Chapter 11 - Series Solutions to Linear Differential Equations - 11.1 A Review of Power Series - Problems - Page 730: 8

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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: $\dfrac{A(x)}{B(x)}=\dfrac{x}{x^2+1}; x_0=0$ This implies that the root of $B(x)=x^2+1=0 \implies x =\pm i$ So, the radius of convergence is $R=|0 \pm i|=1$