The behaviours of convergents in theta-expansions: computational insights based on theta-expansions algorithm using the Maple software

Continued fractions arise naturally in long division and the theory of the approximation to real numbers by rational numbers.  This research considered the implementation on the convergent of $\theta$-expansions of real numbers of $x\in(0,\theta)$ with $0<\theta<1$.  The convergent of $\theta$-expansions are also called as $\theta$-convergent of continued fraction expansions.  This study aimed to establish the properties for a family of $\theta$-convergent in $\theta$-expansions.  The idea of discovering the behaviours of $\theta$-convergent evolved from the concept of regular continued fraction (RCF) expansion and sequences involved in $\theta$-expansions.  The $\theta$-expansions algorithm was used to compute the values of $\theta$-convergent with the help of Maple software.  Consequently, it proved to be an efficient method for fast computer implementation.  The growth rate of $\theta$-convergent was investigated to highlight the performance of $\theta$-convergent.  The analysis on $\theta$-convergent revealed the convergent that gives a better approximation yielding to fewer convergence errors.  This whole paper thoroughly derived the behaviours of $\theta$-convergent, which measure how well a number $x$ is approximated by its convergents for almost all irrational numbers.