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Bulletin of the Iranian Mathematical Society

, Volume 44, Issue 5, pp 1303–1314 | Cite as

A Riemann-Type Theorem for Segmentally Alternating Series

  • Michał Banakiewicz
  • Bruce Hanson
  • Pamela PierceEmail author
  • Franciszek Prus-Wiśniowski
Original Paper

Abstract

We show that given any divergent series \(\,\sum a_n\,\) with positive terms converging to 0 and any interval \(\,[\alpha ,\,\beta ]\subset \overline{\mathbb R}\), there are continuum many segmentally alternating sign distributions \(\,(\epsilon _n)\,\) such that the set of accumulation points of the sequence of the partial sums of the series \(\,\sum \epsilon _na_n\,\) is exactly the interval \(\,[\alpha ,\,\beta ]\). We add some remarks on various segmentations of series with mixed sign terms in order to strengthen a sufficient criterion for convergence of such series.

Keywords

Series segmentation Sign distribution Riemann rearrangement theorem 

Mathematics Subject Classification

Primary 40A05 Secondary 26A99 

References

  1. 1.
    Auerbach, H.: Über die Vorzeichenverteilung in unedlichen Reihen. Studia Math. 2, 228–230 (1930)CrossRefGoogle Scholar
  2. 2.
    Knopp, K.: Theory and Application of Infinite Series. Dover Publications Inc., New York (1990)zbMATHGoogle Scholar
  3. 3.
    Schramm, M., Troutman, J., Waterman, D.: Segmentally alternating series. Am. Math. Mon. 121, 717–722 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Michał Banakiewicz
    • 1
  • Bruce Hanson
    • 2
  • Pamela Pierce
    • 3
    Email author
  • Franciszek Prus-Wiśniowski
    • 4
  1. 1.Studium of MathematicsWest Pomeranian University of TechnologySzczecinPoland
  2. 2.Department of Mathematics, Statistics, and Computer ScienceSt. Olaf CollegeNorthfieldUSA
  3. 3.Department of Mathematics and Computer ScienceThe College of WoosterWoosterUSA
  4. 4.Instytut MatematykiUniwersytet SzczecińskiSzczecinPoland

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