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Find a power series expression $\sum_{n=0}^\infty A_n z^n $ for $ \frac{1}{z^2-\sqrt2 z +2} $

I'm completely stuck on this question. I know how to manipulate power series but I've never had to find one from such a complicated function. Also we're asked to find the radius of convergence although this I think I can do. Also I know that maybe the Taylor series expansion could help but differentiating that function leads to an awful mess.

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$$ z^2-\sqrt{2}\,z+2=(z-a)(z-b)\text{ where }a=\frac{\sqrt2+\sqrt6\,i}{2},\ b=\frac{\sqrt2-\sqrt6\,i}{2}. $$ Use partial fraction decomposition to write $$ \frac{1}{z^2-\sqrt{2}\,z+2}=\frac{A}{z-a}+\frac{B}{z-b}. $$ Now expand in power series $1/(z-a)$ and $1/(z-b)$.

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