Chapter 11 - Series Solutions to Linear Differential Equations - 11.1 A Review of Power Series - Problems - Page 730: 10

$2$

The radius of convergence of a power series represented by $\dfrac{A(x)}{B(x)}$ can be defined as the distance from $min(|x_0-y|)$, where y is root of $B(x)$, that is, $B(y)=0$. We have: $\dfrac{A(x)}{B(x)}=\dfrac{x^2-3}{x^2-2x+5}; x_0=0$ This implies that the root of $B(x)=x^2-2x+5=0 \implies x =1\pm 2i$ So, the radius of convergence is $R=min|1 -(1\pm 2i)|=2$