$ \lim_{n \to \infty} \frac{\sqrt[n]{n*(n+1)...(2n)}}{n}$

You can use the Riemann Integral to compute the limit. Since \begin{eqnarray} &&\lim_{n \to \infty} \ln\frac{\sqrt[n]{n(n+1)...(2n)}}{n}\\ &=&\lim_{n \to \infty}\frac1n\sum_{k=1}^n\ln(1+\frac kn)\\ &=&\int_0^1\ln(1+x)dx\\ &=&2\ln2-1 \end{eqnarray} one has $$ \lim_{n \to \infty} \frac{\sqrt[n]{n(n+1)...(2n)}}{n}=e^{2\ln2-1}=\frac{4}{e}. $$