TITLE: Evaluate $\lim_{n \rightarrow \infty} \frac {[(n+1)(n+2)\cdots(n+n)]^{1/n}}{n}$
QUESTION [7 upvotes]: Evaluate $$\lim_{n \rightarrow \infty~} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$
Attempt: Let $$y=\lim_{n \rightarrow \infty} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$
$$\implies \log y = \lim_{n \rightarrow \infty} \dfrac {1} {n} [\log (n+1) +\cdots+log(n+n)-log(n)] $$
How do I move forward?
Thank you very much for your help.

REPLY [3 votes]: You can try this easy one:
$\lim_{n\to \infty}\dfrac{1}{n}{[(n+1)(n+2)...(n+n)]}^{\frac{1}{n}}=\lim _{n\to \infty}\dfrac{1}{n}[n^n(1+\frac{1}{n})(1+\frac{2}{n})...(1+\frac{n}{n})]^{\frac{1}{n}}$
$=\lim_{n\to \infty}{\prod_{k=1}^n}(1+\frac{k}{n})^{\frac{1}{n}}$
Let $y={\prod_{k=1}^n}(1+\frac{k}{n})^{\frac{1}{n}}\implies \ln y=\frac{1}{n}\sum _{k=1}^n\ln (1+\frac{k}{n})=\int _0^ 1\ln(1+x)\operatorname{dx}$
$\implies y=\frac{4}{e}$