We seek to show that if $(M_{n})_{n \geqslant 0}$ is a sequence of strictly positive real numbers such that the series $\sum_{n \geqslant 1} \frac{M_{n-1}}{M_{n}}$ converges, there exists a function $f \in \mathcal{C}_{c}^{\infty}(\mathbb{R})$ not identically zero such that for all $n \geqslant 0$, $\|f^{(n)}\|_{\infty} \leqslant M_{n}$.
Conclude regarding the initially posed question.