Elementary methods for evaluating Jordan's sums and analogous Euler's type sums and for setting a sigma sum theorem

TL;DR

This paper provides elementary proofs for Jordan's sums and explores Euler's sums, including a new formula for series with equal weight, enhancing understanding of these classical mathematical series.

Contribution

It introduces a complete elementary proof for Jordan's sums and derives a novel formula for Euler-type sums with equal weights.

Findings

Elementary proofs for Jordan's sums established.

New formula for Euler's sums with same weight derived.

Enhanced understanding of classical series relationships.

Abstract

We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.