$\lim_{n \to \infty}(1+\frac{1}{n^2})(1+\frac{2}{n^2})...(1+\frac{n}{n^2})=e^{\frac{1}{2}}$. Here is the beginning of a proof:
Suppose $0<k \leq n$,
$1+\frac{1}{n}<(1+\frac{k}{n^2})(1+\frac{n+1-k}{n^2})=1+\frac{n+1}{n^2}+\frac{k(n+1-k)}{n^4}\leq 1+\frac{1}{n}+\frac{1}{n^2}+\frac{(n+1)^2}{4n^4}$.
I'm confused by the second inequality above. 

Hint: By AM-GM Inequality
$$\frac{k+ n+1-k}{2} \geq \sqrt{k(n+1-k)}$$
