Real Analytic function .

Question

Let $f$ a real analytic function at $x=a$ i.e. having power series expansion about the point $x=a$. Can we say about radius of convergent of power series ? Like its equal to the distance form $a$ to nearest singularity of $f$ in complex analysis. I have a example $\frac{1}{1+x^2}$ which is analytic at every point of $\mathbb R$ but it’s power series about $x=0$ has radius of convergent $1$. So is there any results to estimate it’s radius of convergent? As in this example I think it is related to complex function $\frac{1}{z^2+1}$. Thank you.