How to find the limit of the sequence

Question

I want to find the limit of the sequence $((1+\frac{1}{n})(1+\frac{2}{n})\ldots (1+\frac {n}{n}))^{\frac{1}{n}}$. Applying GM$\leq AM$ we can show that limit of this sequence is less than or equal to $\frac{3}{2}$. Also we can show that this limit is greater than or equal to $1$. But this will not help. How to proceed? I am not getting any idea.

Answer 1

$$\lim_{n\to +\infty}\frac{1}{n}\sum_{k=1}^{n}\log\left(1+\frac{k}{n}\right)=\int_{0}^{1}\log(1+x)\,dx = \log(4)-1 $$ hence by exponentiating $$ \lim_{n\to +\infty}\sqrt[n]{\prod_{k=1}^{n}\left(1+\frac{k}{n}\right)}=\color{red}{\frac{4}{e}}.$$

Answer 2

Note that by Stirling's approximation, $(n!)^{1/n}\sim n/e$. Therefore $$\left(\left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\ldots \left(1+\frac {n}{n}\right)\right)^{\frac{1}{n}} =\left(\frac{(2n)!}{n!n^n}\right)^{1/n}\sim \frac{{(2n/e)^2}}{(n/e)\cdot n}=\frac{4}{e}.$$