A note on series and sequences (Q1093105)

A classical theorem by Pringsheim asserts that if \((a_ n)\) is nonnegative, nonincreasing and if \(\sum a_ n\) converges, then \(na_ n\to 0\), as \(n\to \infty\). In this note, some extensions and improvements of this result are considered which are then used to derive some properties of hyperconvex sequences analogous to those for convex sequences.