Let $F$ be a linear functional on the space of test functions defined by $F(f)=\sum_{n\ge 0} f^{(n)}(n)$. show that $F$ is of infinite order.
Attempt: for any $f$ we have that $supp(f)$ is compact. Hence for some $N$, we have that $f(n)=0$ for any $n>N$. Hence the derivatives vanish as well. But I am ensure where to go from here since the norm is independent of the test function? (Here the norm is the one in the definition of the order of a functional)