Finding the largest region on which Laurent series converges

Question

I need to find the largest region on which $$f(z) = z^4 sin(1/z)$$ defined on $$\mathbb{C}/{0}$$ converges. So for the Laurent series, I got $$z^3 - \frac{z}{3!} + \sum_{k\geqslant2}\frac{-1^{2k+1}}{(2k+1)!z^{2k-3}}$$ and found that zero is an essential singularity. But I'm unsure on how to find the largest region of convergence. From other examples that I've seen, it can usually be found by looking at the largest annulus on which the function is holomorphic but I can't seem to see that in this problem. Any help is appreciated.

Answer 1

$\sin z$ is an entire function and its power series converges for all complex numbers $z$. The Laurent series is obtained from this series by changing $z$ to $\frac 1 z$ and multiplying by $z^{4}$. Hence this series converges for all $z \neq 0$.