# If $p_n$ is prime does $\frac{p_{n+1}}{p_n}$ or $\frac{p_n}{p_{n+1}}$ tend towards some number for large $n$? [duplicate]

If $p_n$ is the nth prime number does $\frac{p_{n+1}}{p_n}$ approach any particular number when $n$ is large? What about $\frac{p_n}{p_{n+1}}$? Is it possible to prove that the limits converge or diverge with elementary methods?
• $p_n \sim n\ln n$ by PNT, hence $1$ for both questions. – Paolo Leonetti Nov 30 '16 at 14:44