Kummer test and regular variation

Abstract

We establish relations among the Kummer test, certain generalization of discrete regular variation, and regular variation on time scales. More precisely, we give a new interpretation (including new proof) of the Kummer test which detects convergence of series. We show that the limit relation from the Kummer test can be rewritten in terms of recently introduced concept of refined regularly varying sequences with respect to an auxiliary sequence $$\tau $$. The theory of such sequences can be developed by transforming them into the new time scale $${\mathbb {T}}=\tau ({\mathbb {N}})$$, which then enables us to utilize the existing results for regularly varying functions on time scales. Replace this sentence by “In particular, the Karamata type theorem and the representation for refined regularly varying sequences yield not only the Kummer test, but provide also asymptotic formulae for the partial sums of series and the representation for the sequences satisfying the Kummer test.