Find the Laurent Series centered at $z=0$ for all the analitical region of $\frac{\sinh(z) - z}{z^4}$

Question

Find the Laurent Series centered at $z=0$ for all the analitical region of

$\frac{\sinh(z) - z}{z^4}$

Answer 1

Step 1: consider the Taylor- series for $sinh.$

Step 2: subtract $z$.

Step 3: now divide by by $z^4.$

Whoops, the Laurent Series centered at $z=0$ is completed.

This series converges on $ \mathbb C \setminus \{0\}.$