A note on Abel's partial summation formula

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.