A091933 Decimal expansion of e^3.

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

Offset

2,1

Comments

Also where x^(1/x^(1/3)) is a maximum. - Robert G. Wilson v, Oct 22 2014

Links

Harry J. Smith, Table of n, a(n) for n = 2..20000

Index entries for transcendental numbers

Formula

From Peter Bala, Jan 12 2022: (Start)

e^3 = Sum_{n >= 0} 3^n/n!. Faster converging series include

e^3 = 18*Sum_{n >= 0} 3^n/(p(n-1)*p(n)*n!), where p(n) = n^2 - 3*n + 5 and

e^3 = -162*Sum_{n >= 0} 3^n/(q(n-2)*q(n-1)*n!), where q(n) = n^3 + 8*n - 3.

e^3 = 22 - Sum_{n >= 0} 3^(n+4)/((n+3)^2*(n+4)^2*n!) and

22/e^3 = 1 - 2*Sum_{n >= 0} (-3)^(n+2)*n^2/(n+3)!.

e^3 = lim_{n -> oo} f(n+2)*f(n)/(n^2*f(n+1)^2), where f(n) = n^(n^2). Compare with e = lim_{n -> oo} g(n+1)/(n*g(n)), where g(n) = n^n. (End)

Example

exp(3) = e^3 = 20.0855369231876677409285296545817178969879... - Harry J. Smith, Apr 30 2009

Maple

Digits:=100: evalf(exp(3)); # Wesley Ivan Hurt, Jul 07 2014

Mathematica

RealDigits[E^3, 10, 100][[1]] (* Alonso del Arte, Jul 07 2014 *)

Prog

(PARI) default(realprecision, 20080); x=exp(3)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b091933.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

Crossrefs

Cf. A001113, A072334, A092426, A092511, A092512, A092513.

Sequence in context: A334411 A028698 A013667 * A058347 A058547 A230910

Adjacent sequences: A091930 A091931 A091932 * A091934 A091935 A091936

Keyword

easy,nonn,cons

Author

Mohammad K. Azarian, Mar 16 2004

Status

approved