Convergence of $\prod\limits_{k=1}^n \left(1+\frac{1}{k^2}\right) $ How can you determine whether the limit
$$ \displaystyle \lim_{n\to\infty} \prod_{k=1}^n \left(1+\frac{1}{k^2}\right)  $$ 
exists? Thanks for any hints!  

$$\frac{\sinh \pi x}{\pi x}=\prod_{k=1}^{\infty }\left(1+\frac{x^2}{k^2}\right)$$
is the Weierstrass product for the hyperbolic sine function, see here
or here. If you plug $x=1$:
$$\prod_{k=1}^{+\infty}\left(1+\frac{1}{k^2}\right) = \frac{\sinh\pi}{\pi}.$$
