convergence of series of function in distributions

Question

I'm trying to prove that:

$$\lim_{n\to\infty} t_n(x) = -\delta'(x)$$ where

$$t_n(x) = \begin{cases}-n^2, & 0\leq x < \frac1n\\ n^2, & -\frac1n< x < 0\end{cases}$$ Help will be appreciated!

Answer 1

What is nice with the distributions is that if $f_n \to \delta$ (in the sense of distributions) then $f_n' \to \delta'$ (where $f_n'$ is the distributional derivative, which is the usual derivative if $f_n$ is differentiable).

So integrate $t_n$ to obtain $f_n(x) = -\int_{-1}^x t_n(y)dy$ and show that $f_n \to \delta$ in the sense of distributions, ie that for any $\varphi \in C^\infty_c $ : $$\lim_{n \to \infty } \int_{-\infty}^\infty \varphi(x) f_n(x)dx = \varphi(0)$$