One of the past comp question
Suppose $\sum a_n z^n$ has a radius of convergence $R_1$ with $0< R_1 < \infty$, and $\sum b_n z^n$ has a radius of convergence $R_2$ with $0< R_2 < \infty$. Prove that $\sum \frac {a_n}{b_n} z^n$ has a radius of convergence $R_3$ satisfying $ R_3<=\frac {R_1} {R_2}$
I think the idea is to prove the series $\sum \frac {a_n}{b_n} z^n$ diverges when $|z|> \frac {R_1} {R_2} $
For that we use rational density theorem and manipulate the terms to get the desired result. I don't think this method is standard way of doing it. I was wondering if someone like to give me another mind blowing approach. Thanks in advance.