Question

For each positive integer $n$, let
$y_{n} = \frac{1}{n}\left(\left(n+1\right)\left(n+2\right)\cdots\left(n+n\right)\right)^{\frac{1}{n}}$
For $x \,\in\, ℝ$ let [L] be the greatest integer less than or equal to $x$. If $\displaystyle\lim_{n \to \infty} y_{n} = L$, then the value of [L] is _____ .

Answer 1

$y_{n}=\left\{\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right) \ldots\left(1+\frac{n}{n}\right)\right\}^{\frac{1}{n}}$
$y_{n}=\displaystyle\prod_{r=1}^{n}\left(1+\frac{r}{n}\right)^{\frac{1}{n}} \log y_{n}=\frac{1}{n} \displaystyle\sum_{r=1}^{n} \ln \left(1+\frac{r}{n}\right)$
$\Rightarrow \displaystyle\lim _{n \rightarrow \infty} \log y_{n}=\displaystyle\lim _{x \rightarrow \infty} \displaystyle\sum_{r=1}^{n} \frac{1}{n}\left(1+\frac{r}{n}\right)$
$\Rightarrow \log L=\int\limits_{0}^{1} \ln (1+x) d x $
$\Rightarrow \log L=\log \frac{4}{e}$
$ \Rightarrow L=\frac{4}{e} \Rightarrow [L]=1$