If this series $S_x$ converges for some $x$, can we find its closed form expression?
$$ S_x=\sqrt{x +\sqrt{x^2+\sqrt{x^3+\sqrt{x^4+\cdots}}}} $$
For $x=1$, it's easy and we have,
$$ S_1 = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}} \implies S_1^2=S_1+1 \implies S_1=\frac{1+\sqrt5}{2}$$
But I can't solve in an expression of $x$ for general case.
If the closed form is not known, can we prove that it does not exist ?