Sum of an infinite series, approx 0.6434
In mathematics, Cahen's constant is defined as the value of an infinite series of unit fractions with alternating signs:
(sequence A118227 in the OEIS)
Here
denotes Sylvester's sequence, which is defined recursively by
![{\displaystyle {\begin{array}{l}s_{0}~~~=2;\\s_{i+1}=1+\prod _{j=0}^{i}s_{j}{\text{ for }}i\geq 0.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc1466297d59fa16b196b37b25f4ad2a1b01bdc)
Combining these fractions in pairs leads to an alternative expansion of Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence. This series for Cahen's constant forms its greedy Egyptian expansion:
![{\displaystyle C=\sum {\frac {1}{s_{2i}}}={\frac {1}{2}}+{\frac {1}{7}}+{\frac {1}{1807}}+{\frac {1}{10650056950807}}+\cdots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc06a2318c69c1d90b826ed1fd3aaa3aa837fa0)
This constant is named after Eugène Cahen [fr] (also known for the Cahen–Mellin integral), who was the first to introduce it and prove its irrationality.
Continued fraction expansion
The majority of naturally occurring[2] mathematical constants have no known simple patterns in their continued fraction expansions. Nevertheless, the complete continued fraction expansion of Cahen's constant
is known: it is
where the sequence of coefficients
0, 1, 1, 1, 2, 3, 14, 129, 25298, 420984147, ... (sequence
A006279 in the
OEIS)
is defined by the recurrence relation
All the partial quotients of this expansion are squares of integers. Davison and Shallit made use of the continued fraction expansion to prove that
is transcendental.
Alternatively, one may express the partial quotients in the continued fraction expansion of Cahen's constant through the terms of Sylvester's sequence: To see this, we prove by induction on
that
. Indeed, we have
, and if
holds for some
, then
where we used the recursion for
in the first step respectively the recursion for
in the final step. As a consequence,
holds for every
, from which it is easy to conclude that
.
Best approximation order
Cahen's constant
has best approximation order
. That means, there exist constants
such that the inequality
has infinitely many solutions
, while the inequality
has at most finitely many solutions
. This implies (but is not equivalent to) the fact that
has irrationality measure 3, which was first observed by Duverney & Shiokawa (2020).
To give a proof, denote by
the sequence of convergents to Cahen's constant (that means,
).[5]
But now it follows from
and the recursion for
that
![{\displaystyle {\frac {a_{n+2}}{a_{n+1}^{2}}}={\frac {a_{n}\cdot s_{n-1}}{a_{n-1}^{2}\cdot s_{n-2}^{2}}}={\frac {a_{n}}{a_{n-1}^{2}}}\cdot {\frac {s_{n-2}^{2}-s_{n-2}+1}{s_{n-1}^{2}}}={\frac {a_{n}}{a_{n-1}^{2}}}\cdot {\Big (}1-{\frac {1}{s_{n-1}}}+{\frac {1}{s_{n-1}^{2}}}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56bac7333418e3894d32c49c6181fc1a890bbecd)
for every
. As a consequence, the limits
and ![{\displaystyle \beta :=\lim _{n\to \infty }{\frac {q_{2n+2}}{q_{2n+1}^{2}}}=2\cdot \prod _{n=0}^{\infty }{\Big (}1-{\frac {1}{s_{2n+1}}}+{\frac {1}{s_{2n+1}^{2}}}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e14acabe1d70b28df74e53e4711c3db695fa79c8)
(recall that
) both exist by basic properties of infinite products, which is due to the absolute convergence of
. Numerically, one can check that
. Thus the well-known inequality
![{\displaystyle {\frac {1}{q_{n}(q_{n}+q_{n+1})}}\leq {\Big |}C-{\frac {p_{n}}{q_{n}}}{\Big |}\leq {\frac {1}{q_{n}q_{n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/964bd17d02b57dd01f41b5dd4563dc19ddfdf481)
yields
and ![{\displaystyle {\Big |}C-{\frac {p_{n}}{q_{n}}}{\Big |}\geq {\frac {1}{q_{n}(q_{n}+q_{n+1})}}>{\frac {1}{q_{n}(q_{n}+2q_{n}^{2})}}\geq {\frac {1}{3q_{n}^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/794cad9d4c119942756df3f107682e7d4d480d2b)
for all sufficiently large
. Therefore
has best approximation order 3 (with
), where we use that any solution
to
![{\displaystyle 0<{\Big |}C-{\frac {p}{q}}{\Big |}<{\frac {1}{3q^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeccf78b8b5e8e9b0728275d9f5ef95f5f0c5cc)
is necessarily a convergent to Cahen's constant.
Notes
- ^ A number is said to be naturally occurring if it is *not* defined through its decimal or continued fraction expansion. In this sense, e.g., Euler's number
is naturally occurring. - ^ Sloane, N. J. A. (ed.), "Sequence A006279", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
References
- Cahen, Eugène (1891), "Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues", Nouvelles Annales de Mathématiques, 10: 508–514
- Davison, J. Les; Shallit, Jeffrey O. (1991), "Continued fractions for some alternating series", Monatshefte für Mathematik, 111 (2): 119–126, doi:10.1007/BF01332350, S2CID 120003890
- Borwein, Jonathan; van der Poorten, Alf; Shallit, Jeffrey; Zudilin, Wadim (2014), Neverending Fractions: An Introduction to Continued Fractions, Australian Mathematical Society Lecture Series, vol. 23, Cambridge University Press, doi:10.1017/CBO9780511902659, ISBN 978-0-521-18649-0, MR 3468515
- Duverney, Daniel; Shiokawa, Iekata (2020), "Irrationality exponents of numbers related with Cahen's constant", Monatshefte für Mathematik, 191 (1): 53–76, doi:10.1007/s00605-019-01335-0, MR 4050109, S2CID 209968916
External links
- Weisstein, Eric W., "Cahen's Constant", MathWorld
- "The Cahen constant to 4000 digits", Plouffe's Inverter, Université du Québec à Montréal, archived from the original on March 17, 2011, retrieved 2011-03-19
- "Cahen's constant (1,000,000 digits)", Darkside communication group, retrieved 2017-12-25