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 := max{n ∈ N | n x} denotes the floor function (Gauss bracket). Theorem 1.1 Let ∞ j=1 b j be a convergent series with b j 0 for all natural numbers j ∈ N := {1, 2, 3,. .. } and b j > 0 for infinitely many j ∈ N. Let a := ∞ j=1 b j ∈ R denote its limit and s n := n j=1 b j denote the corresponding partial sums. If n s n = n a for almost all n ∈ N, i. e. for all but finitely many n ∈ N, then a is irrational. Proof: From the given assumptions on b j it follows that s n < a for all n ∈ N and therefore n s n < n a (n ∈ N). We write n s n = =n s n + R n with 0 R n < 1 and n a = =n a + R n with 0 R n < 1. Therefore we have for all n ∈ N