A note on series with recursively defined terms

TL;DR

This paper investigates the convergence properties of infinite series with terms defined recursively, emphasizing the importance of the derivative of the defining function at zero and introducing a comparison test applicable even without differentiability.

Contribution

It introduces a new comparison test for recursively defined series that works without requiring the differentiability of the defining function.

Findings

The derivative at zero influences series convergence.

A comparison test applicable to non-differentiable functions.

Insights into the role of the defining function's properties.

Abstract

In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even if the defining function of the series is not differentiable.