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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)].

MSC:

11A67 Other number representations

Citations:

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).

References:

[1] Békéssy, A., Bemerkungen zur Engleschen Darstellung reeler Zahlen, Ann. Univ. Sci. Budapest.Eötvös Sect. Math.1 (1958), 143-151. · Zbl 0108.26804
[2] Deheuvels, P., L’encadrement asymptotique des elements de la série d’Engel d’un nombre réel, C. R. Acad. Sci. Paris295 (1982), 21-24. · Zbl 0488.10050
[3] Engel, F., Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.
[4] Erdös, P., Rényi, A., and Szüsz, P., On Engel’s and Sylvester’s series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.1 (1958), 7-32. · Zbl 0107.27002
[5] Hardy, G.H. and Wright, E.M., An Introduction to the Theory of Numbers, Oxford University Press, 1985.
[6] Mays, M.E., Iterating the division algorithm, Fibonacci Quart.25 (1987), 204-213. · Zbl 0621.10008
[7] Pierce, T.A., On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly36 (1929), 523-525. · JFM 55.0305.06
[8] Remez, E. Ya., On series with alternating signs which may be connected with two algorithms of M. V. Ostrogradskii for the approximation of irrational numbers, Uspekhi Mat. Nauk6 (5) (1951), 33-42, (MR #13,444d). · Zbl 0045.02102
[9] Rényi, A., A new approach to the theory of Engel’s series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.5 (1962), 25-32. · Zbl 0232.10028
[10] Rosser, J.B. and Schoenfeld, L., Approxamate formulas for some functions of prime numbers, Illinois J. Math.6 (1962), 64-94. · Zbl 0122.05001
[11] Shallit, J.O., Metric theory of Pierce expansions, Fibonacci Quart.24 (1986), 22-40. · Zbl 0598.10057
[12] Shallit, J.O., Letter to the editor, Fibonacci Quart.27 (1989), 186.
[13] Sierpinski, W., O kilku algorytmach dla rozwijania liczb rzeczywistych na szeregi, C. R. Soc. Sci. Varsovie4 (1911), 56-77, (In Polish; reprinted in French translation as Sur quelques algorithmes pour développer les nombres reéls en séries, in Sierpinski, W., Oeuvres Choisies, Vol. I, PWN, Warsaw, 1974, pp. 236-254.).
[14] Valeyev, K.G. and Zlebov, E.D., The metric theory of an algorithm of M. V. Ostrogradskij, Ukrain. Mat. Z.27 (1975), 64-69. · Zbl 0309.10020
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