For every $\epsilon \in [0, 1)$ and $n$ in $\mathbb N$, let $g(n, \epsilon)$ be a function $\mathbb R \to \mathbb R$.

Suppose for all $\epsilon \in [0, 1)$, $g(n, \epsilon)$ is $C^n$ smooth and that for each $n$, $g(n, \epsilon)$ converges uniformly in $C^n$ norm to $0$ as $e \to 1-$. Suppose further that for every $\epsilon$, $\lim_{n \to \infty}$ $ \sum_{n = 1}^N g(n, \epsilon) \epsilon^n$ converges uniformly.

If $F := \lim_{\epsilon \to 1-} \sum_{n = 1}^N g(n, \epsilon) \epsilon^n$ exists in the uniform sense, is $F$ necessarily $C^\infty$ smooth?