This is an exercise from Stein-Shakarchi.
Suppose that $f$ is holomorphic in an open set containing the closed unit disc, except for a pole at $z_{0}$ on the unit circle. Show that if $f(z) = \sum_{0}^{\infty}a_{n}z^{n}$ in the open unit disc then$$\lim_{n\rightarrow +\infty} \frac{a_{n}}{a_{n+1}} = z_{0}$$
Any hint ?
Hint: Let
$$h(z) = \sum_{k=1}^m \frac{c_k}{(z-z_0)^k}$$
be the principal part of the pole.
Consider $g(z) = f(z) - h(z)$. That is holomorphic in a neighbourhood of the closed unit disk by assumption. That tells you something about the Taylor coefficients of $g$.
The Taylor coefficients of $f$, the $a_n$, are the sum of the Taylor coefficients of $g$ and of $h$. Conclude with the above that the coefficients of $g$ are negligible in the limit.
