Find $\lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} $ using Riemann integral Wonder how to determine this limit by the use of Riemann integral. The limit is as follows:
$$\lim_{n \to +\infty} \frac{1}{n}\sqrt[n]{\frac{(2n)!}{n!}} $$
My instructor told me that the usage of Riemann integral gives spectacular result. Checked Rudin, but did not find any valuable references.
I am very interested in seeing how this "spectacular result" emanates.
Help/advices/solutions very, very appreciated!

Applying Stirling's formula, $n! \displaystyle\operatorname*{\sim}_{n\to\infty} \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$, I got the result $\frac{4}{e}$.
