# Infinite product with zero value

For an infinite product $$\prod a_k$$ to converge we need

1. at most finitely many zero factor, let be $$m$$ the maximum index of them
2. $$c=\lim_{n\to \infty}\prod_{k=m+1}^n a_k$$ must exists, and
3. $$c\ne 0$$.

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)?

• The definition (for my understanding) is from Knopp, Theory and Application of Infinite Series, p. 218, archive.org/details/theoryandapplica031692mbp/page/n232 Commented Jan 18, 2020 at 11:08
• In my book, it is assumed that no zeros occur and the limit is neither zero, perhaps because of the relation we get if we take the logarithm of the product which leads to a sum. This logarithm-step of course does not work if we have a $0$. Even more restrictive, in this case we must have positive factors. But it is probably a matter of taste. Commented Jan 18, 2020 at 11:15
• My personal understanding is that when you take the logarithm of an infinite product converging stricto sensu, it turns to a converging series. Products that tend to zero are simply less interesting.
– user65203
Commented Jan 18, 2020 at 11:25
• @Peter It appears to me that Bronstein has the same definition: Ein unendliches Produkt heißt genau dann konvergent, wenn entweder (1.186) mit $b\ne 0$ vorliegt, oder diese Situation lässt sich nach Weglassen von endlich vielen Faktoren erreichen, die alle gleich null sind. Anderenfalls heißt das Produkt divergent. Commented Jan 18, 2020 at 11:31
• and the following paragraph is "Ein konvergentes unendliches Produkt ist genau dann gleich null, wenn ein Faktor gleich null ist.", and that is exacly what @JackM referred to Commented Jan 18, 2020 at 11:32

Yes, this is the definition in all books covering infinite products. We say $$\prod \frac{n}{n+1}$$ "diverges to $$0$$", and is not included when we say an infinite product "converges". The reason for this definition is that it is useful, for example in complex analysis.

One example: (there are many others) $$\sin z= z \prod_{n=1}^\infty \left(1-{\frac {z^{2} } {\pi^2n^{2} } } \right)$$ where (for all complex $$z$$) it is a convergent infinite product. Therefore, we may read the zeros of $$\sin$$ from it directly. With infinite products that possibly diverge to $$0$$, we cannot do that.

As all mathematics students know, the principle $$ab = 0 \Longrightarrow (a=0\text{ or }b=0)$$ is very useful. We want to keep this useful fact for infinite products also! The is the first (and simplest, of many) reasons that convergence of infinite products is defined this way.

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

• is there a translation into English? Commented Jul 4, 2021 at 1:16

Yes, it is one common definition, because (as Gerald Edgar said) it captures certain natural useful features. On the other hand, as in the question, it certainly has some features that can generate undesirable counterexamples. Try as I might, I’ve not been able to come up with a formalization that includes all the features I want (and no more).

So, unlike other formalizations (like epsilon-delta for continuity), there is definitely an ‘artisanal’ aspect to infinite products.