Difficult question from some test somewhere (I forget).
$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$
$x$ is, of course, an integer.
Difficult question from some test somewhere (I forget).
$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$
$x$ is, of course, an integer.
Write $\frac{x^4 - 1}{x^4+1} = \frac{(x-a_1)\ldots(x-a_4)}{(x-b_1)\ldots(x-b_4)}$ where $a_j$ are the roots of $x^4-1$ and $b_j$ are the roots of $x^4+1$. Then the partial product $ \prod_{x=2}^n \frac{x^4 - 1}{x^4+1} = \frac{\Gamma(n+1-a_1)\ldots \Gamma(n+1-a_4) \Gamma(2-b_1) \ldots \Gamma(2-b_n)}{\Gamma(2-a_1) \ldots \Gamma(2-a_4) \Gamma(n+1-b_1) \ldots \Gamma(n+1-b_4)}$ Now (carefully) take the limit as $n \to \infty$.