The 'infinite product' ...

(1)

... diverges if the corresponding 'infinite sum'...

(2)

... diverges. In this case the 'infinite sum' is...

(3)

... and we can use the 'integral test' to verify its convergence. If we have...

(4)

... and the integral from to diverges. If we have...

(5)

... and the integral from to converges. The conclusion is that the 'infinite product'...

(6)

... converges for and diverges for ...

Merry Christmas from Italy