Let $p_n$ be the $n$th prime, $n\ge 1$. I was wondering if the following product converges $$P=\prod _{k=1}^{\infty} \frac{p_{k+1}}{\sqrt{p_k p_{k+2}}}$$
I have computed some numeric examples and the result $1.224$ keeps the same $3$ decimals when $k$ goes from $10000$ on, which makes me think it converges, but I don't know how to prove or disprove it.
Any idea?
Thank you for your attention
Edit
The result here that $$\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$$ and the comment made by Donald Splutterwit that the general term of the product can be written as $$\sqrt{\frac{3}{2}}\sqrt{\frac{p_n}{p_{n+1}}}$$ let me conclude that the product converges to $$P=\sqrt{\frac{3}{2}}$$