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