From Schmidt's simultaneous approximation theorem we deduce transcendence results concerning series of rational numbers. The denominators of these numbers are from finitely many linear recursive sequences and have to satisfy a divisibility as well as a growth condition. (In an appendix the second author studies the connections between these two kinds of hypothesis.) For the numerators we need some growth conditions too. We study also the implications of Mahler's analytic transcendence method from 1929 to the arithmetical questions considered mainly.
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