Paper Title
Growth Rate of Convergent For -Continued Fraction Expansions With Comparison to RCF Expansions

Abstract
Motivated by problems in random continued fraction expansions, we describe the design and implementation on -expansions of a real number, in where . Initially, for such a number , we compute the numerical results for a few samples of -expansions on irrational number by using a Maple software. Throughout this research, we compare and reveal the different ways between -expansions and regular continued fraction (RCF) expansions. Obviously, lies between and for -expansions, while for RCF expansion is always equal to which tend to give a different impact on the growth rate of convergent for these two expansions. We will critically examine the growth rate of both expansions as to observe the value of various affect the performance of their growth rate. Henceforth, we investigate the factors that determine the different impacts on the value of convergent for these expansions. At the end of this research, we found that from the numerical computation on our samples of -expansions, there are a few cases, which are and give the best approximations within the range of . Whereas for RCF expansions, generally of its continued fractions will always be the best approximations and the last value of is definitely equal to the exact value. Index Terms - -expansions, best approximations, convergent, regular continued fraction.