In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit. 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. However, convergence is a stronger condition: not all series whose terms approach zero converge. The simplest counter example is the harmonic series A summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is called an abelian theorem for M, from the prototypical Abel's theorem. More interesting and in general more subtle are partial converse results, called tauberian theorems, from a prototype proved by Alfred Tauber. Here partial converse means that if M sums the series ?, and some side-condition holds, then ? was convergent in the first place; without any side condition such a result would say that M only summed convergent series (making it useless as a summation method for divergent series).

Данное издание не является оригинальным. Книга печатается по технологии принт-он-деманд после получения заказа.