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The Ratio Test and the Root Test for absolute convergence/divergence of series of numbers $ \sum _{n=0}^{\infty }a_n $ ∑n=0∞an are frequently discussed and proved independently in Calculus courses. The Root Test is stronger (verifies convergence for more series) than the Ratio Test. This relation inspires introduction of some intermediate strength tests (stronger than the Ratio Test and weaker than the Root Test) that we call Power Mean Tests (they, in particular, include the Arithmetic Mean Test). We show the connection between the Root, the Power Mean and the Ratio Tests. We also note that all these tests are related to the test that we formulate and call Generalised f-mean Test (or Kolmogorov–Nagumo–de Finetti mean Test). We provide an example of an infinite series, where the Arithmetic Mean test is the test that should be used for convergence verification, because the Root and the Ratio Tests are not easy to apply. We conclude this work with the statement emphasising why it is ‘fair’ to include the summarising Corollary 3.1 in our Calculus course. [ABSTRACT FROM AUTHOR]

A well-known puzzle involving the repeated flying of a swallow between two cyclists is extended to investigate the effect of nonuniform speeds. The resulting generalizations provide examples that can be used in the context of the teaching aspects of the manipulation of infinite series, including the use of telescoping sums and products. [ABSTRACT FROM AUTHOR]

The purpose of the study was to explore students' consistency in their notions encountering different methods in solving the same task of comparing values of certain infinite series. The participants were 93 male students aged 16–17. Applying cognitive conflict approach and the data were collected through a questionnaire administrated in two stages. In the first stage, students were exposed to a challenging question involving comparison of two infinite series. Then, they encountered all types of various responses consisting of different results provided by the participants, and they were asked to perform the same task again. Analysis of the data, focusing on the changes provided revealed two important points: (a) the students were generally consistent in their notions, even in their incorrect answers, and (b) the students' intuitive notions about infinite series were more complicated than those about infinite sets and sequences. [ABSTRACT FROM AUTHOR]