Theory of Series

In analysis, series occupies a special place and is used in many fields not only in analytical science but also in various subjects of mathematics, including differential equations, computational mathematics, and in applied sciences, including signal processing, and mathematical statistics.Addition is not only the first operation introduced by mankind, but also the most used because it is very easy to understand and clear in its representation. Of course, finite additions that generalize two numbers of additions are distinct in their representation and meaning, but infinite additions, i.e., series, have long been frontiers for mathematicians because of their opacity in meaning and representation. Many mathematicians have explored this frontier but have been traps in mathematics because of the variety of ways in which different arguments and approaches to deal with them. The introduction of convergence in series has become a key starting point for exploring this frontier. Starting from this starting point, this book is written for the purpose of summarizing series theory and the relation with continuity, differentiation and integration, which are the main instruments in analysis.