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October 26 Math Colloquium

Georges Grekos, Université de Saint-Etienne

Convergence of real sequences and series according to an ideal (filter)

In Analysis, one says that a real sequence (an) tends to a if and only if, for any Ε > 0, the set of indices n such that |an −a| > Ε is finite. Changing finite into negligible gives interesting generalizations. In the ’50ies, some authors replaced finite by of zero density. Here, at first approach, density is synonymous to probability. This gave rise to the so called statistical convergence. The family of sets of zero density is an “ideal” (-hereditary family, closed under finite unions). It is in fact an ideal, as defined in Algebra, in the ring (P(N),Δ,∩). Its dual concept is that of filter. The filter associated to an ideal I is the family of complementary sets of all sets belonging to the ideal. Thus a filter is a family closed under superset and finite intersection. I shall use in this talk general ideals (filters) and present:

1) A published work [Math. Bohemica 141 (2016), 483-494] on ideal generalizations of Olivier’s theorem. This theorem appeared in 1827 and says that if the sequence (an) is monotone and the series Σn≥1 an an convergent, then nan tends to 0 as n goes to infinity.

2) A work in progress studying different possible definitions of the Cauchy convergence of a series according to an ideal.

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