Is there a generalization of the fundamental theorem of algebra for power series?
One generalization of the fundamental theorem of algebra to entire functions is given by the Little Picard theorem, which can be phrased as follows:
If $f$ is a non-constant entire function and $w\in \mathbb C$, then the equation $f(z)=w$ always has a solution, except perhaps for a single value of $w$.
This statement generalizes the fundamental theorem of algebra, which can be phrased as follows:
If $f$ is a non-constant polynomial function and $w\in \mathbb C$, then the equation $f(z)=w$ always has a solution.