A072334 Decimal expansion of e^2.

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, 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, 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

Offset

1,1

Comments

Also where x^(1/sqrt(x)) is a maximum. - Robert G. Wilson v, Oct 22 2014

References

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.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

John Cosgrave, e^2 is irrational.

Index entries for transcendental numbers.

Formula

Equals Sum_{n>=0} Sum_{k>=0} 1/(n!*k!). - Fredrik Johansson, Apr 21 2006

Equals Sum_{n>=0} 2^n/n!. - Daniel Hoyt Nov 20 2020

From Peter Bala, Jan 13 2022: (Start)

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

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

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

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

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

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)

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

Example

7.389056098930650...

Mathematica

RealDigits[E^2, 10, 100][[1]] (* Vincenzo Librandi, Apr 05 2020 *)

Prog

(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
(Magma) SetDefaultRealField(RealField(100)); Exp(1)^2; // Vincenzo Librandi, Apr 05 2020

Crossrefs

Cf. A001204 (continued fraction).

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

Sequence in context: A019726 A011330 A093587 * A368654 A358184 A361604

Adjacent sequences: A072331 A072332 A072333 * A072335 A072336 A072337

Keyword

nonn,cons

Author

N. J. A. Sloane, Jul 15 2002

Status

approved