If $p(n)$ be the $n$-th prime number, is $\lim_{n\to \infty} {p(n)}^\frac{1}{n}$ exist and how we can find it if exist?

Question

In my research, I need a counterexample of sequences, and $p(n)$, the $n$- prime number sequence is an important sequence to me. Generally for an arbitrary sequence of the real and positive numbers, such as $a(n)$, any classification of the $\lim_{n\to \infty} {a(n)}^\frac{1}{n}$ will be useful to me.
However, according to my calculation, this limit must be greater than $1.76151$.

Answer 1

The Prime Number Theorem implies $p(n) \sim n \log n$, so $p(n)^{1/n} \to 1$.

Answer 2

The $n$th prime is roughly $n\log n$ therefore $$\underset{n\to \infty }{\text{lim}}(n \log n)^{1/n}=1$$