A note on Abel’s partial summation formula

摘要

AbstractSeveral 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 with a strong order unit, it is shown that the convergence of the series$$sum x_{n}$$∑xnimplies the convergence in density of the sequence$$(nx_{n})_{n}$$(nxn)nto 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 the Tomi?–Weyl inequality of submajorization.]]>