Real Analysis – Convergence of Nested Radicals

Question

In general if we have $$\sqrt{a_0+\sqrt{a_1+\sqrt{a_2….}}}$$
Are there easy ways of finding out if it converges, like from infinite products and series? Are there ways for converting the question of convergence of the radical to a one about series, like the logarithm does for products?

Answer 1

According to the Herschfelds Convergence theorem, the sequence converges iff $a_n^{2^{-n}}$ converges, if $a_i>0 $. We can prove this by noting that, $$\sqrt{a_0+\sqrt{a_1+ \cdots + \sqrt{a_n}}}>{a_n^{2^{-n}}}$$ and also if $a_n<M^{2^{n}}$ then, $$\sqrt{a_0+\sqrt{a_1+ \cdots + \sqrt{a_n}}}<\sqrt{M}\sqrt{1+\sqrt{1 + \cdots}}$$ which is known to converge, and the sequence is non decreasing, showing that it converges. Much thanks to muzzlator for pointing out the theorem.