Convergent or divergent? About a series with primes numbers

Question

Let $p_n$ denote the $n$-prime number. Is the series $$\sum \frac{p_n p_{n-1}...p_1}{(2+p_n p_{n-1}...p_1)}$$ convergent ?

Answer 1

No, it is not convergent. The $n$th term approaches $1$.

Answer 2

$a_n:= \dfrac{p_np_{n-1}.....p_1}{2+p_np_{n-1}.....p_1}=$

$1- \dfrac{2}{2+p_np_{n-1}.....p_1}.$

Note : $a_{n+1} >a_n.$

Let $n_0 \in \mathbb{Z^+} $.

For $n > n_0:$

$a_n >a_{n_0}$, hence bounded below.

Sum is divergent