Iterated Radical Expansions and Convergence
S Finch�- arXiv preprint arXiv:2410.02114, 2024 - arxiv.org
arXiv preprint arXiv:2410.02114, 2024•arxiv.org
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.
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.
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.
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