Limit of a sequence including infinite product. $\lim\limits_{n \to\infty}\prod_{k=1}^n \left(1+\frac{k}{n^2}\right)$ I need to find the limit of the following sequence:
$$\lim\limits_{n \to\infty}\prod_{k=1}^n \left(1+\frac{k}{n^2}\right)$$

$$\left(1+\frac{k}{n^2} \right)\left(1+\frac{n-k}{n^2} \right)=\left(1+\frac{1}{n} +\frac{k(n-k)}{n^4} \right)$$
Then we have
$$\left(1+\frac{1}{n}  \right) \leq \left(1+\frac{1}{n} +\frac{k(n-k)}{n^4} \right) \leq \left(1+\frac{1}{n} +\frac{1}{n^2} \right)=\left(1 +\frac{n+1}{n^2} \right)$$
Therefore
$$\left(1+\frac{1}{n}  \right) ^\frac{n}{2} \leq  \prod_{k=1}^n\left(1+\frac{k}{n^2}\right) \leq \left(1 +\frac{n+1}{n^2} \right)^\frac{n}{2}$$
and squeeze it.
