Define $\langle T, \cdot \rangle: C_0 ^\infty (\mathbb R) \to \mathbb C$ such that $\langle T,\varphi \rangle = \sum_{k=0}^\infty \varphi^{(k)} (\sqrt 2)$.
I'm trying to proof that $\langle T, \cdot \rangle$ does not define a distribution.
I have the following hint: Consider the function $\varphi = e^{x - \sqrt 2}$.
Since $\varphi ^{(k)} (\sqrt 2) = 1$ for all $k$, we'll have $\sum_{k=0}^\infty \varphi^{(k)} (\sqrt 2) = \infty$. However $e^{x - \sqrt 2} \notin C_0 ^\infty (\mathbb R)$. I hear that I can approximate this function by test functions, althoug I don't know how to do it.
Help?