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 {an} be a sequence of positive real numbers monotonically decreasing to 0 such that \Sigma an = oo. Let snj, n = 1,2,..., j = 0,...,n! -1 be real numbers such that

sumj = 0, J\equiv d(mod (n-1)!)n!-1snj = sn-1,d'  n = 2,3,...,  0 \leq d \leq (n-1)!-1.

Then there exists signs \epsilon(n) = ± 1, n = 1,2,... such that

sumk = 1, k\equiv j(mod n!)oo\epsilon(k)ak = snj

for n = 1,2,... and 0 \leq j \leq n!-1. Under the same assumptions on {an}, 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,

sumn\equiv b(mod m)\epsilon(n)an = 0.

This deduction shows that the result:

sumn = 1oo|an| < oo,  Am\equivsumn\equiv 0(mod m)an = 0

for all m = 1,2,... ==> a1 = a2 = ... = 0, is sharp when {|an|} is monotonic. An interesting consequence of the main theorem its that there is a non-trivial power series \Sigma anzn which vanishes for every z = e2\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

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