Can the radius of convergence of a quotient of two analytic power series increase? Suppose $f$ and $g$ are two analytic power series centred at $z=0$, both with radius of convergence equal to $R >0$, and assume also that g is nowhere vanishing in B($0$,R), so that the quotient $f/g$ is defined and analytic there. 
My question is: can $f/g$ be analytic in a domain strictly larger than $B(0,R)$?
In my opinion, if the quotient extends analytically to a larger domain, we should have that also $f$ and $g$ extend, contradicting the fact that $B(0,R)$ is their domain of holomorphy, but I am not able to prove this.

To extend slightly on Martin R's comment, if $f = g h,$ where $h$ is an entire function, the radius of convergence of the ratio goes to infinity.
