I'm trying to prove the following
Suppose $f$ is a holomorphic function in an open set containing $D\setminus \{z_0\}$, where $D$ is the closed unit disc, and $f$ has a pole at $z_0$. Then, if $\sum\limits_{n = 0}^\infty a_nz^n$ is the power series expansion of $f$ near $0$, $\lim \dfrac{a_n}{a_{n+1}}$ exists and is equal to $z_0$.
Through a rotation, we may suppose that $z_0 = 1$. Also,
$\lim\inf \dfrac{a_{n+1}}{a_n} \leq \lim\sup |a_n|^{\frac 1n} = 1\leq \lim\sup\dfrac{a_{n+1}}{a_n}$,
so it's sufficient to prove the existence of the ratio test limit.