A019774 Decimal expansion of sqrt(e).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• cons

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

Examples

			1.6487212707001281468486507878141635716537761007101480115750...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Michael Penn, On means of binomial coefficients., YouTube video, 2020.
• Michael I. Shamos, A catalog of the real numbers, (2007). See p. 522.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000354, A001113, A058281 for continued fraction for sqrt(e), A019775.

Programs

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 *)

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

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

Equals i^(-i/Pi), where i denotes the imaginary unit. - Stefano Spezia, Mar 01 2025