Infinitely nested radicals In a recent paper it was stated (and maybe proved) that we can solve any polynomial equation with nested radicals.
Here "nested radicals" means expression such as:
$$
\sqrt[n]{a+b\sqrt[n/p]{a+b\sqrt[n/p]{a+b \cdots}}}
$$
i.e. with infinite radicals nested each other.
This means that every algebraic number can be expressed by a sort of ''generalized radicals'' and, since every such number can also be expressed by a series, I've searched if there is some way to transform infinitely nested radicals into series.
Searching on the web I've find nothing interesting, so my question is:

there is some canonical way to transform an infinitely nested radical
  in a series?

 A: 
...canonical way to transform an infinitely nested radical in a series?

There is no such way in general. In fact, general non-periodic nested radicals allow no alternative ways to express them (except for trivial identity transformations, which still lead to a nested radical expression).
If we are talking about periodic nested radicals, connected to polynomial equations, such as in the OP, it might indeed be possible to express their limit as a series (if they converge). But as far as I know, there is no better way to do that, than to consider the original algebraic equation.
The theory of nested radicals still has a long way to grow. There is not much known about them, except for the convergence theorem. I suggest reading this article which gives a very good review on the topic of nested radicals and other such expressions.
Also, there are two papers by Dixon Jones, investigating a more general case of continued powers (which also incorporates infinite series and products, continued fractions, etc):
His papers can be found here: first and second.
Mathworld entry here provides other references and examples.
