Showing given function is real analytic on R

Question

I was reading about real analytic functions and I understood that to show a function is real analytic on $\mathbb{R}$, we need to show that there is a power series centered at $x_0$ which converges to an open interval around $x_0$ $\forall x_0 \in \mathbb{R}$
The example I am working on is the function $$f(x) = \frac{1}{1+x^2}$$ I can show that it converges for expansion around 0 by expanding it as a geometric series and this converges in (-1,1) when centered at x=0. How do I show it for other x $\in \mathbb{R}$ ?
I think I remember a fact that if power series centered around x=0 converges in (-1,1) then power series centered at any point in this interval would also converge(can someone give verify if this is indeed true?). But that still leaves us with the rest of the number line.

Answer 1

Try just centering a geometric series around a different point. For example, take $$\frac{1}{1-(-(x-x_0)^2))}$$

If this does have power series that converges, what would it look like? This would mean that you can get a radius 1 of converge around any point that you desire.