Residues of the Gamma function

Question

I am trying to make sense of a proof that the poles of $\Gamma(z)$ are at $z=-n$ and have residue $\frac{(-1)^n}{n}$. The proof reduces $\Gamma(z)$ to the sum of an (entire) incomplete gamma function and a meromorphic part given by $$\sum_{n=0}^\infty\frac{(-1)^n}{n!(z+n)}$$ My issue is the expansion $$\frac{1}{z}-\frac{1}{1+z}+\frac{1}{2(z+2)}- ...$$ does not resemble a Laurent Series since each term is centered around a different point. Can someone explain why it is still valid to take the coefficient of $(z-z_0)^{-1}$ in this form in order to find the residue at $z=z_0$?