## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  648.40001
Autor:  Erdös, Paul; Joó, I.; Székely, L.A.
Title:  Some remarks on infinite series. (In English)
Source:  Stud. Sci. Math. Hung. 22, No.1-4, 395-400 (1987).
Review:  The authors prove four theorems.
(1) Suppose an > 0, an \geq an+1, sumoo1an = oo. Then for every c > 0, sum a2n, sumook = 1ank(c) are equiconvergent, where nk(c)(c) is the minimal m such that kc \leq summj = 1aj.
(2) Suppose an > 0, sum an = oo. (i) If (an) has a majorant (bn) in \ell2 with bn \geq bn+1 for n \geq 1, then there exists a sequence of natural numbers N0 = 0, Ni\nearrow oo, such that (*)  sumNi+1j = Ni+1aj \geq sumNi+2j = N_{i+1+1}aj (i = 0,1,2,...); (ii) If an \geq an+1 for n \geq 1 then there exists a series sum bn having no decomposition (*) and 1/3 < an/bn < 3.
(3) Suppose an > 0, sum an = oo. If sum a2n < oo then X = def{c: sumook = 1ank(c) = oo} is of measure zero, and if sum a2n = oo, then Y = def{c: sumook = 1ank(c) < oo} is meagre (i.e. of first category).
(4) X can be residual, and Y can be of cardinality continuum.
Reviewer:  B.Crstici
Classif.:  * 40A05 Convergence of series and sequences
Keywords:  decomposition of series

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