Is $\prod_{p}{\frac{p-1}{p+1}}=0$?

Question

Is it true that the $$\prod_{p}{\frac{p-1}{p+1}}=0$$ where the product runs over the prime numbers $p$?

Answer 1

There is a theorem stating that $$ \prod_{i=0}^{\infty} (1-a_i)=0 \iff \sum_{i=0}^{\infty}a_i=\infty \quad\quad\quad0 <a_i<1$$

$$$$

Therefore, yes, it converges to zero sice $\frac{p-1}{p+1}=1-\frac{2}{p+1}$ and $\sum_p \frac{1}{p} = \infty$.