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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 391.10038

**Autor: ** Borosh, I.; Chui, C.K.; Erdös, Paul

**Title: ** On changes of signs in infinite series. (In English)

**Source: ** Anal. Math. 4, 3-12 (1978).

**Review: ** The main theorem of this paper is the following: Theorem2: Let **{**a_{n}**}** be a sequence of positive real numbers monotonically decreasing to 0 such that \Sigma a_{n} = oo. Let s_{nj}, n = 1,2,..., j = 0,...,n! -1 be real numbers such that **sum**_{j = 0, J\equiv d(mod (n-1)!)}^{n!-1}s_{nj} = s_{n-1,d'} n = 2,3,..., 0 \leq d \leq (n-1)!-1. Then there exists signs \epsilon(n) = ± 1, n = 1,2,... such that

**sum**_{k = 1, k\equiv j(mod n!)}^{oo}\epsilon(k)a_{k} = s_{nj} for n = 1,2,... and 0 \leq j \leq n!-1. Under the same assumptions on **{**a_{n}**}**, a consequence (Theorem 1) of the above theorem is that there exist signs \epsilon(n) = ± 1, n = 1,2,... such that for every integer m \geq 1 and every integer 0 \leq v \leq m-1,

**sum**_{n\equiv b(mod m)}\epsilon(n)a_{n} = 0. This deduction shows that the result:

**sum**_{n = 1}^{oo}|a_{n}| < oo, A_{m}\equiv**sum**_{n\equiv 0(mod m)}a_{n} = 0 for all m = 1,2,... ==> a_{1} = a_{2} = ... = 0, is sharp when **{**|a_{n}|**}** is monotonic. An interesting consequence of the main theorem its that there is a non-trivial power series \Sigma a_{n}z^{n} which vanishes for every z = e^{2\pi i\theta}, \theta rational. Five interesting problems are also posed by the authors.

**Reviewer: ** M.S.Rangachari

**Classif.: ** * 11B83 Special sequences of integers and polynomials

40A05 Convergence of series and sequences

11B39 Special numbers, etc.

**Keywords: ** changes of signs in infinite series; sequence of positive real numbers

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag