I am currently doing the following problem :---
Let $f$ be holomorphic on $\{z\in \Bbb C:|z|<2,z\not=1\}$. Let $f$ has an representation of the form $f(z)=\sum_{n=0}^{\infty}a_nz^n,|z|<1$. What can we say about the sequence $\{a_n\}$ ?
What I can guess is that if $f$ has a simple pole at $z=1$, then $\displaystyle\lim_{n\to \infty}a_n=-\text{Res}_{z=1}(f)$.
Also if $f$ has a removable singularity at $z=1$ then this limit is actually $0$ and if $f$ has essential singularity at $z=1$, then $\displaystyle \lim_{n\to \infty} a_n$ doesn't exist.
But I can not prove these facts. I don't know what will be the limit in case $f$ has a pole of order $n\geq 2$ at $z=1$. Any help will be appreciated.