How to prove that the series
$$\sum_{k=-\infty}^{\infty}a_k\delta^{(k)}(x-k)$$
converges in $D'$ for all values of $a_k$?
I Understand that the partial sum $$ s_n=\sum_{k=-n}^{n}a_k\delta^{(k)}(x-k)$$ has a compact support $\Rightarrow s_n$ has finite order.
What should I do next?