001204 (continued fraction).

%Y Cf. A001113, A091933, A092426, A092511, A092512, A092513.

%K nonn,cons

%O 1,1

%A _N. J. A. Sloane_, Jul 15 2002