Convergence of infinite product of prime reciprocals?
Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
Solution 1:
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