Convergence in distributions

Question

Let $\varphi ∈D$ be a test function on $\Bbb{R}$. Is the sequence $f_n(x)=\frac{\varphi(nx)}{n}$ convergent in the test function space $D$? What is the limit? Please provide a hint to start.

Answer 1

Convergence in the test function space requires convergence of all derivatives. By the chain rule,

$$f^{(k)}_n(x)= n^{k-1} \varphi^{(k)}(nx).$$

What does this do for $n\to\infty$?