Are there any known criteria in order for $\sqrt{a_1+\sqrt{a_2-\sqrt{a_3+\sqrt{a_4-\sqrt{a_5+\ldots}}}}}$ to converge?

Question

Question: Let $a_n$ be a positive nonzero integer. Are there any known criteria in order for $$\sqrt{a_1+\sqrt{a_2-\sqrt{a_3+\sqrt{a_4-\sqrt{a_5+\ldots}}}}}$$ to converge ?

The motivation here was the initial question: Should we believe $$\sqrt{2+\sqrt{3-\sqrt{5+\sqrt{7-\sqrt{11+\ldots}}}}}$$ converges ? One could equally be curious about $$\sqrt{2+\sqrt{4-\sqrt{6+\sqrt{8-\sqrt{10+\ldots}}}}}$$ and compare that to the nested radical constant. Also observe $$\sqrt{1+\sqrt{2-\sqrt{3+\sqrt{4-\sqrt{5+\ldots}}}}}$$ appears to be a complex number in stark contrast to the nested radical constant. So I guess we should be concerned knowing if $\sqrt{a_1+\sqrt{a_2-\sqrt{a_3+\sqrt{a_4-\sqrt{a_5+\ldots}}}}}$ is a real number.

Note Vijayaraghavan special case of Herschfeld's theorem on nested radicals. But this does not apply to alternating plus and minus. Click here for Herschfeld's paper on nested radicals

Answer 1

A search for "nested radical" brings up this article in mathworld:

http://mathworld.wolfram.com/NestedRadical.html

It only generally talks about radicals where all the signs are positive, unlike your case.

It mentions a theorem that mignt be relevant:

Herschfeld's convergence theorem:

If $0 < p < 1$ and all $x_i \ge 0$ then $\lim_{n \to \infty} x_0 +(x_1+(x_2+(...x_n^p)^p)^p)^p $ exists if and only if $ (x_n)^{p^n}$ is bounded.

The references are

Herschfeld, A. "On Infinite Radicals." Amer. Math. Monthly 42, 419-429, 1935.

Jones, D. J. "Continued Powers and a Sufficient Condition for Their Convergence." Math. Mag. 68, 387-392, 1995.

Your question involves $p = \frac12$.