Regularization of Divergent Series and Tauberian Theorems

The concepts of convergence and divergence, while not defined explicitly until early 19th century, have been studied since the third century BC. Throughout this time, mathematicians have largely focused upon convergent sequences and series, having seemingly little analytical reason to study series that diverged. However, as the area of mathematical analysis developed over the past few centuries, studying properties of divergent series and methods that sum divergent series has produced results that are very useful in the areas of harmonic analysis, number theory, and combinatorics/probability theory. This paper is essentially an investigation of such methods–called summation methods–and includes classical examples, as well as proofs of general properties about these methods. The first section of the paper contains definitions and basic terminology used throughout the paper, as well as a brief commentary on summation methods and the divergent series to which they are applied. Since all the summation methods within this paper will all essentially involve the process of averaging, a definition of weighted averaging is also provided. Within section two of the paper, we present classical examples of summation methods, starting with the identity summation method. Next, the summation method involving standard averaging by arithmetic means, called the Cesàro method, is discussed. After this point, more involved summation methods are discussed at length, including Hölder methods, higher order Cesàro means, and the Euler method. For each of these summation methods, its corresponding collection of averaging weights can be organized as an infinite matrix, and is listed explicitly as such when possible. These matrices are afterwards referred to as “averaging” matrices. In the latter part of this section, we present three theorems pertaining to averaging matrices–the first states that the product of any two averaging matrices will also be an averaging matrix. The second theorem gives a necessary and sufficient condition that ensures a matrix will be “regular”–that is, it will not affect the limit of series that is convergent in