The question is somewhat ambiguously given in the title but here it is again;
Consider an everywhere convergent power series around $z_0$ $$ f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n.$$ If I introduce a parameter $k$ is the power series $$(f(z))^k = \left(\sum_{n=0}^\infty a_n(z-z_0)^n\right)^k$$ also automatically everywhere convergent for any $k$? Can $k$ be any complex number or does this only hold for reals? (if it does hold!)
I presume the answer is yes since for any $z$, $f(z)$ is just "a number" so raising this to a power isn't going to change the convergence properties of the series, but I'd like it if somebody could confirm this for me.