|
%I #75 Mar 26 2022 10:16:48
%S 1,4,4,4,6,6,7,8,6,1,0,0,9,7,6,6,1,3,3,6,5,8,3,3,9,1,0,8,5,9,6,4,3,0,
%T 2,2,3,0,5,8,5,9,5,4,5,3,2,4,2,2,5,3,1,6,5,8,2,0,5,2,2,6,6,4,3,0,3,8,
%U 5,4,9,3,7,7,1,8,6,1,4,5,0,5,5,7,3,5,8,2,9,2,3,0,4,7,0,9,8,8,5,1,1,4,2,9,5
%N Decimal expansion of e^(1/e).
%C e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).
%C Let w(n+1)=A^w(n); then w(n) converges if and only if (1/e)^e <= A <= e^(1/e) (see the comments in A073230) for initial value w(1)=A. If A=e^(1/e) then lim_{n->infinity} w(n) = e. - _Benoit Cloitre_, Aug 06 2002; corrected by _Robert FERREOL_, Jun 12 2015
%C x^(1/x) is maximum for x = e and the maximum value is e^(1/e). This gives an interesting and direct proof that 2 < e < 4 as 2^(1/2) < e^(1/e) > 4^(1/4) while 2^(1/2) = 4^(1/4). - _Amarnath Murthy_, Nov 26 2002
%C For large n, A234604(n)/A234604(n-1) converges to e^(1/e). - _Richard R. Forberg_, Dec 28 2013
%C Value of the unique base b > 0 for which the exponential curve y=b^x and its inverse y=log_b(x) kiss each other; the kissing point is (e,e). - _Stanislav Sykora_, May 25 2015
%C Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e). - _Yuval Paz_, Dec 29 2018
%C The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (1796-1863) in 1850. - _Amiram Eldar_, Jun 17 2021
%H Alois P. Heinz, <a href="/A073229/b073229.txt">Table of n, a(n) for n = 1..10000</a>
%H Ovidiu Furdui, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.124.5.465">Problem 11982</a>, The American Mathematical Monthly, Vol. 124, No. 5 (2017), p. 465; <a href="https://doi.org/10.1080/00029890.2019.1547601">A Limit of a Power of a Sum</a>, Solution to Problem 11982 by Roberto Tauraso, ibid., Vol. 126, No. 2 (2019), p. 187.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap19.html">exp(1/e)</a>.
%H Jonathan Sondow and Diego Marques, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from151to164.pdf">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae, Vol. 37 (2010), pp. 151-164; see Definition 4.1 on p. 158.
%H Jacob Steiner, <a href="https://doi.org/10.1515/crll.1850.40.208">Über das größte Product der Theile oder Summanden jeder Zahl</a>, Crelle, Vol. 40 (1850), pp. 208; <a href="https://eudml.org/doc/147456">alternative link</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SteinersProblem.html">Steiner's Problem</a>.
%F Equals 1 + Integral_{x = 1/e..1} (1 + log(x))/x^x dx = 1 - Integral_{x = 0..1/e} (1 + log(x))/x^x dx. - _Peter Bala_, Oct 30 2019
%F Equals Sum_{k>=0} exp(-k)/k!. - _Amiram Eldar_, Aug 13 2020
%F Equals lim_{x->oo} (Sum_{n>=1} (x/n)^n)^(1/x) (Furdui, 2017). - _Amiram Eldar_, Mar 26 2022
%e 1.44466786100976613365833910859...
%p evalf[110](exp(exp(-1))); # _Muniru A Asiru_, Dec 29 2018
%t RealDigits[ E^(1/E), 10, 110] [[1]]
%o (PARI) exp(1)^exp(-1)
%Y Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).
%Y Cf. A093157, A103476.
%Y Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.
%K cons,nonn
%O 1,2
%A _Rick L. Shepherd_, Jul 22 2002
|
