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

<math>1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots =\sum_{n=1}^\infty\frac{1}{n}.<math>

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

<math>1 - 1 + 1 - 1 + \cdots<math>

the value 0.

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 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). See abelian and tauberian theorems.

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