1,1

e is sometimes called Euler's constant, also Napier's constant.

Also, decimal expansion of sinh(1)+cosh(1) - Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 15 2006

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 (jsondow(AT)alumni.princeton.edu), Sep 04 2006

Mohammad K. Azarian, An Expansion of e, Problem # B-765, Fibonacci Quarterly, Vol. 32, No. 2, May 1994, p. 181. Solution appeared in Vol. 33, No. 4, Aug.1995, p. 377. [From Mohammad K. Azarian (azarian(AT)evansville.edu), Feb 08 2009]

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

E. Maor, e: The Story of a Number, Princton Univ. Press, 1994.

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

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. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679-680.

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

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

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

N. J. A. Sloane, Table of 50000 digits of e labeled from 1 to 50000 [based on the ICON Project link below]

Dave's Math Tables, e

X. Gourdon, Plouffe's Inverter, e to 1.250 billion digits

X. Gourdon and P. Sebah, The constant e and its computation

ICON Project, e to 50000 places

R. Nemiroff and J. Bonnell, The first 5 million digits of the number e

J. J. O'Connor & E. F. Robertson, The number e

S. Plouffe, A million digits

E. Sandifer, How Euler Did It, Who proved e is irrational?

Jean-Louis Sigrist, Le premier million de decimales de e. [From Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 28 2009]

G. Villemin's Almanach of Numbers, Constant"e"

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Uniform Sum Distribution

Eric Weisstein's World of Mathematics, Factorial Sums

Wikipedia, E(mathematical constant

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

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

exp(1)= (16/31*(sum((1/2)^n*(1/2*n^3+1/2*n+1)/n!,n=1..infinity) +1))^2. Robert Israel confirmed that above formula is correct, saying: "In fact, sum(n^j*t^n/n!, n=0..infinity) = 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) ." [From Alexander R. Povolotsky (pevnev(AT)juno.com), Jan 04 2009]

2.71828182845904523536028747135266249775724709369995957496696762772407663\

0353547594571382178525166427427466391932003059921817413596629043572900334\

295260595630738132328627943490763233829880753195251019...

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:

a := N[E, 500]; For[n = 1, n < 250, n++, Print[Floor[10^(n - 1)*a] - Floor[10^(n - 2)*a]*10]] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Feb 17 2006

(PARI) { default(realprecision, 50080); x=exp(1); for (n=1, 50000, d=floor(x); x=(x-d)*10; write("b001113.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 15 2009]

Cf. A002034, A122214, A122215, A122216, A122217, A122416, A122417.

Sequence in context: A021372 A111714 A060302 this_sequence A094121 A105178 A112257

Adjacent sequences: A001110 A001111 A001112 this_sequence A001114 A001115 A001116

nonn,cons,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

Maple code from James A. Sellers (sellersj(AT)math.psu.edu), Feb 13 2001