On the Convergence of the Catalan Generating Function

TL;DR

This paper proves the convergence of the Catalan number generating function directly from its recursion formula, addressing a problem posed by Marshal Hall.

Contribution

It provides a direct proof of convergence for the Catalan generating function based on its recursion, which was previously unresolved.

Findings

Confirmed convergence of the Catalan generating function

Provided a new proof method based on recursion

Addresses a longstanding problem by Marshal Hall

Abstract

We solve a problem of Marshal Hall by proving the convergence of the catalan number generating function directly on the basis of its recursion formula.