New bounds on the length of finite Pierce and Engel series. (English) Zbl 0727.11003

The authors give estimates on the length of Engel series expansions for rational \(x\). The upper and lower bounds are widely separated. The method of estimation of the length is extended to alternating Engel series which is also known as Pierce expansions. For a large variety of results on Engel series as a special case of a general series expansion, see the reviewer [Representations of real numbers by infinite series, Lect. Notes Math. 502. Berlin etc.: Springer-Verlag (1976; Zbl 0322.10002)].


11A67 Other number representations


Zbl 0322.10002
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Online Encyclopedia of Integer Sequences:

Worst cases for Pierce expansions (numerators).
Worst cases for Pierce expansions (denominators).
Numerators of worst case for Engel expansion.
Denominators of worst case for Engel expansion.
Engel expansion of Pi.
Engel expansion of the golden ratio, (1 + sqrt(5))/2 = 1.61803... .
Engel expansion of the Euler-Mascheroni constant gamma A001620 = 0.57721566... .
Engel expansion of zeta(3) = 1.20206... .
Engel expansion of sqrt(5) = 2.23606...
Engel expansion of sqrt(10) = 3.16227...
Engel expansion of 2^(1/3) = 1.25992.
Engel expansion of 3^(1/3) = 1.44225.
Engel expansion of log(2).
Engel expansion of log(3) = 1.09861... .
Engel expansion of log(10) = 2.30259...
Engel expansion of 1/log(2) = 1.4427...
Engel expansion of 1/log(10) = 0.434294....
Engel expansion of Pi^2 = 9.8696...
Engel expansion of Pi^2/6, or zeta(2) = 1.64493.
Engel expansion of sqrt(Pi) = 1.77245... .
Engel expansion of Gamma(1/3) = 2.6789385....
Engel expansion of Gamma(2/3) = 1.35412.
Engel expansion of gamma^2, (gamma is the Euler-Mascheroni constant A001620) = 0.333178.
Engel expansion of 1/gamma, (gamma is the Euler-Mascheroni constant A001620) = 1.73245.
Engel expansion of log(1/gamma) (where gamma is the Euler-Mascheroni constant A001620) = 0.549539...
Engel expansion of 1/e = 0.367879... .
Engel expansion of 1/e^2 = 0.135335... .
Engel expansion of log(Pi) = 1.14473... .
Engel expansion of e^Pi = 23.14069... .
Engel expansion of Pi^e = 22.4592.
Engel expansion of e^gamma (gamma is the Euler-Mascheroni constant A001620) = 1.78107.
Engel expansion of -log(log(2)) = 0.36651292... .
Pierce expansion of 1/sqrt(2).
Pierce expansion of 1/e^2.
Pierce expansion of 1/zeta(2).
Pierce expansion of log(2).
Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).
Engel expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
Engel expansion of beta = 3/(2*log(alpha/2)); alpha = A195596.
Engel expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.
Engel expansion of the positive root of x^x^x^x = 2.
Maximum value of n-th row of A268057.
Engel expansion of exp(Pi/4).


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