We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$ For $n \in \mathbf{N}^*$, set $a_n = \frac{1}{n+1}$ and $b_n = \frac{n}{n+1}$. Establish the pointwise convergence of the sequence of applications $(\Psi_n)_{n \in \mathbf{N}^*}$, from $]0, \pi]$ to $\mathbf{R}$, defined by: $$\forall n \in \mathbf{N}^*, \forall t \in ]0, \pi], \Psi_n(t) = \ln(a_n^2 \cos^2 t + b_n^2 \sin^2 t)$$ Deduce $f''(0)$.
We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by:
$$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
For $n \in \mathbf{N}^*$, set $a_n = \frac{1}{n+1}$ and $b_n = \frac{n}{n+1}$.
Establish the pointwise convergence of the sequence of applications $(\Psi_n)_{n \in \mathbf{N}^*}$, from $]0, \pi]$ to $\mathbf{R}$, defined by:
$$\forall n \in \mathbf{N}^*, \forall t \in ]0, \pi], \Psi_n(t) = \ln(a_n^2 \cos^2 t + b_n^2 \sin^2 t)$$
Deduce $f''(0)$.