Divergent Series

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Related to Summability theory: 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.

References in periodicals archive ?
As we have seen in Feichtinger and Weisz [7, 8], it plays an important rule in summability theory, too.
Over the years, and under different names, statistical convergence has been discussed in number theory [25], trigonometric series [36], and summability theory [15,17].
Harder and Hicks [11], Rhoades [29, 31], Osilike [26] and Singh et al [37] used the method of the summability theory of infinite matrices to prove various stability results for certain contractive definitions.
Shah, Summability Theory and Applications, Prentice-Hall of India, New Delhi, 1988;
Later on it was further investigated from sequence space point of view and linked with summability theory by several authors.
Later on it was further investigated from sequence point of view and linked with the summability theory by Fridy [8] and many others.
In Feichtinger and Weisz [10] we applied the homogeneous Herz spaces in summability theory. [[??].sub.q]([[??].sup.d]) contains all measurable functions f for which
Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy (1985), Salat (1980), Connor (1999) and many others.