For each positive integer n , let y _ n = (1)/(n) ( ( n + 1 )( n +...

{y}_{n} = {\left\{\left(1 + \dfrac{1}{n}\right)\left(1 + \dfrac{2}{n}\right) \ldots . \left(1 + \dfrac{n}{n}\right)\right\}}^{\dfrac{1}{n}}$<br />${y}_{n} = \mathop{∏}_{r = 1}^{n}{\left(1 + \dfrac{r}{n}\right)}^{\dfrac{1}{n}}$<br />$\ln \left({y}_{n}\right) = \dfrac{1}{n}\mathop{∑}_{r = 1}^{n}\ln \left(1 + \dfrac{r}{n}\right)$<br />$⇒ \lim_{n \to \infty} \ln ( {y}_{n} ) = \lim_{n \to \infty} \mathop{∑}_{r = 1}^{n}\dfrac{1}{n}\ln \left(1 + \dfrac{r}{n}\right)$<br />$⇒ \ln \left(L\right) = {∫}_{0}^{1}\ln \left(1 + x\right)dx$<br />$⇒ \ln \left(L\right) = \ln \left(\dfrac{4}{e}\right)$<br />$⇒ L = \dfrac{4}{e}$<br />\left[L\right] = 1$