Infinite Sequences and Series

Up to this point in the proposed new perspective for understanding “what is calculus?” the domain associated with both integration and differentiation has mostly been confined to continuous functions. As a concluding chapter of this opus, the focus is directed to a discussion of discrete variables with an examination of the domain of sequences and series; then a re-definition of important functions, in particular trigonometric and exponential functions, in term of infinite series, and a broad look at the concept of infinity as both a cardinal and an ordinal number.

This chapter begins by defining the concept of sequences and both the mathematical limitations and the heuristic expectations that are fundamental to a quantitative, as well as a qualitative, development of the question “is the sequence of counting numbers unending?” and the related question “if there is such a “last” number, to which the name “infinity” has been given, what are its properties?” In the preceding chapters one observed that infinite concepts applied not only to being “infinitely large”, but also to being “infinitely small”. To this latter category the term “infinitesimal” was applied. In this chapter, the further concept, referred to as different “orders” of infinity, will be encountered. Emphasis will be placed on a concept that this author prefers to associate with the heuristic of being “infinitely dense”, in contradistinction to one of being “infinitely large”.