# Residues for a Laurent series with two centres

I have worked out that the Laurent series for $$\left(z^4sin(\frac{1}{z}) + (z+1)^4sin(\frac{1}{z+1})\right)$$ is given by:

$$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \left( \frac{1}{z^{2n-3}} + \frac{1}{(z+1)^{2n-3}} \right)$$

I want to know the residue of the function at $$0$$ and $$-1$$. However I'm confused how to find this as the series is not of the normal form $$\sum_{n=-\infty}^\infty a_n(z-z_0)^n$$ How do I calculate these residues?