%I M1727 N0684 #293 Aug 30 2024 11:11:35

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

%T 4,9,7,7,5,7,2,4,7,0,9,3,6,9,9,9,5,9,5,7,4,9,6,6,9,6,7,6,2,7,7,2,4,0,

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

%N Decimal expansion of e.

%C e is sometimes called Euler's number or Napier's constant.

%C Also, decimal expansion of sinh(1)+cosh(1). - _Mohammad K. Azarian_, Aug 15 2006

%C If m and n are any integers with n > 1, then |e - m/n| > 1/(S(n)+1)!, where S(n) = A002034(n) is the smallest number such that n divides S(n)!. - _Jonathan Sondow_, Sep 04 2006

%C Limit_{n->infinity} A000166(n)*e - A000142(n) = 0. - _Seiichi Kirikami_, Oct 12 2011

%C Euler's constant (also known as Euler-Mascheroni constant) is gamma = 0.57721... and Euler's number is e = 2.71828... . - _Mohammad K. Azarian_, Dec 29 2011

%C One of the many continued fraction expressions for e is 2+2/(2+3/(3+4/(4+5/(5+6/(6+ ... from Ramanujan (1887-1920). - _Robert G. Wilson v_, Jul 16 2012

%C e maximizes the value of x^(c/x) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. This explains why elements of A000792 are composed primarily of factors of 3, and where needed, some factors of 2. These are the two primes closest to e. - _Richard R. Forberg_, Oct 19 2014

%C There are two real solutions x to c^x = x^c when c, x > 0 and c != e, one of which is x = c, and only one real solution when c = e, where the solution is x = e. - _Richard R. Forberg_, Oct 22 2014

%C This is the expected value of the number of real numbers that are independently and uniformly chosen at random from the interval (0, 1) until their sum exceeds 1 (Bush, 1961). - _Amiram Eldar_, Jul 21 2020

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.

%D E. Maor, e: The Story of a Number, Princeton Univ. Press, 1994.

%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 52.

%D G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places. Math. Tables and Other Aids to Computation 4, (1950). 11-15.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

%H N. J. A. Sloane, <a href="/A001113/b001113.txt">Table of 50000 digits of e labeled from 1 to 50000</a> [based on the ICON Project link below]

%H Mohammad K. Azarian, <a href="http://www.fq.math.ca/Scanned/32-2/elementary32-2.pdf">An Expansion of e, Problem # B-765</a>, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. <a href="http://www.fq.math.ca/Scanned/33-4/elementary33-4.pdf">Solution</a> appeared in Vol. 33, No. 4, Aug. 1995, p. 377.

%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/21-24-2012/azarianIJCMS21-24-2012.pdf">Euler's Number Via Difference Equations</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.

%H L. E. Bush, <a href="http://www.jstor.org/stable/2311358">The William Lowell Putnam Mathematical Competition</a>, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 18-33, problem 3.

%H Ed Copeland and Brady Haran, <a href="https://www.youtube.com/watch?v=xOXsDfMMTjs">A proof that e is irrational</a>, Numberphile video (2021).

%H Dave's Math Tables, <a href="http://www.math2.org/math/constants/e.htm">e</a>

%H X. Gourdon, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/expof1.txt">e to 1.250 billion digits</a>

%H X. Gourdon and P. Sebah, <a href="http://numbers.computation.free.fr/Constants/E/e.html">The constant e and its computation</a>

%H ICON Project, <a href="http://www.cs.arizona.edu/icon/oddsends/e.htm">e to 50000 places</a>

%H Roger Mansuy, <a href="https://images.math.cnrs.fr/Un-intrigant-poeme-mathematique.html">Un intrigant poème... mathématique</a>, Images des Mathématiques, CNRS, 2023. In French.

%H R. Nemiroff and J. Bonnell, <a href="http://antwrp.gsfc.nasa.gov/htmltest/gifcity/e.5mil">The first 5 million digits of the number e</a>

%H Remco Niemeijer, <a href="http://programmingpraxis.com/2012/06/19/digits-of-e/">Digits Of E, programmingpraxis</a>

%H J. J. O'Connor & E. F. Robertson, <a href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/e.html">The number e</a>

%H Michael Penn, <a href="https://www.youtube.com/watch?v=mOBv_t4GuxI">e is irrational</a>, YouTube video, 2020.

%H Simon Plouffe, <a href="http://www.cecm.sfu.ca/projects/ISC/records.html">A million digits</a>

%H G. W. Reitwiesner, <a href="https://doi.org/10.1007/978-1-4757-3240-5_34">An ENIAC determination of pi and e to more than 2000 decimal places</a>, Pi, A Source book, pp 277-281, 2000.

%H E. Sandifer, How Euler Did It, <a href="https://www.maa.org/sites/default/files/pdf/editorial/euler/How%20Euler%20Did%20It%2028%20e%20is%20irrational.pdf">Who proved e is irrational?</a>, MAA Online (2006)

%H D. Shanks and J. W. Wrench, Jr., <a href="http://dx.doi.org/10.1090/S0025-5718-69-99858-5">Calculation of e to 100,000 decimals</a>, Math. Comp., 23 (1969), 679-680.

%H Jean-Louis Sigrist, <a href="http://jlsigrist.com/e.html">Le premier million de décimales de e</a>

%H J. Sondow, <a href="http://arxiv.org/abs/0704.1282"> A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly, 113 (2006), 637-641 (article) and 114 (2007), 659 (addendum).

%H J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II</a>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

%H G. Villemin's Almanach of Numbers, <a href="http://translate.google.com/translate?hl=en&amp;sl=fr&amp;u=http://villemin.gerard.free.fr/Wwwgvmm/Analyse/ExpoVal.htm">Constant "e"</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e.html">e</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/eDigits.html">e Digits</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UniformSumDistribution.html">Uniform Sum Distribution</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/eApproximations.html">e Approximations</a>

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/E_%28mathematical_constant%29">E (mathematical constant)</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x).

%F e is the unique positive root of the equation Integral_{u = 1..x} du/u = 1.

%F exp(1) = ((16/31)*(1 + Sum_{n>=1} ((1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!)))^2. _Robert Israel_ confirmed that the above formula is correct, saying: "In fact, Sum_{n=0..oo} n^j*t^n/n! = P_j(t)*exp(t) where P_0(t) = 1 and for j >= 1, P_j(t) = t (P_(j-1)'(t) + P_(j-1)(t)). Your sum is 1/2*P_3(1/2) + 1/2*P_1(1/2) + P_0(1/2)." - _Alexander R. Povolotsky_, Jan 04 2009

%F exp(1) = (1 + Sum_{n>=1} ((1+n+n^3)/n!))/7. - _Alexander R. Povolotsky_, Sep 14 2011

%F e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2. - _Rok Cestnik_, Jan 19 2017

%F From _Peter Bala_, Nov 13 2019: (Start)

%F The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,..., where R(n,x) is the n-th row polynomial of A269953.

%F e = 2 + Sum_{n >= 0} (-1)^n*(n+2)!/(d(n+2)*d(n+3)), where d(n) = A000166(n).

%F e = Sum_{n >= 0} (x^2 + (n+2)*x + n)/(n!(n + x)*(n + 1 + x)), provided x is not zero or a negative integer. (End)

%F Equals lim_{n -> oo} (2*3*5*...*prime(n))^(1/prime(n)). - _Peter Luschny_, May 21 2020

%F e = 3 - Sum_{n >= 0} 1/((n+1)^2*(n+2)^2*n!). - _Peter Bala_, Jan 13 2022

%F e = lim_{n->oo} prime(n)*(1 - 1/n)^prime(n). - _Thomas Ordowski_, Jan 31 2023

%F e = 1+(1/1)*(1+(1/2)*(1+(1/3)*(1+(1/4)*(1+(1/5)*(1+(1/6)*(...)))))), equivalent to the first formula. - _David Ulgenes_, Dec 01 2023

%F From _Michal Paulovic_, Dec 12 2023: (Start)

%F Equals lim_{n->oo} (1 + 1/n)^n.

%F Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)

%F Equals Product_{k>=1} (1 + 1/k) * (1 - 1/(k + 1)^2)^k. - _Antonio Graciá Llorente_, May 14 2024

%e 2.71828182845904523536028747135266249775724709369995957496696762772407663...

%p Digits := 200: it := evalf((exp(1))/10, 200): for i from 1 to 200 do printf(`%d,`,floor(10*it)): it := 10*it-floor(10*it): od: # _James A. Sellers_, Feb 13 2001

%t RealDigits[E, 10, 120][[1]] (* _Harvey P. Dale_, Nov 14 2011 *)

%o (PARI) default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 15 2009

%o (Haskell) -- See Niemeijer link.

%o a001113 n = a001113_list !! (n-1)

%o a001113_list = eStream (1, 0, 1)

%o [(n, a * d, d) | (n, d, a) <- map (\k -> (1, k, 1)) [1..]] where

%o eStream z xs'@(x:xs)

%o | lb /= approx z 2 = eStream (mult z x) xs

%o | otherwise = lb : eStream (mult (10, -10 * lb, 1) z) xs'

%o where lb = approx z 1

%o approx (a, b, c) n = div (a * n + b) c

%o mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)

%o -- _Reinhard Zumkeller_, Jun 12 2013

%Y Cf. A002034, A003417 (continued fraction), A073229, A122214, A122215, A122216, A122217, A122416, A122417.

%Y Expansion of e in base b: A004593 (b=2), A004594 (b=3), A004595 (b=4), A004596 (b=5), A004597 (b=6), A004598 (b=7), A004599 (b=8), A004600 (b=9), this sequence (b=10), A170873 (b=16). - _Jason Kimberley_, Dec 05 2012

%Y Powers e^k: A092578 (k = -7), A092577 (k = -6), A092560 (k = -5), A092553 - A092555 (k = -2 to -4), A068985 (k = -1), A072334 (k = 2), A091933 (k = 3), A092426 (k = 4), A092511 - A092513 (k = 5 to 7).

%K nonn,cons,nice,core

%O 1,1

%A _N. J. A. Sloane_