I need examples of the following facts:
1) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges $\nRightarrow \prod_{j=0}^{+\infty}(1+|a_{j}|)$ converges
2) $\prod_{j=0}^{+\infty}(1+a_{j})$ converges $\nRightarrow \sum_{j=0}^{+\infty}a_{j}$ converges
3) $\sum_{j=0}^{+\infty}a_{j}$ converges $\nRightarrow \prod_{j=0}^{+\infty}(1+a_{j})$ converges
where $a_{j} \in \mathbb{C}$.
Any hint ?
You should know (if not yet, then from now on) that $\prod\limits_{n=1}^\infty (1+\lvert a_n\rvert)$ converges if and only if $\sum\limits_{n=1}^\infty \lvert a_n\rvert$ converges. That makes the first rather easy, picking a conditionally convergent series has a good chance of making $\prod (1+a_n)$ converge but not $\prod (1+\lvert a_n\rvert)$. If the first try doesn't succeed, try another time.
For the second and third non-implications, note that
$$\prod_{n=1}^\infty (1+a_n) \text{ converges } \iff \sum_{n=1}^\infty \log (1+a_n) \text{ converges},$$
provided none of the $a_n$ has the value $-1$ (when $\log (1+a_n)$ is not defined), and we take the principal branch of the logarithm when $\lvert a_n\rvert < 1$.
So if we write $\lambda_n = \log (1+a_n)$, we have $a_n = e^{\lambda_n}-1 = \lambda_n + \frac{1}{2}\lambda_n^2 + O(\lambda_n^3)$.
One can then choose a sequence $\lambda_n$ such that $\sum\limits_{n=1}^\infty \lambda_n$ converges (conditionally), $\sum\limits_{n=1}^\infty \lambda_n^2$ diverges, and $\sum\limits_{n=1}^\infty \lvert \lambda_n\rvert^3$ converges.
The third is very similar, since
$$\log (1+a_n) = a_n - \frac{1}{2}a_n^2 + O(a_n^3),$$
one can use the sequence $\lambda_n$ (or $i\lambda_n$) from the second as $a_n$.