Boundedness, Monotonicity, and Convergence of Square and Cubic Radical Sequences

Abstract

This paper provides an in-depth insight into basic operations with inequalities that are carefully crafted for the illustration of monotonicity and boundedness characterization of the general recursive radical sequence. Essentially such properties are necessary and sufficient in the proof of their convergence. Furthermore, other hybrid forms of this sequence type are extensively explored taking into consideration the leading term to appropriately obtain well-defined intervals, where convergence is permitted, using simple but highly effective and concise short proves without any recourse to the application of other mathematical techniques and particularly in general the use of prove by mathematical inductions which are best tailored for recursive sequences involving the square root. In furtherance to this approach, the paper gave an illustrative prove by showing an extension of the method to cubic radicals which are also recursively defined as a sequence of real numbers in terms of its leading term.