Find the value of $\,\, \lim_{n \to \infty}\Big(\!\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\!\Big)^{\!1/n} $ What is the limit of:
$$
\lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big)
\cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}?
$$
By computer, I guess the limit is equal to $\dfrac{4}{e}$, but I have no idea about proving that.
Thank you for your any help.

Let $f(n)=[\prod_{i=0}^n (1+r/n)]^{1/n}$ Then, $\ln f(n)=1/n\sum_{r=0}^n\ln (1+r/n)\Rightarrow \lim_{n\to \infty}\ln f(n)=\int_{0}^1 \ln (x+1) dx=\ln 2-\int_{0}^1 \dfrac{x}{x+1}dx=2\ln 2-1\Rightarrow \lim_{n\to \infty} f(n) =4/e$
