OFFSET
1,2
COMMENTS
Also where x^(x^(-2)) is a maximum. - Robert G. Wilson v, Oct 22 2014
e^(1/2) maximizes the value of x^(c/(x^2)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..20000
Michael Penn, On means of binomial coefficients., YouTube video, 2020.
FORMULA
sqrt(e) = Sum_{n>=0} 1/(2^n*n!) = Sum_{n>=0} 1/(2n)!!. - Daniel Forgues, Apr 17 2011
sqrt(e) = 1 + Sum_{n>0} Product_{i=1..n} 1/(2n). - Ralf Stephan, Sep 11 2013
Continued fraction representation: sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). See A000354 for details. - Peter Bala, Jan 30 2015
sqrt(e) = (1/2)*( 1 + (3 + (5 + (7 + ...)/6)/4)/2 ) = 1 + (1 + (1 + (1 + ...)/6)/4)/2. - Rok Cestnik, Jan 19 2017
sqrt(e) = 2*Sum_{n >= 0} 1/((1 - 4*n^2)*(2^n)*n!). - Peter Bala, Jan 16 2022
sqrt(e) = (16/31)*(1 + Sum_{n>=1} (1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!). - Alexander R. Povolotsky, Jul 01 2022
sqrt(e) = Sum_{n >= 0} (n + 1/2)/(2^n*n!). - Peter Bala, Jun 29 2024
EXAMPLE
1.6487212707001281468486507878141635716537761007101480115750...
MAPLE
evalf(sqrt(exp(1)), 120); # Muniru A Asiru, Aug 16 2018
MATHEMATICA
RealDigits[N[Sqrt[E], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)
PROG
(PARI) default(realprecision, 20080); x=sqrt(exp(1)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019774.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved