Infinite Product: Value Not Equal to Zero or One?

[AntonioM]

when does infinite product have a value not equal to zero or one?

 

[Char. Limit]

I assume you mean a convergent infinite product, because a divergent infinite product has the simple example

\prod^{\infty}_{i=2} \left(1 + \frac{1}{i}\right)

For a convergent infinite product, I offer up this example:

\prod^{\infty}_{i=1} \left(\frac{1}{1-\left(\frac{1}{p_i}\right)^2}\right) = \frac{\pi^2}{6}

That is, incidentally, the Euler Product.

EDIT: Forgot to mention, p_i means the i^th prime number.