Uniform convergence of of series of functions

Question

Let $p\in \mathbb{N}$ and $(a_n)_{n\in \mathbb{Z}}$ a sequence of complexe numbers such that

$$|a_n|\le (1+|n|^p)\quad \quad \forall n\in \mathbb{Z}$$

Could you please help me to show that $$\sum_{n\in \mathbb{Z}} a_n e^{2i\pi n x } \phi’(x)$$ converge uniformly for all $\phi \in D(\mathbb{R})$.

Thanks in advance