In mathematics, a divergent series is a series that does not converge.

If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. The simplest example of a divergent series whose terms do approach zero is the harmonic series

Divergent series can sometimes be assigned a value by using a summability method. For example, Cesàro summation assigns the divergent series

the value .

For convergent series, a good summability method M agrees with the actual limit of the series. Such a result is called an abelian theorem for M, because the prototype was Abel's theorem. More interesting and in general more subtle are partial converse results, called tauberian theorems because of a prototype proved by Alfred Tauber . Here partial converse means that if M sums the series Σ, and some side-condition holds, then Σ was convergent in the first place; without any side condition such a result would say that M only summed convergent series (making it an essentially useless summation method).

It is a result of Banach that there are many, in fact universal summation methods that apply to series of bounded complex terms. This is an application of functional analysis, showing that a suitable linear operator to a space of convergent sequences exists. It is not very constructive. The subject of divergent series, as a domain of mathematical analysis, is primarily concerned with explicit and natural techniques such as Abel summation , Cesàro summation and Borel summation , and their relationships. The advent of Wiener's tauberian theorem marked an epoch in the subject, introducing unexpected connections to Banach algebra methods in Fourier theory .

Reference

Divergent Series by G. H. Hardy