For any integer \(n\geq3\),

\[\left\lfloor\dfrac{1}{1-\displaystyle\prod_{k=n}^\infty\left(1-\dfrac{1}{F_k}\right)}\right\rfloor=F_{n-2}\]
where \(F_n\) is the \(n\)-th Fibonacci number.
#FibonacciNumber #Fibonacci #FibonacciSequence #FibonacciSeries #Product #FloorFunction #Floor #InfiniteProduct

Is \(\left\lfloor\dfrac{n!}{e}\right\rfloor,\ \forall n\in\mathbb{N}\) always even? or equivalently, is \(\dfrac{1}{2}\left\lfloor\dfrac{n!}{e}\right\rfloor,\ \forall n\in\mathbb{N}\) always an integer?🤔 🔗 https://www.youtube.com/watch?v=wrHxeHJDTk4&t=604s&ab_channel=MichaelPenn

#floorfunction #function #even #EulerNumber #Factorial #NaturalNumber #MichaelPenn