Questions mathématiques diverses

Question 83:
Déterminer la limite suivante :
$\lim\limits_{n \to +\infty}\left({\prod\limits_{k=1}^{n}{\tan\left({\dfrac{k\pi}{2n}}\right)}}\right)^{\tfrac{1}{n}}$

Réponse 83:
Posons $P_n=\left({\prod\limits_{k=1}^{n}{\tan\left({\dfrac{k\pi}{2n}}\right)}}\right)^{\tfrac{1}{n}}$ $$\begin{align*} \ln(P_n)&=\dfrac{1}{n}\sum\limits_{k=1}^{n}{\ln\left({\tan\left({\dfrac{k\pi}{2n}}\right)}\right)}\\ &=\dfrac{1}{n}\sum\limits_{k=1}^{n}{\ln\left({\tan\left({\dfrac{\pi}{2}.\dfrac{k}{n}}\right)}\right)} \end{align*}$$ $$\begin{align*} &\lim\limits_{n \to +\infty}\ln(P_n)\\ &=\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\tan\left({\dfrac{\pi}{2}x}\right)}\right)dx}\\ &=\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\tan\left({\dfrac{\pi}{2}(1-x)}\right)}\right)dx} (\text{Voir Question 82})\\ &=\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\tan\left({\dfrac{\pi}{2}-\dfrac{\pi}{2}x}\right)}\right)dx}\\ &=\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\cot\left({\dfrac{\pi}{2}x}\right)}\right)dx} \end{align*}$$ $$\begin{align*} &\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\tan\left({\dfrac{\pi}{2}x}\right)}\right)dx}+\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\cot\left({\dfrac{\pi}{2}x}\right)}\right)dx}\\ &=\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}{\ln\left({\cot\left({\dfrac{\pi}{2}x}\right)\tan\left({\dfrac{\pi}{2}x}\right)}\right)dx}\\ &=\lim\limits_{\epsilon \to 0}\int_{\epsilon}^{1-\epsilon}\ln(1)dx=0\\ &\lim\limits_{n \to +\infty}\ln(P_n)=0 \Longrightarrow \boxed{\lim\limits_{n \to +\infty}P_n=e^0=1} \end{align*}$$