The theorems you describe only consider products of the form $\prod(1+a_n)$ with $a_n \geqslant 0$ and $\prod(1-b_n)$ with $0\leqslant b_n < 1$.
In general, we have infinite products where $a_n$ could be any real or complex number. The strongest condition that guarantees convergence of $\prod(1+a_n)$ is, of course, absolute convergence. This reverts back to your first theorem.
The product $\prod(1+a_n)$ where $a_n \in \mathbb{R} \text{ or }
\mathbb{C}$ is said to be absolutely convergent if $\prod (1+|a_n|)$
is convergent. The product converges absolutely if and only if
$\sum|a_n|$ is convergent. Also, convergence of $\prod (1+|a_n|) $
implies convergence of $\prod (1 +a_n)$.
In the absence of absolute convergence, the product may still converge conditionally but, as far as I know, there are no all-encompassing necessary and sufficient conditions.
There are some ad hoc theorems for conditional convergence of products where the sign of $a_n$ is unrestricted. For example:
If the series $\sum a_n^2$ is convergent, then the product
$\prod(1+a_n)$ and the series $\sum a_n$ either both converge or both
diverge.
To prove this note that if $\sum a_n^2$ converges then $|a_n| \to 0$ and for sufficiently large $n$ we have $|a_n| < 1/2$ and
$$|\log(1+a_n) - a_n| = \left|\sum_{k=2}^\infty(-1)^{k}\frac{a_n^k}{k} \right| \leqslant \frac{a_n^2}{2}\sum_{k=0}^\infty|a_n|^k = \frac{a_n^2}{2}\frac{1}{1-|a_n|}< a_n^2$$
Thus the series $\sum [\log(1+a_n) - a_n]$ is absolutely convergent by the comparison test, and we have existence of the limit,
$$\lim_{N\to \infty} \log \left(\prod_{n=1}^N (1+a_n)\right) - \sum_{n=1}^Na_n =\lim_{N\to \infty} \prod_{n=1}^N [\log(1+a_n)- a_n ] $$
proving that the product and series must converge or diverge together.
Hint: Write $\tau = a +ib.$ Then $b>0.$ We have
$$e^{2\pi i n\tau} = e^{-2\pi nb}\cdot e^{2\pi i na}.$$
What is the absolute value of the last expression?
Best Answer
Taking for granted that $\exp$ is continuous, we just need to show that $\sum_{k=1}^\infty \log (1+a_k)$ converges absolutely if $\sum_{k=1}^\infty a_k$ does. Note that the assumption implies that $a_k \to 0$ so that the (principal) logarithm is well defined for large enough $k$; without loss $k=1$ is large enough.
For this, note the (imprecise) inequality from e.g. Taylor expansion that $$ |\log(1+a_k)| ≤ 10|a_k|$$ so that $$\sum_{k=1}^K|\log(1+a_k)| ≤ 10\sum_{k=1}^K |a_k| \to 10 \sum_1^\infty |a_k| < \infty$$ This implies that $\sum_{k=1}^\infty \log (1+a_k)$ converges absolutely, and therefore the product $$ \prod_{k=1}^\infty (1+a_k) := \lim_{K\to \infty}\exp \left(\sum_{k=1}^K \log(1+a_k)\right) = \exp \left(\sum_{k=1}^\infty \log(1+a_k)\right)$$ converges.
It is a positive number because $\exp$ restricted to the reals has positive range, and $∏_{k=1}^\infty |1+a_k| = \exp (\sum_{k=1}^\infty\log|1+a_k|)$ (with the same definition of infinite product above) is the exponential of a real number.