A Theorem on Transcendence of Infinite Series II

A theorem on irrationality of infinite series and applications

Here and in the sequel we maintain the convention of [Bad] that all series which appear are supposed to be convergent. Moreover, (an) and (bn), n ≥ 1, always denote sequences of positive integers. Also, for the sake of brevity, we simply say “the series ∑∞ n=1 bn/an is irrational” instead of “the sum of the series ∑∞ n=1 bn/an is an irrational number”. We note that Theorem A is, in a certain se...

On the transcendence of some infinite sums

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...

Irrationality of certain infinite series II

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...

Arithmetic Gevrey Series and Transcendence. a Survey

Some Remarks on Infinite Series

In the present paper we investigate the following problems. Suppose an >O for n_-I and Z a,=-. n=1 N° 1. Does there exist a sequence of natural numbers No =O, Ni l-, such that it decomposes the series monotone decreasingly : In order to state the second problem we define the index nk (c) as the minimum m such that (2) Now the second problem is as follows. are equiconvergent. m kc a j. j=1 N° 2....