ABSTRACT
From a naive point of view, all mathematics can be considered as an intriguing interplay between finite and infinite collections of objects or, if one prefers, between various discrete and continuous structures. This chapter deals with the summation methods and Lebesgue nonmeasurable functions. There are standard operations over series, which are useful in various topics of mathematical analysis (for instance, the product of two series). Also, many standard transformations of a series are known, the main purpose of which is to make the behavior of the series much better in the sense of its convergence. Among such transformations, Abel’s transformation can be treated as a discrete version of integration by parts. In addition, Euler’s transformation for so-called alternating series, Kummer’s transformation, Markov’s transformation, and others are frequently useful for improvement of convergence. The chapter also discusses briefly several set-theoretical aspects of summation theory.
