Is the following infinite product zero?
$$P_1 = 2 \times \mathbf{0} \times 4 \times 5 \times 6 \times \ldots $$
The reason for doubt is that the following infinite product diverges to infinity.
$$P_2 = 2 \times \mathbf{3} \times 4 \times 5 \times 6 \times \ldots $$
This makes me consider the following:
$$ f(x)= x\cdot e^{1/x} $$
at $x\rightarrow 0$,the factor $x$ is not enough to cause the product to be zero.
An infinite product is a sequence give by $a_n = \prod_{k=1}^n p_k$ with $p_k$ the $k$-th term of the product. Since for $k \geq 2$ you have $\prod_{k=1}^n p_k = 2 \cdot 0 \cdot \prod_{k=3}^n p_k$ we know that $ a_n = 0$ for all $n \geq 2$ so this sequence converges to $0$ and rapidly.