Iterated Radical Expansions and Convergence

We treat three recurrences involving square roots, the first of which arises from an infinite simple radical expansion for the Golden mean, whose precise convergence rate was made famous by Richard Bruce Paris in 1987. A never-before-seen proof of an important formula is given. The other recurrences are non-exponential yet equally interesting. Asymptotic series developed for each of these two examples feature a constant, dependent on the initial condition but otherwise intrinsic to the function at hand.