For each positive integer $n$, let $y_{n}=\frac{1}{n}((n+1)(n+2) \ldots(n+n))^{\frac{1}{n}}$ For $x

Question

For each positive integer $n$, let $y_{n}=\frac{1}{n}((n+1)(n+2) \ldots(n+n))^{\frac{1}{n}}$ For $x \in \mathbb{R}$, let $[x]$ be the greatest integer less than or equal to $x$. If $\lim _{n \rightarrow \infty} y_{n}=L$, then the value of $[L]$ is

Professor PC Thomas Classes · Answer

(1) : We have\ $\log y_{n}=\frac{1}{n}\left\{\begin{array}{c}\log \left(1+\frac{1}{n}ight)+\log \left(1+\frac{2}{n}ight) \ +\ldots+\log \left(1+\frac{n}{n}ight)\end{array}ight\}$\ $	herefore \lim _{n ightarrow \infty} \log y_{n}=\lim _{n ightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \log \left(1+\frac{r}{n}ight)$\ $=\int_{0}^{1} \log (1+x) d x=\int_{1}^{2} \log x d x$\ $=[x \log x-x]_{1}^{2}=2 \log 2-1=\log \frac{4}{e}$\ $	herefore \quad L=4 / e$ Thus, $[L]=1$.\