Approximation for the \(n^\text{th} \) prime number

The Prime Number Theorem states that \(\displaystyle \pi(x) \sim \frac{x}{\log{x}}\), where \(\displaystyle \pi(x)\) is the number of primes not exceeding \(\displaystyle x\), that is, \(\displaystyle \lim _{x\to \infty }{\frac {\;\pi (x)\;}{\frac {x}{\log(x)}}}=1\). Let \(\displaystyle p_{n}\) be the \(\displaystyle n\)th prime number. Prove that, as a corollary of the Prime Number Theorem, \(\displaystyle p_{n} \sim n\log{n}\), that is, \(\displaystyle p_{n} \to n\log{n}\) as \(\displaystyle n \to \infty\).

Note by Alexander Israel Flores Gutiérrez
1 year, 5 months ago

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