TITLE: $y=(1+\frac{1}{n^3})(1+\frac{2^3}{n^3})...(1+\frac{n^3}{n^3})$
QUESTION [5 upvotes]: If $$y=\left(1+\frac{1}{n^3}\right)\left(1+\frac{2^3}{n^3}\right)...\left(1+\frac{n^3}{n^3}\right)$$ then what is $$\lim_{n\rightarrow \infty } y^{\frac{1}{n}}$$ ?
Can anyone guide me for this? Hints are appreciated. Thanks in advance.

REPLY [6 votes]: In order to complete the previous answer, you may also notice that once
$$ \lim_{n\to +\infty}\sqrt[n]{y_n}=\exp\int_{0}^{1}\log(1+x^3)\,dx \tag{1}$$
is established, by integration by parts and partial fraction decomposition it follows that:
$$ \lim_{n\to +\infty}\sqrt[n]{y_n}= \color{red}{\large 4\,e^{\frac{\pi}{\sqrt{3}}-3}}.\tag{2}$$