# A note on Abel's partial summation formula

## TL;DR

This paper explores applications of Abel's partial summation formula in analyzing series convergence in ordered Banach spaces, providing new proofs and extending classical results like the Jensen-Steffensen inequality.

## Contribution

It introduces novel applications of Abel's formula to convergence in density and offers new proofs of classical inequalities within the framework of ordered Banach spaces.

## Key findings

- Convergence of series implies density convergence of scaled sequences in certain Banach spaces.
- New proof of Jensen-Steffensen inequality using Abel's partial summation.
- Extension of Tomic-Weyl inequality to trace and submajorization contexts.

## Abstract

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.