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
Examples
exp(3) = e^3 = 20.0855369231876677409285296545817178969879... - _Harry J. Smith_, Apr 30 2009
Links
Programs
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Maple
Digits:=100: evalf(exp(3)); # Wesley Ivan Hurt, Jul 07 2014
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Mathematica
RealDigits[E^3, 10, 100][[1]] (* Alonso del Arte, Jul 07 2014 *)
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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
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)
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