Question

For each positive integer $n$, let
$y_n=\frac{1}{n}(n+1)(n+2)...(n+n{)}^{1/n}$
For $x\in R$, let [x] be the greatest integer less than or equal to $x$. If $\underset{n\to \infty }{lim}{y}_n=L$, then the value of $\left[L\right]$ is _________.

Answer 1

$y_n{\{(1+\frac{1}{n})(1+\frac{2}{n})...(1+\frac{n}{n})\}}^{\frac{1}{n}}$

$y_n=\prod _{r=1}^n{(1+\frac{r}{n})}^{1/n}$

$log\left(y_n\right)=\frac{1}{n}\sum _{r=1}^{n}ln(1+\frac{r}{n})$

$\Rightarrow \underset{n\to \infty }{lim}log\left(y_n\right)=\underset{x\to \infty }{lim}\sum _{r=1}^n\frac{1}{n}ln(1+\frac{r}{n})$

$\Rightarrow logL={\int }_0^{1}ln(1+x)dx$

$\Rightarrow logL={|x.ln(1+x\left)\right|}_0^1-|\frac{1}{1+x}.x{|}_0^1$

$\Rightarrow logL=\log \frac{4}{e}$

$\Rightarrow L=\frac{4}{e}$

$\Rightarrow \left[L\right]=1$.