A001113 Decimal expansion of e.
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, 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, 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
Offset: 1
Examples
2.71828182845904523536028747135266249775724709369995957496696762772407663...
References
- Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 400.
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 250-256.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers, p. 85.
- E. Maor, e: The Story of a Number, Princeton 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.
- 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).
- Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 1 and 2, equations 1:7:4, 2:5:4 at pages 13, 20.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.
Links
- N. J. A. Sloane, Table of 50000 digits of e labeled from 1 to 50000 [based on the ICON Project link below]
- 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.
- Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.
- L. E. Bush, The William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 68, No. 1 (1961), pp. 18-33, problem 3.
- Ed Copeland and Brady Haran, A proof that e is irrational, Numberphile video (2021).
- 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
- Brady Haran and James Grime, Incredible Formula, Numberphile YouTube video, 2016.
- ICON Project, e to 50000 places
- Roger Mansuy, Un intrigant poème... mathématique, Images des Mathématiques, CNRS, 2023. In French.
- R. Nemiroff and J. Bonnell, The first 5 million digits of the number e
- Remco Niemeijer, Digits Of E, programmingpraxis
- J. J. O'Connor & E. F. Robertson, The number e
- Michael Penn, e is irrational, YouTube video, 2020.
- Simon Plouffe, A million digits
- G. W. Reitwiesner, An ENIAC determination of pi and e to more than 2000 decimal places, Pi, A Source book, pp 277-281, 2000.
- E. Sandifer, How Euler Did It, Who proved e is irrational?, MAA Online (2006)
- D. Shanks and J. W. Wrench, Jr., Calculation of e to 100,000 decimals, Math. Comp., 23 (1969), 679-680.
- Jean-Louis Sigrist, Le premier million de décimales de e.
- Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 339.
- Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641 (article) and 114 (2007), 659 (addendum).
- Jonathan Sondow and K. Schalm, 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, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
- G. Villemin's Almanach of Numbers, Constant "e"
- Eric Weisstein's World of Mathematics, e
- Eric Weisstein's World of Mathematics, e Digits
- Eric Weisstein's World of Mathematics, Factorial Sums
- Eric Weisstein's World of Mathematics, Uniform Sum Distribution
- Eric Weisstein's World of Mathematics, e Approximations
- Wikipedia, E (mathematical constant)
- Index entries for "core" sequences
- Index entries for transcendental numbers
Crossrefs
Cf. A002034, A003417 (continued fraction), A073229, A122214, A122215, A122216, A122217, A122416, A122417.
Programs
-
Haskell
-- See Niemeijer link. a001113 n = a001113_list !! (n-1) a001113_list = eStream (1, 0, 1) [(n, a * d, d) | (n, d, a) <- map (\k -> (1, k, 1)) [1..]] where eStream z xs'@(x:xs) | lb /= approx z 2 = eStream (mult z x) xs | otherwise = lb : eStream (mult (10, -10 * lb, 1) z) xs' where lb = approx z 1 approx (a, b, c) n = div (a * n + b) c mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f) -- Reinhard Zumkeller, Jun 12 2013 -
Maple
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 Sellers, Feb 13 2001
-
Mathematica
RealDigits[E, 10, 120][[1]] (* Harvey P. Dale, Nov 14 2011 *)
Formula
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)*(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
exp(1) = (1 + Sum_{n>=1} ((1+n+n^3)/n!))/7. - Alexander R. Povolotsky, Sep 14 2011
e = 1 + (2 + (3 + (4 + ...)/4)/3)/2 = 2 + (1 + (1 + (1 + ...)/4)/3)/2. - Rok Cestnik, Jan 19 2017
From Peter Bala, Nov 13 2019: (Start)
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.
e = 2 + Sum_{n >= 0} (-1)^n*(n+2)!/(d(n+2)*d(n+3)), where d(n) = A000166(n).
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)
Equals lim_{n -> oo} (2*3*5*...*prime(n))^(1/prime(n)). - Peter Luschny, May 21 2020
e = 3 - Sum_{n >= 0} 1/((n+1)^2*(n+2)^2*n!). - Peter Bala, Jan 13 2022
e = lim_{n->oo} prime(n)*(1 - 1/n)^prime(n). - Thomas Ordowski, Jan 31 2023
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
From Michal Paulovic, Dec 12 2023: (Start)
Equals lim_{n->oo} (1 + 1/n)^n.
Equals x^(x^(x^...)) (infinite power tower) where x = e^(1/e) = A073229. (End)
Equals Product_{k>=1} (1 + 1/k) * (1 - 1/(k + 1)^2)^k. - Antonio Graciá Llorente, May 14 2024
Equals lim_{n->oo} Product_{k=1..n} (n^2 + k)/(n^2 - k) (see Finch). - Stefano Spezia, Oct 19 2024
e ~ (1 + 9^((-4)^(7*6)))^(3^(2^85)), correct to more than 18*10^24 digits (Richard Sabey, 2004); see Haran and Grime link. - Paolo Xausa, Dec 21 2024.
Comments