Question

Prove que ( 1 − 1 /2) (1 − 1 /3) (1 − 1/ 4) · · · (1 − 1 /n) = (1/ n) , para todo n ∈ N, n ≥ 2.

Answer 1

$\Large\displaystyle\begin{gathered}\sf\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \left(1-\frac{1}{4}\right)\cdots\left(1-\frac{1}{n}\right) =\frac{1}{n} \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf\left(\frac{1}{\diagup\!\!\!2}\right)\left(\frac{\diagup\!\!\!2}{\diagup\!\!\!3}\right) \left(\frac{\diagup\!\!\!3}{\diagup\!\!\!4}\right)\diagup\!\!\!\cdots\left(\frac{\diagup\!\!\!(n-1)}{n}\right) =\frac{1}{n} \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf \frac{1}{n} =\frac{1}{n} \ \ \ (Ok)! \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \left(1-\frac{1}{4}\right)\cdots\left(1-\frac{1}{n}\right) =\frac{1}{n} \ , \ \forall n\in \mathbb{N}\ , n \geq 2\end{gathered}$