Convergence of infinite product $\prod_{n=2}^\infty (1- \frac 1n) $

Question

I am revising Complex Analysis and I am a bit confused. I have a couple of results from lectures which say that $\prod_{n=1}^\infty (1+a_n)$ converges if and only if the sum $\sum_{n=1}^\infty \log(1+a_n) $ converges absolutely.

And also, the infinite product converges absolutely if and only if the $|a_n|$ are summable.

Consider the example: $$\prod_{n=2}^\infty \left(1- \frac 1n\right) $$ It can easily be shown that the partial product $\prod_{n=2}^{N} (1- \frac 1n) = \frac 1N $ which tends to zero as $N\to \infty$

Maybe I'm interpreting the theorems wrong, but the sums:

$\sum_{n=2}^\infty |\log(1-\frac 1n)| $ and $\sum_{n=1}^\infty |-\frac 1n| $ both diverge, so from the results I listed at the start, the product cannot converge, but it does - to zero.

I'm getting quite confused here so I would appreciate some help!

Answer 1

If partial products tends to zero as $N\to \infty$ we say that infinite product diverges to $0$.

Answer 2

The right answer to your question is that the statement should read as follows:

$\displaystyle\prod_{n=1}^\infty (1+a_n)$ converges to a non-zero real number if and only if the sum $\displaystyle\sum_{n=1}^\infty \log(1+a_n) $ converges absolutely.

As you have rightly observed the infinite product in your example does indeed converge to $0$. There are some people who would argue that the infinite product diverges to $0$, since the corresponding infinite sum diverges. However, this is incorrect, especially if you view the infinite product as $$\displaystyle\prod_{n=1}^\infty (1+a_n) = \lim_{N \to \infty} b_N, \text{ where } b_N = \displaystyle\prod_{n=1}^N (1+a_n)$$

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EDIT Also, to emphasize the fact that converging to zero is the right term to use, I quote the following proposition from Stein and Shakarchi. Page 141:

"Proposition $3.1$: If $\Sigma \left\vert a_n \right\vert < \infty$, then the product $\prod(1+a_n)$ converges. Moreover, the product converges to $0$ if and only if one of its factors is $0$."

Hence, converges to zero is the right term to use. When the sum diverges, one needs to be careful as to what happens to the product and argue whether the product converges to zero or diverges accordingly.