I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists N\in\mathbb{N},\,\exists B>0:\forall k\in\mathbb{N},\,|c_k|\leq Bk^N$$ I have managed to prove the $\Leftarrow$ direction by showing that $u$ would be a linear functional which is sequentially continuous, so clearly an element of $\mathcal{S}'(\mathbb{R})$.
However, I'm struggling with the remaining implication ($\Rightarrow$), which I intuitively believe to be correct. My idea was to proceed by contradiction: assume $\forall N\in\mathbb{N},\,\forall B>0,\,\exists k\in\mathbb{N}:|c_k|> Bk^N$ and then try to find a $\phi\in\mathcal{S}(\mathbb{R})$ such that $|u(\phi)|=|\sum_{k=1}^{\infty}c_k\phi(k)|=\infty$, which would then imply that $u\notin\mathcal{S}'(\mathbb{R})$. Unfortunately this has not gotten me very far. Any ideas?
We go back to the definition of continuity of a linear functional on the Schwartz space: this means that there exists an integer $N$ and a constant $K$ such that for each $\varphi\in\mathcal S(\mathbb R)$, $$|T(\varphi)|\leqslant K\sum_{i,j=1}^N\sup_{x\in\mathbb R}\left|x^i\varphi^{(j)}(x)\right|.$$ Consider $\varphi$ a test function such that $\varphi(x)=1$ if $|x|\leqslant 1/2$ and $0$ if $|x|\gt 1$. Then use the displayed equation with $\varphi_k(x):=\varphi(x+k)$.