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