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 $$\tau $$ τ . The theory of such sequences can be developed by transforming them into the new time scale $${\mathbb {T}}=\tau ({\mathbb {N}})$$ T = τ ( 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.

中文翻译：

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Kummer 检验和常规变异

我们建立了 Kummer 检验、离散规则变化的某些概括和时间尺度上的规则变化之间的关系。更准确地说，我们对检测级数收敛的 Kummer 检验给出了新的解释（包括新的证明）。我们表明，Kummer 检验的极限关系可以根据最近引入的关于辅助序列 $$\tau $$ τ 的细化规则变化序列的概念来重写。可以通过将它们转换为新的时间尺度 $${\mathbb {T}}=\tau ({\mathbb {N}})$$ T = τ ( N ) 来发展这些序列的理论，然后使我们能够将现有结果用于在时间尺度上有规律地变化的函数。将这句话替换为“特别是，