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A Riemann-type theorem for segmentally alternating series. (English) Zbl 1420.40001

Summary: 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.

MSC:

40A05 Convergence and divergence of series and sequences
26A99 Functions of one variable
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[1] Auerbach, H.: Über die Vorzeichenverteilung in unedlichen Reihen. Studia Math. 2, 228-230 (1930) · JFM 56.0200.02 · doi:10.4064/sm-2-1-228-230
[2] Knopp, K.: Theory and Application of Infinite Series. Dover Publications Inc., New York (1990) · JFM 54.0222.09
[3] Schramm, M., Troutman, J., Waterman, D.: Segmentally alternating series. Am. Math. Mon. 121, 717-722 (2014) · Zbl 1322.40001 · doi:10.4169/amer.math.monthly.121.08.717
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