A Riemann-Type Therem for Segmentally Alternating Series

We show that given any divergent series ∑an∑an with positive terms converging to 0 and any interval [α,β]⊂R¯¯¯¯[α,β]⊂R¯, there are continuum many segmentally alternating sign distributions (ϵn)(ϵn) such that the set of accumulation points of the sequence of the partial sums of the series ∑ϵnan∑ϵnan is exactly the interval [α,β][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.