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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]

We use freshman calculus to prove thatfor θ ∈ (0, π) andn= 0, 1, 2, … ; in particular, we obtain a simple unified method for evaluating the following infinite series: [ABSTRACT FROM PUBLISHER]

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her own variations. [ABSTRACT FROM PUBLISHER]

In this note, a class of integrals on the unit hypercube are considered and series solutions presented. The material is at a level appropriate for use with undergraduates, either as part of a course on multivariable calculus, or as a short independent research project. Suitable exercises for use in these contexts are given. [ABSTRACT FROM AUTHOR]

The formula for the Laplace transform of an exponential function, , can be derived with equal ease if the parameters a and s are complex numbers. This leads to formulas for the Laplace transforms of eatsin (bt) and eatcos (bt) (where a and b are complex) and to calculations of certain inverse Laplace transforms without the need to consider Laplace transforms of derivatives or convolutions. Simpler proofs also follow from coupling the formula , for which a simple proof is given, with the fact that the operator commutes with certain familiar infinite series. [ABSTRACT FROM AUTHOR]

We compute closed forms for some parametric integrals. The tools used are MacLaurin developments and other infinite series. Finally, Stirling numbers appear. [ABSTRACT FROM PUBLISHER]

The study we report here examines parts of what Chevallard calls the institutional dimension of the students' learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duval's Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concept's significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests. [ABSTRACT FROM AUTHOR]

In this note, we construct infinite series using M-bonacci numbers in a manner similar to that used in previous studies and investigate the convergence of the series to an integer. Our results generalize the ones obtained for Fibonacci numbers. [ABSTRACT FROM AUTHOR]

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals. [ABSTRACT FROM AUTHOR]

This note could find classroom use in an introductory course on complex analysis. Using some of the most significant theorems from complex analysis, the main result provides a simple method for transforming many elementary functions (defined on the complex plane) into everywhere continuous functions that are differentiable only on a nowhere dense set. Accordingly, such continuous functions are termed ‘practically nowhere differentiable’. The twofold pedagogical value of this method is that (1) students can readily generate examples of everywhere continuous, practically nowhere differentiable functions that do not require any direct appeal to infinite series, and (2) the often dynamical difference between the behaviour of functions of a complex variable and functions of a real variable is showcased. [ABSTRACT FROM AUTHOR]

A class of infinite series is evaluated with the aid of Taylor's theorem and a comparison is made with other methods. [ABSTRACT FROM AUTHOR]

An interesting class of infinite series and definite integrals are evaluated with the aid of Taylor's theorem. [ABSTRACT FROM AUTHOR]

One of the most astonishing properties when studying numerical series is that the sum is not commutative, that is the sum may change when the order of its elements is altered. In this note an example is given of such a series. A well-known mathematical proof is given and a MATLAB© program used for different rearrangements of the series converging to different numbers. It is noted that the code follows the mathematical idea of the proof of the property. Developing programs showing the main ideas in the proofs of the theorems may be a good tool in the understanding of such theorems and proofs. [ABSTRACT FROM AUTHOR]

Simple examples are really encouraging in the understanding of rearrangements of infinite series, since many texts and teachers provide only existence theorems. In the absence of examples, an existence theorem is just a statement and lends little confidence to understanding. Iterated sums of double series seem to have a similar spirit of rearrangements of series though they are not the same. Therefore learning about iterated sums of double series will help the understanding of the rearrangements of series. Without simple examples, which is the case with many texts, theorems about iterated sums do not contribute much to our understanding. Many students and perhaps some teachers would be discouraged by this in the learning process. In this note, several inspiring examples are presented to show that infinite sums indeed behave very differently from finite sums. [ABSTRACT FROM AUTHOR]

A number of examples of infinite series are given whose sum can be expressed in closed form, sometimes as an integral. This leads naturally to the determination of a closed form for the sum of a less well-known series in terms of an interesting definite integral. Other examples are included, as well as suggestions for further work. [ABSTRACT FROM AUTHOR]

There are many nice problems in the theory of geometric series. The famous problem where atiny bird flies repeatedly between two trains approaching each other on the same track from opposite ends is analysed in this paper. While the common sense solution is very quick and clever, there are very nice things that students in a calculus course can learn by taking the infinite series route. [ABSTRACT FROM AUTHOR]

Dicusses certain finite and infinite series which involve Fibonacci numbers in some way. Determination of approaches to obtaining the sum of the series; Suggestion that readers seek other proofs for the sums; Provision of generalizations for both types of series.

Focuses on Abel's test which is used for testing the convergence of an infinite series. Incorporation of the derivative form of the integral test with Abel's test; Use of Maple, a computer algebra system to make conjectures on the conditions of theorems; Determination of the convergence of a series.

Presents some new ways of using a spreadsheet in teaching foundations of the theory of infinite series. Use of the spreadsheet's ability to model inequalities of the Bolzano-Cauchy type; Provision of students with a kind of visual imagery of the process of convergence or divergence.

In this note it is shown how closed sums of infinite series may be used to construct efficient economized formulas for microstrip electrical impedances. [ABSTRACT FROM AUTHOR]

There are many general techniques useful for summing a variety of infinite series and even more, less general, manipulations that apply tobut afew specific series. The method shown in this note is one that falls in the latter category. It is provided in the spirit of expanding the arsenal of methods that might be tried. The author has found no reference to the technique illustrated. [ABSTRACT FROM AUTHOR]