For the infinite product $\prod_{k=1}^\infty z_k$ to converge, it is necessary that for every $\varepsilon > 0$ a $N$ exists such that for every $k > N$ and for every integer $r \geq 1$: $$|z_{k+1}z_{k+2}\cdots z_{k+r} - 1| < \varepsilon$$
Is it a sufficient condition? if the answer is "NO", How can I change the condition to become sufficient?
For an infinite product to converge, you first take it's ln (or log of any base, but ln is standard) with by the identities of log will turn the product in a infinite sum, if this sum converges, so does de product.
PS: I am not being tecnical here, I would need to prove this fact if so, but I believe that the intuition is more important.