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**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 a_{n} > 0, a_{n} \geq a_{n+1}, **sum**^{oo}_{1}a_{n} = oo. Then for every c > 0, **sum** a^{2}_{n}, **sum**^{oo}_{k = 1}a_{nk(c)} are equiconvergent, where n_{k(c)}(c) is the minimal m such that kc \leq **sum**^{m}_{j = 1}a_{j}.

(2) Suppose a_{n} > 0, **sum** a_{n} = oo. (i) If (a_{n}) has a majorant (b_{n}) in \ell_{2} with b_{n} \geq b_{n+1} for n \geq 1, then there exists a sequence of natural numbers N_{0} = 0, N_{i}\nearrow oo, such that (*) **sum**^{Ni+1}_{j = Ni+1}a_{j} \geq **sum**^{Ni+2}_{j = N_{i+1}+1}a_{j} (i = 0,1,2,...); (ii) If a_{n} \geq a_{n+1} for n \geq 1 then there exists a series **sum** b_{n} having no decomposition (*) and 1/3 < a_{n}/b_{n} < 3.

(3) Suppose a_{n} > 0, **sum** a_{n} = oo. If **sum** a^{2}_{n} < oo then X = ^{def}**{**c: **sum**^{oo}_{k = 1}a_{nk(c)} = oo**}** is of measure zero, and if **sum** a^{2}_{n} = oo, then Y = ^{def}**{**c: **sum**^{oo}_{k = 1}a_{nk(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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag