Divergent Series

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Related to Summation methods: Summability, Convergent series

divergent series

[də′vər·jənt ′sir·ēz]
(mathematics)
An infinite series whose sequence of partial sums does not converge.

Divergent Series

 

a series in which the sequence of partial sums does not have a finite limit. If the general term of the series does not tend to zero, the series diverges, for example, 1 - 1 + 1 - 1 + … + (– 1)n-1. The harmonic series 1 + 1/2 + … + 1/2 + … is an example of a divergent series whose general term tends to zero. There exist numerous classes of divergent series that converge in some generalized sense, since to each such divergent series some “generalized sum” may be assigned that possesses the most important properties of the sum of a convergent series.

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Finally we remark that a symmetric summation method in the time and frequency domain can be defined by