On the transcendence of some operations of infinite series

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a sum, product and quotient of some series of positive rational terms are transcendental numbers. We recall that all the infinite series that we are going to treat are Liouville numbers. At the end this article, we establish an approximation measure of these numbers.