# A convergence theorem for continued fractions of the form $K_{n=1}^{\infty} a_{n}/1$

## TL;DR

This paper establishes a convergence theorem for a specific class of continued fractions, providing conditions under which their odd and even parts converge to the same limit, with illustrative examples.

## Contribution

It introduces new convergence conditions for continued fractions of the form $K_{n=1}^{\infty} a_{n}/1$, ensuring their odd and even parts converge to the same value.

## Key findings

- Derived explicit conditions for convergence of continued fractions
- Proved that under these conditions, odd and even parts converge to the same limit
- Provided examples illustrating the application of the theorem

## Abstract

In this paper we present a convergence theorem for continued fractions of the form $K_{n=1}^{\infty}a_{n}/1$. By deriving conditions on the $a_{n}$ which ensure that the odd and even parts of $K_{n=1}^{\infty}a_{n}/1$ converge, these same conditions also ensure that they converge to the same limit. Examples will be given.