In the special case that , the result is the famous statement:
The ratio of the factorial, that counts all permutations of an ordered set S with cardinality, and the subfactorial (a.k.a. the derangement function) , which counts the amount of permutations where no element appears in its original position, tends to as grows.
As a ratio of ratiosedit
A unique representation of e can be found within the structure of Pascal's Triangle, as discovered by Harlan Brothers. Pascal's Triangle is composed of binomial coefficients, which are traditionally summed to derive polynomial expansions. However, Brothers identified a product-based relationship between these coefficients that links to e. Specifically, the ratio of the products of binomial coefficients in adjacent rows of Pascal's Triangle tends to e as the row number increases. This relationship and its proof are outlined in the discussion on the properties of the rows of Pascal's Triangle.[10][11]
In trigonometryedit
Trigonometrically, e can be written in terms of the sum of two hyperbolic functions,
^Sandifer, Ed (Feb 2006). "How Euler Did It: Who proved e is Irrational?" (PDF). MAA Online. Retrieved 2017-04-23.
^Brown, Stan (2006-08-27). "It's the Law Too — the Laws of Logarithms". Oak Road Systems. Archived from the original on 2008-08-13. Retrieved 2008-08-14.
^Formulas 2–7: H. J. Brothers, Improving the convergence of Newton's series approximation for e, The College Mathematics Journal, Vol. 35, No. 1, (2004), pp. 34–39.
^Formula 8: A. G. Llorente, A Novel Simple Representation Series for Euler’s Number e, preprint, 2023.
^"e", Wolfram MathWorld: ex. 17, 18, and 19, archived from the original on 2023-03-15.
^J. Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729–734.
^J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan Journal 16 (2008), 247–270.
^H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e, The Mathematical Intelligencer, Vol. 20, No. 4, (1998), pp. 25–29.