I recently solved this problem:
Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a $C^{\infty}$ (infinitely many times differentiable) function such that the sequence of derivatives $\{f^{(n)}\}$ converges uniformly on any compact set to a function $g$. Prove that $g(x) = ce^x$ for some constant $c\in\mathbb{R}$.
My question is, what are the possible $C^{\infty}$ functions with a uniformly convergent sequence of derivatives? My initial thought is that it is just polynomials and $ce^x$, but would it be possible for there to be a $C^{\infty}$ function not of that form whose sequence of derivatives converge to $ce^x$?