This post concerns a radical I included in an earlier post that didn't receive much attention, and I hypothesize that this radical has quite an interesting closed form, therefore I'm posting this.
(This, visually, is like the product equivalent of this radical)
I will denote it $R$ and is defined as follows
$$R = x\sqrt{x\sqrt{\sqrt{x\sqrt{\sqrt{\sqrt{x...}}}}}}$$ Evaluating each term's powers we can see that $$R = x\cdot x^{2^{-1}}\cdot x^{2^{-3}}\cdot x^{2^{-6}}\cdot x^{2^{-10}}...$$ The powers of 2 follow the sequence of triangle numbers and therefore, $$R = x\prod_{n=1}^{\infty}x^{2^{-\frac{n(n+1)}{2}}}$$ $$ = x\cdot x^{\sum_{n=1}^{\infty}2^{-\frac{n(n+1)}{2}}}$$ The exponent of $x$ is the sum $$1+ \sum_{n=1}^{\infty}2^{-\frac{n(n+1)}{2}}$$ which seems like it has some identity to elliptic functions similar to how the q-Pochhammer symbol includes the pentagonal numbers for its sum. I don't know how to evaluate this sum further. Any help or solutions would be greatly appreciated