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
Examples
7.389056098930650...
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.
Crossrefs
Programs
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Magma
SetDefaultRealField(RealField(100)); Exp(1)^2; // Vincenzo Librandi, Apr 05 2020
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Mathematica
RealDigits[E^2, 10, 100][[1]] (* Vincenzo Librandi, Apr 05 2020 *)
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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
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
Comments