Infinite product with zero value For an infinite product $\prod a_k$ to converge we need 


*

*at most finitely many zero factor, let be $m$ the maximum index of them

*$c=\lim_{n\to \infty}\prod_{k=m+1}^n a_k$ must exists, and

*$c\ne 0$.



My question is "Why the additional condition 3?"

Consider 
$$\tag{1}
\prod_{k=1}^\infty \frac{n}{n+1}=\frac{1}{2}\frac{2}{3}\frac{3}{4}\cdots
$$
The $n$th partial product would be $1/n$, thus the limit is zero.
The definition above excludes (1) from the converging infinite products, but I do not understand what is bad about (1) converging to zero. There must be some consideration behind it.
EDIT
My question could be read as follows: What are the advantages of this definition? Is there a better (easier to develop by excluding the zero) definition ? Why is the zero limit excluded (even in case there are no zero factors)?

I would like to add a footnote of Pringsheim from 1888, I think the first "complete" work on the topic. He discuss exactly this question and he gives, 132 years ago, the same arguments as @GEdgar: convenience (Bequemlichkeit) and conserving the properties of a finite product.

Pringsheim: Über die Konvergenz unendlicher Produkte. Mathem. Ann. Bd. 33, S. 119-154. 1888.

The book can be found 
https://archive.org/details/mathematischean46behngoog/page/n126


