Say I have a differential equation
and I know $P(x)=0$ at $x=\pm i$. Then there is a theorem which helps us conclude that the series solution centered at $x=0$ for $y$ converges for $|x|<|\pm i-0|=1$.
I have two questions:
What is the name of this theorem? I have been searching my textbook and Googling with no success.
Suppose $P(x)=0$ at $x=1,2$. I want my series to center at $x=0$. Then is my radius of convergence $|x|<1$ or $|x|<2$ or is there some other mess going on?