Polynomials tending to $\exp (t^2/2)$

Question

I need to show this, I got how it should work but I don't find a proper way to demonstrate it. Any thoughts ?

$$ \lim\limits_{N \to + \infty} \prod\limits_{n=0}^{N-1} \left(1+ n \frac{t^2}{N^2} \right) = \exp\left(\frac{t^2}{2}\right) $$

Thank you if you took time considering it

Answer 1

Hint:

Taking the logarithm and developing the logarithm with Taylor, the linear term is

$$\sum_{n=0}^{N-1}n\frac{t^2}{N^2}=\frac{(N-1)N}2\frac{t^2}{N^2}$$ which tends to $t^2/2$. The next terms vanish because of larger powers of $N$ at the denominator.