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

For each positive integer n, let yₙ = [1/n(n+1)(n+2)…(n+n)]^(1/n) For x ∈ R, let [x] be the greatest integer less than or equal to x. If lim_(n→∞) yₙ = L, then the value of [L] is ______.

$y_n = \left[\dfrac{1}{n^n}(n+1)(n+2)\cdots(n+n)\right]^{1/n}$

$[x] = \text{greatest integer} \le x$

$\text{To find: } [L],\qquad L = \lim_{n\to\infty} y_n$

$\dfrac{(n+1)(n+2)\cdots(n+n)}{n^n} = \prod_{k=1}^{n}\left(1+\dfrac{k}{n}\right)$

$y_n = \left[\prod_{k=1}^{n}\left(1+\dfrac{k}{n}\right)\right]^{1/n}$

$\ln y_n = \dfrac{1}{n}\sum_{k=1}^{n}\ln\!\left(1+\dfrac{k}{n}\right)$

$\lim_{n\to\infty}\ln y_n = \int_0^1 \ln(1+x)\,dx$

$\int \ln(1+x)\,dx = (1+x)\ln(1+x) - (1+x)$

$\int_0^1 = (2\ln2 - 2) - (0 - 1) = 2\ln2 - 1$

$\ln L = 2\ln2 - 1 = \ln\dfrac{4}{e}$

$L = \dfrac{4}{e} \approx 1.47$

$\boxed{[L] = 1}$