Per OP's request I made my comment into an answer. I am making this CW; if someone wants to add details of the solutions, feel free to do so. (Of course, if you prefer, you can post them in a separate post, so that you are rewarded by reputation for you effort.)
Such Examples are given in Problems 3.8.5 and 3.8.9 in Wieslawa J. Kaczor, Maria T. Nowak: Problems in mathematical analysis: Volume 1; Real Numbers, Sequences and Series, p.113-114.
Problem 3.8.5 Set
$$a_{2n-1}=\frac1{\sqrt{n}}+\frac1n,\qquad a_{2n}=-\frac1{\sqrt{n}}, n\in\mathbb N.$$
Show that the product $\prod\limits_{n=1}^\infty(1+a_n)$ converges, although the series $\sum\limits_{n=1}^\infty a_n$ diverges.
A solution is given on p.364.
Problem 3.8.9. Prove that the product $\prod\limits_{n=1}^\infty \left(1+(-1)^{n+1}\frac1{\sqrt{n}}\right)$ diverges although the series $\sum\limits_{n=1}^\infty (-1)^{n+1}\frac1{\sqrt{n}}$ converges.
A solution is given on p.365.
Best Answer
If all $a_n \in (0,1)$, $\displaystyle\prod_{n=1}^{+\infty} (1- a_n)$ is non-zero if and only if $\sum_{n=1}^{+\infty} a_n < +\infty$
And we know that $\displaystyle\sum_{n=1}^{+\infty} \frac{1}{p_n} = +\infty$, you can find proofs here