<s>[INST] Region of Convergence for Laurent Expansion Find the Laurent expansion of $\frac{z}{(z+1)(z+2)}$ about the singularity $z=-2$. Specify the region of convergence and the nature of singularity at $z = -2$.
The Laurent expansion I get is 
$1+(z+2)+(z+2)^2+ \ldots + \frac{2}{z+2}$
The singularity is a simple pole. 
But how to find the Region of Convergence. 
 [/INST] The Laurent series can be obtained as
$$\frac{z}{(z+1)(z+2)} = \frac{1}{z+2}\frac{z+2 - 2}{z+2 - 1} \\ = \left(\frac{2}{z+2} - 1\right)\frac{1}{1 - (z+2)} \\ = \left(\frac{2}{z+2} - 1\right)\sum_{n=0}^\infty (z+2)^n$$
and the geometric series on the RHS converges when $|z+2| < 1$.
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