Please evaluate the limit $\lim\limits_{n\rightarrow\infty} \prod\limits_{k=1}^{n}\left(1+\dfrac{k}{n^2}\right)^2$ if it exists.
I have a heuristic argument to solve this limit but cannot prove it rigorously. The idea is that for large $n$, $\left(1+\dfrac{k}{n^2}\right)$ can be approximated by $\left(1+\dfrac{1}{n^2}\right)^k$. So, essentially the limiting value is equal to $\lim\limits_{n\rightarrow \infty} \left(1+\dfrac{1}{n^2}\right)^{2\sum\limits_{k=1}^{n} k}$. This quantity can be easily computed to be $e$. Please help me out to make this argument rigorous.
