I know, a priori, how to show that a sequence of functions converges or not in the sense of distributions, but now I am trying to show whether a sequence of distributions converges or not in the sense of distributions, but there are no examples in my books or lectures.
We denote by $\delta_n$ the distribution defined by $$\delta_n(\varphi)=\varphi(n)$$ and we define the series $$S=\sum_{k=0}^{+∞}{k^4\delta_k}$$
I want to know whether or not $S$ converges in the sense of the distributions (I don't especially need to know towards what). Intuitively, since this series contains a product by a test function, we will have that for a sufficiently large $k$ the series will be null because $\varphi$ has a compact support, its supremum will be reached. Therefore, the series converges to a complex number by this property of the test function, and does not converge to infinity. I do not know if this reasoning is correct, nor how the show it formaly.
It is convergent. Theorem 6.5 part (f) in Rudin's FA shows that $\phi_i \to \phi$ in the space of test function implies that all that $\phi_i$'s have their support contained in one fixed compact set. From this it follows that $\sum k^{4}\phi_n (k) \to \sum k^{4}\phi(k)$ whenever $\phi_n \to \phi$. This proves convergence in the sense of distributions. .