| Infinite series are important in science and mathematics because many functions either arise most naturally in the form of infinite series, or have infinite series representations that are useful for numerical computations. In addition, known infinite series and their characteristics, including their convergence or divergence, are frequently used in proofs of mathematical theorems, particularly in number theory. Historically, developing the definitions and structure of infinite series has played a key role in the rigorization of the calculus. In fact, attempts made at rigorization of the calculus that did not include consideration of infinite series and convergence theory have met failure. Paradoxes that have involved infinite series have motivated mathematicians in directions that have become very important areas in analysis. Infinite series are therefore a worthwhile study, along with their characteristics, the most basic of which is convergence or divergence. This is the topic to be addressed in this paper.;Chapter 1 of this paper will introduce the reader to series in general, Chapter 2 will discuss the different types of series, then Chapter 3 will give formal definitions for infinite sequences and series as well define convergence and divergence. Next, Chapter 4 will demonstrate several well-known convergence and divergence tests in their present-day form, Chapter 5 and 6 will follow the historical development of infinite series and convergence theory, and Chapter 7 will be a conclusion and a look at current-day views on convergence theory and its importance. Sources used in this thesis are books and journal articles on series, real analysis, convergence theory, and history of mathematics. |
