If a series is not a power series, is the notion of radius of convergence still legit?

Question

This notion is defined over complex power series. But what about complex series not in a "power" form? Is the convergence region always form a circle centered at origin in the complex plane?

Answer 1

No, certainly not. For example, $$ \sum_{k=0}^\infty (f(z))^k $$ converges if and only if $|f(z)| < 1$ and that set is (in general) not a disc.

For a concrete example with an interesting domain of convergence, take $$ \sum_{k=0}^\infty \sin^k z. $$