sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$ Does there exist a sequence $a_n$  of complex numbers such that $\sum _{i = 0}^\infty a_n$ converges and the product $\prod _{i = 0}^\infty (1+a_n)$ does not converge to any complex number(not even 0).
And conversely a sequence  $a_n$ s.t. $\prod _{i = 0}^\infty (1+a_n)$ converges to a non zero complex number. But $\sum _{i = 0}^\infty a_n$ does not converge.
We all know that if $\sum _{i = 0}^\infty a_n $ converges absolutely iff the corresp product converges absolutely. So i tried alternating serieses like $ a_n = (-1)^n /n $ and $(-1)^n / \sqrt n $  but that did not work. i tried like $ \sum _{i} \sum _ {k} \frac{ e^{2(pi)i/k}}{k} $but i couldn't go furthur.
Can anybody help?
 A: $\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:
$$\prod_{1}^{\infty}(1+a_n)=\prod_{1}^{\infty}(1+a_{2n})\cdot\prod_{1}^{\infty}(1+a_{2n-1})$$
$$=\prod_{1}^{\infty}(1+a_{2n})\cdot (1+a_{2n-1})$$
$$=\prod \left(1-\frac{1}{1+\sqrt{n}}\right)\left(1+\frac{1}{\sqrt{n}}\right)$$
$$=\ldots$$
$$=\prod\frac{\sqrt{n}(1+\sqrt{n})}{\sqrt{n}(1+\sqrt{n})}$$
$$=\prod_{1}^{\infty}1=1$$
