We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let $$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$ Show that $J_{n} = \frac{\pi}{2}$ for $1 \leqslant n \leqslant 7$ and that $\left(J_{n}\right)_{n \geqslant 7}$ is strictly increasing.
We consider a sequence $\left(X_{n}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables, taking values in $\{1, -1\}$ and such that, for all $k \in \mathbb{N}^{*}$, $P\left(X_{k} = 1\right) = P\left(X_{k} = -1\right) = \frac{1}{2}$. For $n \in \mathbb{N}^{*}$, let
$$J_{n} = \int_{0}^{\infty} \frac{1 - \cos(t) \cos\left(\frac{t}{3}\right) \cdots \cos\left(\frac{t}{2n-1}\right)}{t^{2}} \mathrm{~d}t$$
Show that $J_{n} = \frac{\pi}{2}$ for $1 \leqslant n \leqslant 7$ and that $\left(J_{n}\right)_{n \geqslant 7}$ is strictly increasing.