The paper discusses a method for obtaining lower and upper bounds for series \(\sum_{k=0}^\infty a_k\) where \(a_k= f(k)\) is a rational function of \(k\). The series is split into \(S_n+ R_n\), where \(S_n= \sum_{k=0}^n a_k\) and \(R_n\) is the remainder series. The finite sum \(S_n\) is estimated using interval arithmetic, and the remainder is transformed using \(g(t):= f((n+1)/ t)= t^m \widetilde {g}(t)\), where \(m\) is the degree of the denominator minus the degree of the numerator when \(f(k)\) is written as a quotient of polynomials in \(k\). The function \(\widetilde {g} (t)\) is then estimated with the help of so- called functoids following ideas introduced in K. W. Kaucher and W. L. Miranker [Self-validating numerics for function space problems, Academic Press, New York (1984; Zbl 0548.65028)]and implemented in the present author’s Diplomarbeit at the University of Karlsruhe [Implementierung eines Funktoids und Lösung einiger funktionaler Probleme mit Verifikation]. It is assumed throughout that calculations are done with a computer algebra system. Some numerical results are given for illustrative examples.