An infinite product of complex terms can diverge/converge to zero if
(a) one or more of the terms are zero and all other terms are finite
or
(b) infinitely many terms have $ |z| < 1 $ and atmost finite terms are such that $ 1 < |z| < \infty $
Examples: $$ P_1 = 2e^{i\theta} \cdot 1e^{i2\theta} \cdot \frac{1}{2}e^{i3\theta} \cdot \frac{1}{4}e^{i4\theta} \cdots $$ $$ P_2 = \frac{1}{2}e^{i\theta} \cdot \frac{3}{4}e^{i\theta} \cdot \frac{7}{8}e^{i\theta} \cdot \frac{15}{16}e^{i\theta} \cdots $$
Are there any other reasons for an infinite product $ \prod_{n=0}^\infty z_n $ to converge/diverge to zero? ($ z_n \in \mathbb {C} $)
There is a third possibility. You can have infinitely many terms with $|z|<1$ and infinitely many terms with $|z|>1$. Example: $$ \frac{1}{1}\bigl(1+\frac{1}{1}\Bigr)\frac{1}{2}\bigl(1+\frac{1}{4}\Bigr)\frac{1}{3}\bigl(1+\frac{1}{9}\Bigr)\dots $$