On Certain Reciprocal Sums

TL;DR

This paper introduces a new sequence derived from convergent series of positive real numbers and investigates its properties specifically for the series sum of reciprocals of integers raised to a power k, with k ≥ 2.

Contribution

It defines a novel sequence associated with convergent series and analyzes its behavior for the p-series, providing new insights into their structure.

Findings

The sequence converges for the series with k ≥ 2.

Explicit formulas or properties of the sequence are derived.

Insights into the nature of reciprocal sums for integer powers.

Abstract

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.