I am learning the distribution theory currently. My professor in class said that,
if we defined $J_1(\varphi):=\sum_{n=0}^\infty \varphi^{(n)}(0)$ and $J_2(\varphi):=\sum_{n=0}^\infty \varphi^{(n)}(n)$, where $\varphi\in C^\infty_0(\mathbb R)$, then both $J_1$ and $J_2$ were not distributions(i.e. $J_1$ and $J_2$ were not continuous linear functional on $C^\infty_0(\mathbb R)$).
Clearly $J_1$ and $J_2$ are linear. Suppose $\varphi_k\to0$ in $C^\infty_0(\mathbb R).$ Then given any $\epsilon \gt0,$ for every positive integer $i$, we may find an $N_i\in\mathbb N$ such that $sup_{x\in\mathbb R} |\varphi_k^{(i)}(x)| \lt {\epsilon\over{2^i}}$ when $k\ge N_i$. Hence when $k\to \infty$, lim$_{k\to\infty}J_1(\varphi_k)=\text{lim}_{k\to\infty}\sum_{n=0}^\infty \varphi^{(n)}_k(0)=\text{lim}_{k\to\infty}\text{lim}_{m\to\infty}\sum_{n=0}^m \varphi^{(n)}_k(0)=$
$\text{lim}_{m\to\infty}\text{lim}_{k\to\infty}\sum_{n=0}^m \varphi^{(n)}_k(0)=0,$ where we can interchange the limits because the series is absolutely convergent. So $J_1$ is continuous. Similarly we can show that $J_2$ is continuous.
So the only possibility that $J_1$ and $J_2$ are not distributions is that $J_1(\varphi_1)=\infty$ and $J_2(\varphi_2)=\infty$ for some $\varphi_1,\varphi_2\in C^\infty_0(\mathbb R).$
My question is, can someone give me examples of $\varphi_1,\varphi_2\in C^\infty_0(\mathbb R)$, such that $$J_1(\varphi_1)=\sum_{n=0}^\infty \varphi_1^{(n)}(0)=\infty$$ and $$J_2(\varphi_2)=\sum_{n=0}^\infty \varphi_2^{(n)}(n)=\infty.$$
Thanks in advance.
$J_2$ is of course a distribution.
For $J_1$, let $\psi \in C^\infty_c, \psi=1$ on $[-1,1]$ then look at $$\lim_{N\to \infty} J_1(\psi \sum_{n=0}^N \frac{x^n}{n!})$$