All principles and methods in mathematical analysis are based on the theory of real numbers and are taken as a starting point. Therefore, it is important to have a correct knowledge for the set of real numbers in order to have a profound knowledge of the principles and methods of analysis. There is a common point between two sets of numbers, called "ordered fields," in which both operations are defined and the size relation between any two numbers holds in the algebraic sense. However, there is an essential difference between the two sets. It cannot be described simply by algebraic...

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.,...