Convergent sums T=n=1P(n)/Q(n) and U=n=1(-1)nP(n)/Q(n), where P(X), Q(X) ∈ Q[X], and Q(X) has only simple rational roots, are investigated. Adhikari, Shorey and the authors have shown that T and U are either rational or transcendental. Simple necessary and sufficient conditions are formulated for the transcendence of T and U if the degree of Q is 3 and 2, respectively.

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