How to find the radius of convergence of the power series $f(x)$

Question

This power series $f(x)$ centered at $x=0$, and thats mean $\sum {a_n x^n }$ $$ f(x) = \frac{{x - 1}}{{x^2 - 2x + 5}}. $$ I found that the radius of convergence for this power series is: $R=\sqrt{5}$.

Answer 1

Yes, the answer is $\sqrt 5$. It follows from the fact that$$\frac{x-1}{x^2-2x+5}=\frac12\left(\frac1{x-1-2i}+\frac1{x-1+2i}\right)$$and that $|1+2i|=|1-2i|=\sqrt 5$.