sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

$\textrm{(1) $\sum a_n$ converges, $\prod a_n$ does not}$

Take $$\{a_n\}:=\left\{\frac{i}{\sqrt{1}},\frac{-i}{\sqrt{1}},\frac{i}{\sqrt{2}},\frac{-i}{\sqrt{2}},\ldots\right\}$$ Then $$\sum a_n=i\sum(-1)^{n-1}b_n$$ where $$\{b_n\}=\left\{\frac{1}{\sqrt{1}},\frac{1}{\sqrt{1}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\ldots\right\}$$ is a sequence of positive real numbet, non-increasing and infinitesimal, hence by Leibniz the series $$\sum (-1)^{n-1}b_n$$ converges, hence $\sum a_n$ converges too.

Now consider the infinite product $\prod (1+a_n)$. For every positive integer $N$, the $2N$-partial product is $$\prod_{1}^{2N}a_j=\left(1+\frac{i}{\sqrt{1}}\right)\left(1-\frac{i}{\sqrt{1}}\right)\left(1+\frac{i}{\sqrt{2}}\right)\left(1-\frac{i}{\sqrt{2}}\right)\cdot\ldots\cdot\left(1-\frac{i}{\sqrt{2N}}\right)$$

$$=\left(1-\frac{i^2}{1}\right)\left(1-\frac{i^2}{1}\right)\ldots\left(1-\frac{i^2}{2N}\right)$$

$$=\left(1+1\right)\left(1+\frac{1}{2}\right)\ldots \left(1+\frac{1}{2N}\right)$$

$$=2\cdot\frac{3}{2}\cdot\frac{6}{3}\ldots\frac{2N+1}{N}$$

$$=2N+1$$ Thus $$\lim_{N\rightarrow +\infty}\prod_{1}^{2N}(1+a_n)=\lim_{N\rightarrow +\infty} 2N+1=+\infty$$ hence also diverges the following $$\lim_{N\rightarrow +\infty}\prod_{1}^{N} (1+a_n)=\prod_{1}^{+\infty}(1+a_n)$$

$\textrm{(2)$\prod (1+a_n)$ converges, $\sum a_n$ does not}$

Take $\{a_n\}$ defined by $$a_{2n-1}:=\frac{1}{\sqrt{n}}$$ $$a_{2n}:=-\frac{1}{1+\sqrt{n}}$$ for every $n\geq 1$. Then $$\sum_{n=1}^{+\infty}a_n=\sum_{n=1}^{+\infty}a_{2n-1}+\sum_{n=1}^{+\infty}a_{2n}$$

$$=\sum_{n=1}^{+\infty}\frac{1}{\sqrt{n}}-\sum_{n=1}^{+\infty}\frac{1}{1+\sqrt{n}}$$

$$=\sum_{n=1}^{+\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{1+\sqrt{n}}\right)$$

$$=\sum_{n=1}^{+\infty}\frac{1}{n+\sqrt{n}}$$

But $n+\sqrt{n}<2n$, hence $\frac{1}{n+\sqrt{n}}>\frac{1}{2n}$, and so $$\sum_{1}^{\infty}\frac{1}{n+\sqrt{n}}\geq\frac{1}{2}\sum_{1}^{\infty}\frac{1}{n}$$ and you get divergence by comparison with harmonic series.

Now, as for the divergence of infinite product, consider: $$\prod_{1}^{\infty}(1+a_n)=\prod_{1}^{\infty}(1+a_{2n})\cdot\prod_{1}^{\infty}(1+a_{2n-1})$$

$$=\prod_{1}^{\infty}(1+a_{2n})\cdot (1+a_{2n-1})$$

$$=\prod \left(1-\frac{1}{1+\sqrt{n}}\right)\left(1+\frac{1}{\sqrt{n}}\right)$$

$$=\ldots$$

$$=\prod\frac{\sqrt{n}(1+\sqrt{n})}{\sqrt{n}(1+\sqrt{n})}$$

$$=\prod_{1}^{\infty}1=1$$