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A series whose partial sums do not have a finite limit of a sequence. They may grow without bound or oscillate. Some divergent series belong to the domain of a summation method, such as Cesàro summation.

A summation method that includes every ordinarily convergent series and assigns it its ordinary sum. The Cesàro theorem for convergent sequences makes first-order Cesàro summation regular. Regularity alone does not justify every rearrangement or multiplication of a divergent series.