Irina Kandayrova

The article demonstrates the research methodology on alternating number series convergence, providing a clear scheme of research. This work covers material sufficient for the section part "Number series" learning, which relates to alternating number series. It is intended for engineering students, however, it will be useful to anyone interested in the theory of series. The article presents recommendations in form of tasks and examples for practical classes preparation, control work, homework, exams. There are also tasks for independent work. The article can be used by teachers conducting practical classes. The author presumes that the reader knows the basic theoretical concepts of infinitely small and large quantities, their comparison, different ways of calculating limits; possesses knowledge necessary for positive number series convergence study.

The theme related to the study of numerical series for convergence is complex and important in the further study of mathematics. Therefore, it is necessary that students could easily and quickly find numerical series convergence signs. In this article on task examples the author presents the application method of Kummer sign from which originally followed well-known sufficient signs of numerical series with positive terms convergence study by d'Alembert, Cauchy, Raabe, Bertrand, Gauss and others. The article material is intended for students of technical specialties and it will be useful to all those who are interested in the theory of series. The article presents recommendations in the form of tasks and examples for exam preparation, practical training, control work and homework. The work material can be used by teachers conducting practical and theoretical classes. The article results from years of teaching experience and it is essentially a plan for practical classes on this subject. The author presumes that the reader possesses the skills of limits calculation, the basic concepts of infinitely small and large quantities theory and their comparison.

To increase the level of mathematical preparation of schoolchildren and university entrants choosing technical and mathematical specialties of universities in school geometry lessons and in the process of preparing for the passing of the USE in mathematics (profile level), special attention should be given to those mathematical questions, without profound knowledge of which successful learning in higher educational institution would be impossible. In this article we give examples of solving problems with parameters. They are offered as increased difficulty problems when passing the USE. One of the differentiated approach possibilities of future students’ preparation is demonstrated who will have to learn the analytical geometry course among other courses. The examined problems show the advantages of graphical solution application with the help of second-order curves. The content of the article may be interesting to teachers, high school students, who are preparing to enter higher educational institutions for technical and mathematical specialties.

The material of the article is intended for students of technical specialties, but it will be useful to anyone who is interested in the theory of improper integrals. It presents recommendations in the form of tasks and examples for exam preparation, practical training, control work, homework. Here you can find the tasks proposed for independent work to consolidate the knowledge. The work material can be used by teachers conducting practical classes. The author assumes that the reader knows the basic concepts of the infinitely small and large quantities theory, their comparison, different ways of limits calculating.

In order to increase the level of mathematical preparation of students of mathematical and technical specialties special attention is paid to the issues of higher mathematics, without a solid knowledge of which it is impossible to become good specialists in their field. Such important topics include the topic of numerical series. Their use is very wide: approximate evaluation of functions and integrals, replacement of complex functions is more straightforward, in particular by polynomials, whose properties are well studied; calculation of the limits, solution of differential equations. The paper deals with the example task to demonstrate the technique of the study of the convergence of numerical series with positive terms and benefits of this technique. The material is designed for students of technical specialties, but will be useful to the one interested in the theory of series. The author presents recommendations in the form of tasks and examples for exam preparation, practical training, control and watershed work, homework. The work material can be used by teachers, leading practical classes. The author assumes that the reader knows the basic concepts of the theory of infinitely small and large quantities, their comparison, different ways of calculating the limits.