We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$ For every non-zero natural number $n$, we consider the functions $$\begin{cases} f_n(x) = \dfrac{n}{1 + n^2 x^2} & \\ g_n(x) = nx^n & \text{if } x \in [0,1] \text{ and zero elsewhere} \\ h_n(x) = n^2 \sin nx & \text{if } x \in [-\pi/n, \pi/n] \text{ and zero elsewhere} \end{cases}$$ a) Verify that they belong to $\mathcal{F}_{sr}$. b) Study the variations of the functions $f_n, g_n$ and $h_n$ then sketch their graphs for $n = 1$ and $n = 2$. c) Study the convergence of the sequences of distributions $(T_{f_n}), (T_{g_n})$ and $(T_{h_n})$.
We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if
$$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For every non-zero natural number $n$, we consider the functions
$$\begin{cases} f_n(x) = \dfrac{n}{1 + n^2 x^2} & \\ g_n(x) = nx^n & \text{if } x \in [0,1] \text{ and zero elsewhere} \\ h_n(x) = n^2 \sin nx & \text{if } x \in [-\pi/n, \pi/n] \text{ and zero elsewhere} \end{cases}$$
a) Verify that they belong to $\mathcal{F}_{sr}$.\\
b) Study the variations of the functions $f_n, g_n$ and $h_n$ then sketch their graphs for $n = 1$ and $n = 2$.\\
c) Study the convergence of the sequences of distributions $(T_{f_n}), (T_{g_n})$ and $(T_{h_n})$.