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.