This contribution, which is a follow-up to author's papers [3] and [4], deals with the series of reciprocals of the quadratic polynomials with double positive integer root. The formula for the sum of this kind of series expressed by means of harmonic numbers are derived and verified by several examples evaluated using the basic programming language of the computer algebra system Maple 16. There is stated another formula using generalized harmonic numbers, too. This contribution can be an inspiration for teachers of mathematics who are teaching the topic Infinite series or as a subject matter for work with talented students.
In this contribution we deal with the sum of the series of reciprocals of the cubic polynomials with different positive integer roots. We derive the formula for the sum of these series and verify it by some examples using the basic programming language of the computer algebra system Maple 16. This paper can be an inspiration for teachers who are teaching the topic Infinite series or as a subject matter for work with talented students.
This contribution is a follow-up to author's previous published papers and deals with the sum of the series of reciprocals of the cubic polynomials with one zero and double non-zero integer root. We derive the formula for the sum of these series and verify it by some examples using the basic programming language of the computer algebra system Maple 16. This contribution can be an inspiration for teachers who are teaching the topic Infinite series or as a subject matter for work with talented students.
This paper is inspired by a text of the book [7] ("Úvod do algebry" in Czech, "Introduction to Algebra" in English) of the authors Ladislav Kosmák and Radovan Potůček. They followed the great work of Professor Otakar Borůvka in the field of the partition theory, groupoids and groups and gave them in the context to contemporary modern algebra. Academian Borůvka have deduced and proved many results concerning the partition theory in his publications.His first works [1] and [2] were published during World War II and his monographs [3] and [4] were released in the post-war years.In this paper we deal with a construction of the smallest common coarser of two set partitions associated with equivalence relations, we give a special relation used in the construction and an illustration of blocks of this coarser.Remark 1. Algebra of binary relations was worked up by Jacques Riguet in 1948 (see [8], the translation into Russian has its origin from 1963). Professor Borůvka was rightfully proud himself on his theory of partitions and an appliance of the algebra of relations has never used. From twenty-seven chapters of the book [4] is six devoted to the general theory of partitions and in the chapter 14 and 21 -24 are paragraphs dealing with partitions in the theory of grupoids and groups.
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