ATOM Volume XIV: Sequences and Series

Secondary school students are often familiar with finite arithmetic and geometric series. Those who attempt a more advanced level of study become introduced to infinite series and some formal techniques of their summation. However, many interesting, non-standard, and important examples remain outside of students’ view and experience.

In this book, while maintaining rigorous approach, we use a more intuitive treatment of the topic. We refer to mostly elementary techniques involving solving algebraic inequalities, linear and quadratic equations. We believe that the ideas we explain and illustrate with many examples can be understood at the secondary school level and help to develop a genuine understanding of the topic. An advanced familiarity with the topic may foster a deeper study of mathematics at the university level.

Some of our problems are connected to Euclidean geometry or reveal other links with topics studied at the secondary school level. We also illustrate how infinite sums may appear while solving some word problems that do not explicitly refer to series and convergence. We talk about some practical applications, such as calculations with an approximation. As well, we introduce some notions and objects that are extremely important in modern mathematics, for example, the Riemann zeta function and the Dirichlet kernel. We hope that reading this book and solving the exercises will stimulate students’ interest and fascination with this amazing area of mathematics.