A New Type of Continued Fraction Expansion

TL;DR

This paper introduces a novel continued fraction expansion for real numbers within a specific interval, establishes its fundamental properties, and proves its convergence, expanding the theoretical framework of continued fractions.

Contribution

The paper defines a new continued fraction expansion for real numbers and proves its convergence, extending the classical theory to a new class of expansions.

Findings

Defined a new continued fraction expansion for real numbers.

Proved the convergence of this new expansion.

Derived basic properties similar to classical continued fractions.

Abstract

In this paper we define a new type of continued fraction expansion for a real number $x \in I_m:=[0,m-1], m\in N_+, m\geq 2$: \[x = \frac{m^{-b_1(x)}}{\displaystyle 1+\frac{m^{-b_2(x)}}{1+\ddots}}:=[b_1(x), b_2(x), ...]_m. \] Then, we derive the basic properties of this continued fraction expansion, following the same steps as in the case of the regular continued fraction expansion. The main purpose of the paper is to prove the convergence of this type of expansion, i.e. we must show that \[x= \lim_{n\rightarrow\infty}[b_1(x), b_2(x), ..., b_n(x)]_m. \]