Power series of a function which is analytic everywhere

Question

I am having doubt regarding the power series of the function $g(x) = \frac{1}{1+x^2}$. Since the function $f(x) = {1+x^2}$ is analytic everywhere and by using the theorem that if $f(x)$ is analytic everywhere and has no real zeroes then the reciprocal of the function $f(x)$ will also be everywhere analytic.
Therefore, I can conclude that the radius of convergence of the power series of $g(x) = \frac{1}{1+x^2}$ should be infinite around every point $x_{0}$ $\in$ $\mathbb{R}$ but then if I write the power series of $\frac{1}{1+x^2}=1-x^2+x^4-x^6+....$ and by using the root test the radius of convergence is coming out to be $1$.
I am confused why these two concepts are contradicting because the power series should have the radius of convergence of $\infty$, but the radius of convergence from the root test is coming to be $1$. I cannot figure out where am I doing wrong? Thanks.