If $Y_n$ is a Poisson random variable with mean $n^{1/2}$

$S_n = \left(\frac{\sqrt{2} Y_n - \sqrt{2n}}{n^{1/4}}\right)$

If we consider a sequence $S_1, S_2, . . . S_n$, provide the the limiting distribution.

Attempt

Mgf of $Y_n$ is as follows:

$M_{Y_n}(t) = exp\sqrt{n}(e^{t} -1)$

So for all $t \in \mathbb{R}$, we have

$M_{\left(\frac{\sqrt{2}}{n^{1/4}}\right)Y_n - \left(\frac{\sqrt{2}n}{n^{1/4}}\right)}(t) = exp\sqrt{n}(e^{\left(\frac{\sqrt{2}}{n^{1/4}}\right)(t-n )-1} )$

Taking the $\lim_{x\to \infty}$ $M_{T_n}(t) = $ will lead to a lot of mess that, I think, will be a Maclaurin Series expansion. Any nudge in the right direction or an easier approach would be appreciated.