Search

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]

In this paper we discuss the important Abel's summation formula, which is a very powerful tool for analysing series of real or complex numbers. We derive from it an integral test which may be useful in cases where the classical integral test may not be applied. We also discuss how this new integral test may be used when one is dealing with alternating series of real or complex numbers. The main goal of this note is to provide a supplementary material for courses dealing with infinite series of real or complex numbers. [ABSTRACT FROM AUTHOR]

In this note we introduce an infinite series which represents an interesting challenge for students with the relevant background. [ABSTRACT FROM AUTHOR]

We present an alternative and pedagogical method to calculate the centre of mass of homogeneous triangular plates by using scaling, symmetry and geometric infinite series. This work also aims to better understand problems that involve concepts of centre of mass of discrete and continuous systems. [ABSTRACT FROM PUBLISHER]

This study aims to broadly examine how commonly various conceptualizations of the definite integral are drawn on by students as they attempt to explain the meaning of integral expressions. Previous studies have shown that certain conceptualizations, such as the area under a curve or the values of an anti-derivative, may be less productive in making sense of contextualized integrals. On the other hand, interpreting the integral using Riemann sum-based conceptions proves much more productive for understanding contextualized integrals. This study investigates how frequently students from a US calculus population drew on these three conceptualizations (as well as others) to interpret the meaning of definite integrals. The results were achieved by asking a large sample of students from two US colleges (n = 150) four open-ended questions regarding the underlying meaning of definite integrals. Data from the student responses show a high prevalence of area and anti-derivative ideas and a relatively low occurrence of multiplicatively based summation ideas for interpreting these integrals. Possible reasons for and implications of the results are discussed. [ABSTRACT FROM AUTHOR]

We use a trigonometric identity and integral calculus to evaluate several infinite series; in particular, we deduce the corresponding partial fraction decomposition of,,and csch x. [ABSTRACT FROM PUBLISHER]

We use a trigonometric identity and elementary calculus to obtain simple proofs of the following identities: and. [ABSTRACT FROM AUTHOR]