<s>[INST] Taylor series for the function $f(z) = \frac{1}{(z-5)(z-7)}$ on a disc centered at point $z_0=3$ I started by expressing the function as sum of two fractions using partial fraction decomposition to get $\frac{-1}{2(z-5)} + \frac{1}{2(z-7)}$
However I could only then end up writing that as the sum of two power series :
$$\frac{1}{10} \sum_{n=0}^\infty \frac{z^n}{3^n} - \frac{1}{14} \sum_{n=0}^\infty \frac{z^n}{7^n}$$ and I'm unsure how to write this in the form of a Taylor series.
 [/INST] Hint: For example,
$$
\frac{1}{z-7} = \frac{1}{(z-3)-4} = -\frac{1}{4} \frac{1}{1-(z-3)/4} = -\frac{1}{4} \sum_{i=0}^\infty \left(\frac{z-3}{4}\right)^i.
$$
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