manuscripta mathematica

, Volume 17, Issue 3, pp 221–226 | Cite as

On three problems for sequences

  • D. Landers
  • L. Rogge


If ξ∈ (0,1) and A=an, nɛℕ is a sequence of real numbers define Sn(ξ,A)∶=Σ{ak∶:k=[nξ]+1 to n}, nɛℕ, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence Sn (ξ,A), nɛℕ, for all ξ in an appropriate set K of real numbers, that the sequence an, nɛℕ, converges to zero. It was shown that such a conclusion is possible if K={ξ,1−ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:
  1. 1)

    does the convergence of Sn (ξ,A), nɛℕ, for a single irrational number ξ imply an→0.

  2. 2)

    does the convergence of Sn(ξ,A), nɛℕ, for finitely many rational numbers ξ∈ (0, 1) imply an→0.

  3. 3)

    does the convergence of Sn (ξ,A), nɛℕ, for all rational numbers ξ∈ (0,1) imply an→0?



Real Number Number Theory Algebraic Geometry Rational Number Topological Group 
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  1. [1]
    BOJANIC, R. and SENETA, E.: A unified theory of regularly varying sequences. Mathematische Zeitschrift. To appear.Google Scholar
  2. [2]
    HIGGINS, R.: A note on a problem in the theory of sequences. Elemente d. Math. (1974), 37–39.Google Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • D. Landers
    • 1
  • L. Rogge
    • 2
  1. 1.Mathematisches Institutder Universität Köln5 Köln 41
  2. 2.Fachbereich StatistikUniversität Konstanz775 Konstanz 16

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