A Riemann-Type Theorem for Segmentally Alternating Series
series segmentation, sign distribution, riemann rearrangement theorem
We show that given any divergent series ∑𝑎𝑛 with positive terms converging to 0 and any interval [𝛼,𝛽]⊂ℝ⎯⎯⎯⎯⎯ , there are continuum many segmentally alternating sign distributions (𝜖𝑛) such that the set of accumulation points of the sequence of the partial sums of the series ∑𝜖𝑛𝑎𝑛 is exactly the interval [𝛼,𝛽] . 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.
Banakiewicz, Michał; Hanson, Bruce; Pierce, Pamela B.; and Prus-Wiśniowski, Franciszek, "A Riemann-Type Theorem for Segmentally Alternating Series" (2018). Bulletin of the Iranian Mathematical Society, 44(5), 1303-1314. 10.1007/s41980-018-0092-z. Retrieved from https://openworks.wooster.edu/facpub/236