↑ 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; pages 34-39.
↑ J. Sondow, A faster product for pi and a new integral for ln pi/2, Amer. Math. Monthly 112 (2005) 729-734.
↑ H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e. The Mathematical Intelligencer, Vol. 20, No. 4, 1998; pages 25-29.
![{\displaystyle {\begin{aligned}e&=\left[2;1,{\textbf {2}},1,1,{\textbf {4}},1,1,{\textbf {6}},1,1,{\textbf {8}},1,1,\ldots ,{\textbf {2n}},1,1,\ldots \right]\\&=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7545eb2db1a146610cada2235197624030f6750)
![{\displaystyle e=\left[\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{k!}}\right]^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee680fb2f6c024d3207b864560743c6502bc0828)
![{\displaystyle e=\left[\sum _{k=0}^{\infty }{\frac {1-2k}{(2k)!}}\right]^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3fed598ba8fe6acd4dae74a3348d7780f1b1e07)
![{\displaystyle e=\left[\sum _{k=0}^{\infty }{\frac {4k+3}{2^{2k+1}\,(2k+1)!}}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d26c23d9900e39fbd0c5a8062a393db51ebc445f)
![{\displaystyle e=-{\frac {12}{\pi ^{2}}}\left[\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\ \cos \left({\frac {9}{k\pi +{\sqrt {k^{2}\pi ^{2}-9}}}}\right)\right]^{-1/3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4dcbdc0c2eb17e0125dbd6001c98254232cd77)
![{\displaystyle e=\prod _{n=1}^{\infty }{\sqrt[{n}]{\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}}}=\left({\frac {2}{1}}\right)^{1/1}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/2}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/3}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/4}\cdots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c354924ce5d3eee5defbbd9246f3c176204217a)
![{\displaystyle e=\lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/310afedad5e855a28eb0d1b232a29031bd682548)
![{\displaystyle e=\lim _{n\to \infty }\left[{\frac {(n+1)^{n+1}}{n^{n}}}-{\frac {n^{n}}{(n-1)^{n-1}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cffa4b98fac483870fb5da1d93940ef414d8319)
