I have $\frac{1}{z+z^2}$ defined on the annulus $|z+1|>1$.
I write:
$f(z) = \frac{1}{z} - \frac{1}{z+1} = -\frac{1}{z+1} + \frac{1}{(z+1)-1} = -\frac{1}{z+1} + \frac{1}{z+1} \cdot \frac{1}{1-\frac{1}{z+1}} = -\frac{1}{z+1} + \frac{1}{z+1} \sum_{n=0}^{\infty}(z+1)^{-n} = -(z+1)^{-1} + \sum_{n=0}^{\infty}(z+1)^{-n-1} = -(z+1)^{-1} + \sum_{n \leq -1}^{}(z+1)^{n} = \sum_{n \leq -2}^{}(z+1)^{n}$
Is this correct?