A Riemann-Type Theorem for Segmentally Alternating Series

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