A Riemann-Type Theorem for Segmentally Alternating Series

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.