In Remmert's Topics in Complex Analysis (pg4),
If we were to call $\prod a_n$ convergent whenever the sequence of partial products had a limit $a$, undesirable pathologies would result: a product would be convergent with value $0$ if just one factor $a_n$ is $0$; for another, $\prod a_n$ could be zero even if not a single factor were zero.
I don't understand what is so pathological about these cases - aren't them just special cases? Or am I missing something.
On
For simplicity, consider the case of a sequence of positive reals. Note that
$$ \log \prod_n a_n = \sum_n \log a_n $$
The theory of infinite products runs very closely parallel to the theory of infinite sums; the fact we distinguish a sum that goes to $-\infty$ translates over to a need to distinguish a product that goes to zero.
It may seem odd to use the phrase "divergent" for a product that converges, but that usage is already established; e.g. the adjective is also used to refer to sums, limits, and integrals that converge to $\infty$ or to $-\infty$.
And, in the complex plane, $\log(0)$ is an extremely poorly behaved notion; we should be much shyer about complex products converging to $0$ than we are about real sums converging to $-\infty$.
For the first, it is obvious that if all the $a_n$ are greater than $1+\epsilon$ the product diverges. In most of our work on series and sequences convergence is determined by "what happens out by infinity". Adding or changing a finite number of terms early in the sequence does not change whether it converges, just the limit it converges to. Here, if we put a $0$ at the head of the sequence, it becomes convergent. Whether this is pathological enough to make you change your definition is in the eye of the beholder. The second doesn't seem so strange to me.