What is value of $\prod_{x\geq 2} \frac{x^2+1}{x^2-1}\;?$ For $x \in \mathbb{Z^+}$ what is:
$$
\prod_{x\geq 2} \frac{x^2+1}{x^2-1}\;?
$$
Numerically it seems to be approximately $3.676$.

Very partial answer:
$$\prod_{x\geq2} \frac{x^2+1}{x^2-1} = 
\prod_{x\geq2} \frac{1+\frac{1}{x^2}}{1-\frac{1}{x^2}}.$$
Then Maple says that both the numerator and denominator converge.  It gives
$$\prod_{x\geq2} 1-\frac{1}{x^2}  = \frac{1}{2}$$
and 
$$\prod_{x\geq2} 1+\frac{1}{x^2}  = \frac{\sinh \pi}{2\pi}.$$
So your product is apparently equal to $(\sinh \pi)/\pi.$
