Let $I \subseteq \mathbb{R}$ be an open interval and let $f_n : I \to \mathbb{R}$ be a sequence of real analytic functions such that $$ F(x, y) := \sum_{n=0}^\infty f_n(x)y^n $$ converges in the $C^1$-norm on $I \times (-\epsilon, \epsilon)$ for some $\epsilon > 0$. Is $F$ real analytic, possibly after taking a smaller $\epsilon$?
We have power series expansions $f_n(x) = \sum_{m=0}^\infty a_{mn}x^n$, but the trouble is that the radius of convergence $R_n$ depends on $n$ and I'm not sure if we can get $\liminf R_n > 0$.
Try $$f_n(x) = \frac{1/n!}{x^2+1/n^2}$$ (assuming $0\in I$)
If $F(x,y)$ was analytic at $(x,y)=(0,0)$ then $F(x,r)$ would be analytic at $x=0$ for $r$ small enough.