Convergent sums $T = {\sum}_{n=1}^{\infty} P(n)/Q(n)$ and $U = {\sum}_{n=1}^{\infty}(-1)^n P(n)/Q(n)$, where $P(X), Q(X) \in {\mathbb Q}[X]$, 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.