  Several applications of Abel's partial summation formula to the convergence
of series of positive vectors are presented. For example, when the norm of the
ambient ordered Banach space is associated to a strong order unit, it is shown
that the convergence of the series $\sum x_{n}$ implies the convergence in
density of the sequence $(nx_{n})_{n}$ to 0. This is done by extending the
Koopman-von Neumann characterization of convergence in density. Also included
is a new proof of the Jensen-Steffensen inequality based on Abel's partial
summation formula and a trace analogue of Tomi\'{c}-Weyl inequality of
submajorization.
