# List of representations of e

The representations of \( e \), the base of the natural logarithm, are the infinite series, limits, continued fractions, infinite products, and related expansions that all converge to the same constant \( e = 2.718281828459045\ldots \).

| Key fact | Detail |
|---|---|
| Series | \( e = \sum_{k=0}^{\infty} 1/k! \), first published by Newton in 1669<sup>[1](https://mathworld.wolfram.com/e.html)</sup> |
| Limit | \( e = \lim_{n\to\infty} (1+1/n)^n \), arising from Jacob Bernoulli's 1683 study of compound interest<sup>[2](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)</sup> |
| Simple continued fraction | \( e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, \ldots] \), given by Euler, with the pattern proved much later<sup>[3](https://wstein.org/edu/Fall2002/124/stein/main-05-2003.pdf)</sup> |
| Convergence of the series | The remainder after \( n \) terms is below \( 1/(n \cdot n!) \); about 20 terms give 18 correct digits and about 120 terms give 200 decimal places<sup>[4](https://www.irejournals.com/paper-details/1723438)</sup><sup> • </sup><sup>[2](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)</sup> |
| Limit error bound | \( e - (1+1/n)^n = e/(2n) + O(n^{-2}) \), so the limit converges slowly<sup>[4](https://www.irejournals.com/paper-details/1723438)</sup> |
| Number-theoretic status | Euler observed in 1737 that its continued fraction would show \( e \) irrational; Liouville showed in 1840 that \( e \) satisfies no quadratic equation; Hermite proved \( e \) transcendental in 1873<sup>[1](https://mathworld.wolfram.com/e.html)</sup> |
| Irrationality measure | \( \mu(e) = 2 \), the least possible value, so \( e \) is "least" transcendental in this sense<sup>[1](https://mathworld.wolfram.com/e.html)</sup> |

## Series and limit representations

The factorial series is the most direct representation:

\[ e = \sum_{k=0}^{\infty} \frac{1}{k!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots \]

Newton published it in 1669, and Euler gave a full treatment in his *Introductio in Analysin infinitorum* (1748), where he established the equality of this series with the limit definition and evaluated \( e \) to 23 decimal places.<sup>[1](https://mathworld.wolfram.com/e.html)</sup><sup> • </sup><sup>[2](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)</sup> The series converges very fast because the remainder after \( n \) terms is below \( 1/(n \cdot n!) \).<sup>[4](https://www.irejournals.com/paper-details/1723438)</sup>

The limit representation

\[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} \]

came first historically: [Jacob Bernoulli](https://www.edgepedia.org/jacob-bernoulli) encountered it in 1683 while examining continuous compound interest and used the binomial theorem to show the limit lies between 2 and 3, the first approximation to \( e \) and the first time a number was defined by a limiting process.<sup>[2](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)</sup> Its convergence is slow: \( e - (1+1/n)^n = e/(2n) + O(n^{-2}) \), so about \( n = 10{,}000{,}000 \) is needed for six correct digits.<sup>[4](https://www.irejournals.com/paper-details/1723438)</sup>

MathWorld lists further series variants, including \( e = \left[\sum_{k=0}^{\infty} (-1)^k/k!\right]^{-1} \), \( e = \sum_{k=0}^{\infty} (2k+1)/(2k)! \), and \( e = 2\sum_{k=0}^{\infty} (k+1)/(2k+1)! \).<sup>[1](https://mathworld.wolfram.com/e.html)</sup> An integral characterization also serves as a definition: \( e \) is the unique positive number with \( \int_1^e \frac{dx}{x} = 1 \).<sup>[1](https://mathworld.wolfram.com/e.html)</sup>

## Continued fractions

**The simple continued fraction.** Applying the continued fraction algorithm to \( e \) gives

\[ e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, \ldots] \]

with the partial quotients after the first following the repeating pattern \( 1, 1, 2n \) occurring in blocks for n = 1, 2, 3, \ldots<sup>[16](https://fibonacci-numbers.surrey.ac.uk/Fibonacci/cfINTRO.html)</sup> Euler wrote down this expansion and observed the pattern but did not publish a proof; the proof, whose underlying idea is due to Hermite, was later given in short form by Cohn.<sup>[3](https://wstein.org/edu/Fall2002/124/stein/main-05-2003.pdf)</sup><sup> • </sup><sup>[5](https://mathworld.wolfram.com/eContinuedFraction.html)</sup> Euler also gave two related expansions,

\[ \frac{e-1}{2} = [0; 1, 6, 10, 14, 18, \ldots] \]

with partial quotients increasing by 4, and \( e - 1 = [1; 1, 2, 1, 1, 4, 1, 1, 6, \ldots] \). He knew that if the pattern in the first continued fraction continued forever, the fraction would never terminate, which would prove \( e \) irrational; this can be seen as the first attempt at such a proof.<sup>[2](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)</sup>

The convergents of the simple continued fraction are the best rational approximations to \( e \) among all fractions with denominators up to the same size, and they satisfy \( |e - p_k/q_k| < 1/q_k^2 \). The first few are \( 3 \), \( 19/7 \), \( 193/71 \), and \( 2721/1001 \).<sup>[3](https://wstein.org/edu/Fall2002/124/stein/main-05-2003.pdf)</sup><sup> • </sup><sup>[4](https://www.irejournals.com/paper-details/1723438)</sup> The same method generalizes: the continued fraction of \( e^{1/k} \) is

\[ e^{1/k} = [1; (k-1), 1, 1, (3k-1), 1, 1, (5k-1), 1, 1, (7k-1), \ldots] \]

for every positive integer \( k \).<sup>[3](https://wstein.org/edu/Fall2002/124/stein/main-05-2003.pdf)</sup>

**Generalized continued fractions.** Many generalized continued fractions (with numerators other than 1) also converge to \( e \). Two proved by elementary methods are

\[ e = 2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{2}{3 + \cfrac{3}{4 + \ddots}}}} \qquad \text{and} \qquad e = 3 + \cfrac{-1}{4 + \cfrac{-2}{5 + \cfrac{-3}{6 + \cfrac{-4}{7 + \ddots}}}} \]

The first is well known; the second was discovered numerically by the Ramanujan Machine project and then proved.<sup>[6](https://arxiv.org/abs/1907.05563)</sup> A third, derived from the ratio of factorials to derangement numbers (subfactorials), reads

\[ e = 2 + \cfrac{2}{2 +} \cfrac{3}{3 +} \cfrac{4}{4 +} \cfrac{5}{5 +} \cdots \]

and converges faster than both the limit \( (1+1/n)^n \) and Euler's simple continued fraction.<sup>[7](https://arxiv.org/abs/2012.12692)</sup> The DLMF records the classical continued fractions for the exponential function itself, valid for all complex \( z \), for example

\[ e^z = 1 + \cfrac{z}{1 -} \cfrac{z}{2 +} \cfrac{z}{3 -} \cfrac{z}{2 +} \cfrac{z}{5 -} \cfrac{z}{2 +} \cdots \]

which specializes to \( e \) at \( z = 1 \).<sup>[8](https://dlmf.nist.gov/4.9)</sup>

**Algorithmic discovery.** The Ramanujan Machine, published in *Nature* in 2021, uses meet-in-the-middle and gradient-descent algorithms to match numerical values and thereby conjecture continued fraction representations of \( e \), \( \pi \), Catalan's constant, and zeta values without prior knowledge of the underlying structure; several of its \( e \) conjectures have since been proved, including by symbolic methods that generate proofs automatically.<sup>[9](https://www.nature.com/articles/s41586-021-03229-4)</sup><sup> • </sup><sup>[10](https://arxiv.org/abs/2004.00090)</sup> [Polynomial](https://www.edgepedia.org/polynomial) continued fraction techniques produce infinite families of inequivalent continued fractions all converging to the same constant, \( e \) among them.<sup>[11](https://arxiv.org/abs/math/0402462)</sup>

## Infinite products

Several infinite products converge to \( e \). Eugène Catalan discovered the product

\[ e = \frac{2}{1} \left(\frac{4}{3}\right)^{1/2} \left(\frac{6 \cdot 8}{5 \cdot 7}\right)^{1/4} \left(\frac{10 \cdot 12 \cdot 14 \cdot 16}{9 \cdot 11 \cdot 13 \cdot 15}\right)^{1/8} \cdots \]

which Nimbran and Levrie decomposed into further products and generalized to products for \( e^n \).<sup>[12](https://indianmathsoc.org/ms/mathstudent-part-2-2016.pdf)</sup> Pippenger's 1980 product, analogous to Wallis's product for \( \pi \), gives

\[ \frac{e}{2} = \left(\frac{2}{1}\right)^{1/2} \left(\frac{2 \cdot 4}{3 \cdot 3}\right)^{1/4} \left(\frac{4 \cdot 6 \cdot 6 \cdot 8}{5 \cdot 5 \cdot 7 \cdot 7}\right)^{1/8} \cdots \]<sup>[1](https://mathworld.wolfram.com/e.html)</sup><sup> • </sup><sup>[12](https://indianmathsoc.org/ms/mathstudent-part-2-2016.pdf)</sup>

A number-theoretic product proved by Robert P. Schneider, a mathematician working in number theory and combinatorics, expresses \( e \) through the golden ratio, the [Möbius function](https://www.edgepedia.org/mobius-function), and the Euler phi function.<sup>[13](https://arxiv.org/abs/1211.2025)</sup>

## Other representations

The Engel expansion of \( e \), a representation as a sum of unit fractions with recursively multiplied denominators, is exceptionally regular: its terms are \( 1, 1, 2, 3, 4, 5, 6, \ldots \), the natural numbers themselves.<sup>[5](https://mathworld.wolfram.com/eContinuedFraction.html)</sup> The factorial series can also be rewritten as a nested expression,

\[ e = 1 + 1 + \frac{1}{2}\left(1 + \frac{1}{3}\left(1 + \frac{1}{4}\left(1 + \frac{1}{5}(1 + \cdots)\right)\right)\right) \]

which yields a nested radical when both sides are raised to a power.<sup>[1](https://mathworld.wolfram.com/e.html)</sup> The decimal digits of \( e \) form sequence A001113 in the [On-Line Encyclopedia of Integer Sequences](https://www.edgepedia.org/on-line-encyclopedia-of-integer-sequences), and the continued fraction partial quotients form A003417; related sequences record continued fractions for \( 2e \), \( e/2 \), \( 4e \), and \( e/4 \), each with its own predictable pattern.<sup>[14](https://oeis.org/A003417/internal)</sup>

## Why the representations matter

The continued fraction is not merely a curiosity: its structure carries number-theoretic information. A continued fraction terminates exactly when the number is rational, and an eventually periodic one belongs to a root of a quadratic equation with integer coefficients (Lagrange's theorem). Since the partial quotients of \( e \) grow without bound but never repeat, the expansion directly shows that \( e \) is irrational and not quadratic.<sup>[2](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)</sup><sup> • </sup><sup>[1](https://mathworld.wolfram.com/e.html)</sup> Euler proved irrationality this way in 1737; Liouville showed in 1840 that \( e \) satisfies no quadratic equation with integer coefficients; and Hermite settled the question in 1873 by proving \( e \) transcendental, that is, not the root of any nonzero polynomial with rational coefficients. Waldschmidt, a professor of mathematics at the Institut de mathématiques de Jussieu, describes Hermite's proof as the model on which most later transcendence proofs for constants of analysis are based.<sup>[1](https://mathworld.wolfram.com/e.html)</sup><sup> • </sup><sup>[15](https://polyjmath.org/volumes/polyjmath-4-14.html)</sup>

The representations also differ sharply in computational value. The factorial series delivers about 18 correct digits from 20 terms, while the limit \( (1+1/n)^n \) gains only about half a digit per doubling of \( n \); the derangement continued fraction converges faster than Euler's simple continued fraction, which in turn beats the compound-interest limit.<sup>[4](https://www.irejournals.com/paper-details/1723438)</sup><sup> • </sup><sup>[7](https://arxiv.org/abs/2012.12692)</sup> A quantitative measure of how well \( e \) can be approximated comes from Sondow's bound: for any integers \( p, q \) with \( q > 1 \), \( |e - p/q| > 1/(S(q)+1)! \), where \( S(q) \) is the Smarandache function, the smallest \( m \) such that \( q \) divides \( m! \).<sup>[1](https://mathworld.wolfram.com/e.html)</sup>

## References

1. [Weisstein, Eric W. "e." MathWorld](https://mathworld.wolfram.com/e.html)
2. [O'Connor, J. J. and Robertson, E. F. "The number e." MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//HistTopics/e/)
3. [Stein, W. "7.3 The Continued Fraction of e," course notes, Harvard, 2002–2003](https://wstein.org/edu/Fall2002/124/stein/main-05-2003.pdf)
4. [Muhati, Linet. "Five Approaches to Computing e: Derivations, Error Bounds and History," IRE Journals 10(3), 2026](https://www.irejournals.com/paper-details/1723438)
5. [Weisstein, Eric W. "e Continued Fraction." MathWorld](https://mathworld.wolfram.com/eContinuedFraction.html)
6. [Lu, Zhentao. "Elementary proofs of generalized continued fraction formulae for e," arXiv:1907.05563 (2019)](https://arxiv.org/abs/1907.05563)
7. [Lynch, Peter. "Derangements and Continued Fractions for e," arXiv:2012.12692 (2020)](https://arxiv.org/abs/2012.12692)
8. [DLMF §4.9 Continued Fractions, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.9)
9. [Raayoni, G. et al. "Generating conjectures on fundamental constants with the Ramanujan Machine," Nature 590, 67–73 (2021)](https://www.nature.com/articles/s41586-021-03229-4)
10. [Dougherty-Bliss, R. and Zeilberger, D. "Automatic Conjecturing and Proving of Exact Values of Some Infinite Families of Infinite Continued Fractions," arXiv:2004.00090 (2020)](https://arxiv.org/abs/2004.00090)
11. [McLaughlin, J. and Wyshinski, N. "Real Numbers With Polynomial Continued Fraction Expansions," arXiv:math/0402462; Acta Arithmetica 116(1), 63–79 (2005)](https://arxiv.org/abs/math/0402462)
12. [Nimbran, A. S. and Levrie, P. "Infinite Products for e^n via Catalan's Product," The Mathematics Student 85(3–4), 21–24 (2016)](https://indianmathsoc.org/ms/mathstudent-part-2-2016.pdf)
13. [Schneider, Robert P. "A Golden Product Identity for e," Mathematics Magazine 87(2), 132–134 (2014); arXiv:1211.2025](https://arxiv.org/abs/1211.2025)
14. [OEIS A003417, Continued fraction for e](https://oeis.org/A003417/internal)
15. [Waldschmidt, Michel. "An introduction to the strategy of transcendence proofs," Hardy–Ramanujan Journal](https://polyjmath.org/volumes/polyjmath-4-14.html)
16. [Continued Fractions - An introduction](https://fibonacci-numbers.surrey.ac.uk/Fibonacci/cfINTRO.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Real and complex number constructions › Continued fractions and Diophantine approximation of reals*

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