Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$ Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$.
Is the first step here to notice that
$$ \frac{1}{z-2} + \frac{1}{z-3} = \frac{2z-5}{(z-2)(z-3)}$$
and compute the Laurent series as a single term rather than in two parts?

Hint: This is the first step
$$\frac{1}{z-2} =\frac{1}{z}\Big(\frac{1}{1-\frac{2}{z}}\Big)  \ \text{and}\ \ \frac{1}{3} \Big(\frac{1}{1-\frac{z}{3}}\Big)\\ \frac{1}{z-3} = \frac{1}{z} \Big(\frac{1}{1-\frac{3}{z}}\Big)\ \ \text{and}\ \ \frac{1}{2} \Big(\frac{1}{1-\frac{z}{2}}\Big)$$
