$\lim_{z \to z_0}\sum_{k=1}^{\infty} a_k (z-z_0)^{-k+m}= \infty$ if there exist at least one $k> m$ such that $a_k \neq 0$?

Question

Let $z_0, a_k \in \mathbb{C}$ and $m \in \mathbb{N}$. How can I formally prove that $\lim_{z \to z_0}\sum_{k=1}^{\infty} a_k (z-z_0)^{-k+m}= \infty$ if there exist at least one $k> m$ such that $a_k \neq 0$?

Answer 1

Counterexample: Take $z_0=0$. Note that $$\sin\left(\frac1z\right)=\frac1z-\frac16\frac1{z^3}+\dots .$$So that series does not tend to infinity as $z\to0$.