Why do we say some infinite products "diverge" when the limit is zero, a finite value?

Question

Infinite series (sums) are discussed a lot, but infinite products less so. Despite having tried hard to find reading material on infinite products, I have only found lecture notes and not proper texts.

Having read these, eg "Background Notes Theory of Infinite Products", I am puzzled as to why infinite products that do have a limit of 0 are said to "diverge".

Here is one example:

$$ \prod_{n=2}^{\infty}(1-\frac{1}{n}) $$

As usual we take the partial sum and consider the limit.

$$ \require{cancel} \begin{align} \prod_{n=2}^{N}(1-\frac{1}{n}) &= \prod_{n=2}^{N}(\frac{n-1}{n}) \\ \\ &= \frac{1}{\cancel{2}} \times \frac{\cancel{2}}{\cancel{3}} \times \frac{\cancel{3}}{4} \times \ldots \times \frac{N-1}{N} \\ \\ &= \frac{1}{N} \end{align} $$

As $N \rightarrow \infty$, this partial sum tends to 0.

Because this limit is finite, albeit 0, I feel this should be considered convergent. What key insight am I missing?

Update: according to lecture notes, "diverge to zero" means there are an infinite number of factors that are zero, so this specific example doesn't "diverge to zero".

Answer 1

The simplest way to explain this convention is that it ensures that an infinite product $\prod r_i$ of positive reals converges iff its logarithm $\sum \log r_i$ converges in the usual sense, which reduces the theory of infinite products (of positive reals) to the theory of infinite sums. If an infinite product diverges to $0$ then its logarithm diverges to $-\infty$. The divergence of the infinite product you describe turns out to be equivalent to the divergence of the harmonic series, for example (Example 5 in the notes you've linked).

This convention also has the desirable property that it makes convergence closed under taking inverses: $\prod r_i$ converges to $\ell$ iff $\prod r_i^{-1}$ converges to $\ell^{-1}$.

More abstractly, you might say that when we deal with infinite products we restrict our attention to the positive reals $\mathbb{R}_{+}$, because taking products involving $0$ is not very interesting. So $0$ is excluded and takes on the role of a "point at infinity" (and again this can be motivated by thinking about the logarithm $\log : \mathbb{R}_{+} \to \mathbb{R}$).

(Considering the nonzero reals is not much of an upgrade over the positive reals, because for convergence only finitely many terms can be negative anyway. There are reasons to consider the nonzero complex numbers though, e.g. when considering the Euler product of the Riemann zeta function for complex values of $s$.)