Let's consider $\mathbb{T}:=\mathbb{R}/2\pi \mathbb{Z}$ and let's denote $\mathcal{E}(\mathbb{T})$ to be the space of $C^\infty$ functions defined on $\mathbb{T}$, where the topology on $\mathcal{E}(\mathbb{T})$ is defined via the following: $\phi_n \rightarrow \phi$ in $\mathcal{E}(\mathbb{T})$ iff $\phi_n^{(k)} \rightarrow \phi^{(k)}$ uniformly on $\mathbb{T}$.
Now, we consider the space of distributions $\mathcal{E}'(\mathbb{T})$.
The question that I have to solve is to check, if for every sequence of complex numbers $(a_n)_{n \in \mathbb{Z}}$ the limit $\sum_{n=-N}^N a_n e^{inx}$ exists in $\mathcal{E}'(\mathbb{T})$.
So, as far as I understand, we need to show that for every $\phi \in \mathcal{E}(\mathbb{T})$ the following limit exits $$\lim_{N \rightarrow \infty} \int_\mathbb{T} \sum_{n=-N}^N a_n e^{inx} \phi(x) dx$$
(where we work on this integral since we consider the action of a distribution $\sum_{n=-N}^N a_n e^{inx}$ on a test function $\phi$). Could anyone share some light how we can show that something like this won't converge to any distribution $u \in \mathcal{E}'(\mathbb{T})$ for some $\phi \in \mathcal{E}(\mathbb{T})$ and $a_n$? I suspect it doesn't have to in general, since this formula is a similar one like the one we work with Fourier series, but neither of the ideas that I had were successful.
I would appreciate any help here.
Take $\phi(x)=\sum_{n\ge 1} 2^{-n} e^{-inx}$ and $a_n=2^n 1_{n\ge 1}$ you'll have $$\int_0^{2\pi} \sum_{n=-N}^N a_n e^{inx}\phi(x)dx=2\pi N$$