The Transcendence of Certain Infinite Series
نویسندگان
چکیده
منابع مشابه
Irrationality of certain infinite series
In this paper a new direct proof for the irrationality of Euler's number e = ∞ k=0 1 k! is presented. Furthermore, formulas for the base b digits are given which, however, are not computably effective. Finally we generalize our method and give a simple criterium for some fast converging series representing irrational numbers.
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In a recent paper a new direct proof for the irrationality of Euler's number e = ∞ k=0 1 k! and on the same lines a simple criterion for some fast converging series representing irrational numbers was given. In the present paper, we give some generalizations of our previous results. 1 Irrationality criterion Our considerations in [3] lead us to the following criterion for irrationality, where x...
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In this paper we investigate the infinite convergent sum T = ∑∞ n=0 P (n) Q(n) , where P (x) ∈ Q[x], Q(x) ∈ Q[x] and Q(x) has only simple rational zeros. N. Saradha and R. Tijdeman have obtained sufficient and necessary conditions for the transcendence of T if the degree of Q(x) is 3. In this paper we give sufficient and necessary conditions for the transcendence of T if the degree of Q(x) is 4...
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Let ξ ∈ (0, 1) be an irrational with aperiodic continued fraction expansion: ξ = [0; u0, u1, u2, . . .], and suppose the sequence (un)n≥0 of partial quotients takes only values from the finite set {a1, a2, . . . , ak} with 1 ≤ a1 < a2 < · · · < ak, k ≥ 2. We prove that if the frequency of a1 (or ak) in (un)n≥0 is at least 1/2, and (un)n≥0 begins with arbitrarily long blocks that are almost squa...
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ژورنال
عنوان ژورنال: Rocky Mountain Journal of Mathematics
سال: 2005
ISSN: 0035-7596
DOI: 10.1216/rmjm/1181069744