Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that \begin{equation}\frac{X_n-n\mu}{\sqrt{n}\sigma}\rightarrow\mathcal{N}(0,1),\end{equation} for some known $\mu$ and unknown $\sigma$. Given the function \begin{equation}F[z,s]=\sum_{n=0}^\infty z^{-n} M_{X_n}(s),\end{equation} is it possible to extract $\sigma$ without the use of inverse transforms?

For example: \begin{equation}F[z,s]=\frac{zs}{1-e^s+zs}.\end{equation} Answer: $\sigma^2=\frac{1}{12}$.