Suppose an arbitrary transcendental equation like $$\phi(u,z) = 0 \quad .$$ To find "analytically" the solution $u=u(z)$ that satisfies this I consider an expansion around small $z$ and produce a series expansion.
First inquiry: Do transcendental equations always allow some kind of expansion?
Second inquiry: The series produced by a transcendental equation is always convergent or can it be divergent?
I am interested to know some examples where a transcendental equation produces a divergent series (zero radii of convergence) or if there is some theorem that prohibits this.
I am asking this because I looked at two simple scenarios where the series has a finite radius of convergence and I am curious whether this is a general thing.
The scenarios I looked where $$ u e^u = z,$$ which produces a convergent series around $z=0$ and we also know the exact answer as the Lambert W function, $u = W(z)$.
And also, $$ -z e^u + \ln u = 0 \Rightarrow e^{-ze^u} u =1$$ which does not produce a named function but can be represented either by a convergent series around $z=0$ or by the continued exponential $$ u = e^{ze^{ze^{ze^{...}}}}$$