Find the order of the distribution $\langle u,\phi\rangle=\sum_{k=1}^\infty\partial^k\phi(1/k)$

Question

Find the order of the distribution $u$ in $\mathscr{D}^\prime((0,\infty))$ $$\langle u,\phi\rangle=\sum_{k=1}^\infty\partial^k\phi(1/k)$$

I guess it does not have finite order, but I have no idea to show that