Showing MGF of Poisson converges to MGF of N(0,1) I'm trying to finish up a proof of the CLT for the Poisson distribution, but am having some trouble evaluating a limit. I've shown that the moment generating function for the standardized Poisson is $$M_X (t) = \exp \left \{-t\sqrt{\lambda}-\lambda+\lambda e^{t/\sqrt{\lambda}} \right \}$$ (this is the form of the MGF which was asked for us to show) But I'm having some trouble evaluating $$\lim_{\lambda \rightarrow \infty} \exp \left \{-t\sqrt{\lambda}-\lambda+\lambda e^{t/\sqrt{\lambda}} \right \} = e^{t^2/2}$$ Any suggestions for my mathematically-challenged mind?
 A: Figured it out with the help of my roommate. Here's the solution if anyone is curious:
$$
\begin{align*}
M_X (t) &= \exp \left \{-t\sqrt{\lambda}-\lambda+\lambda e^{t/\sqrt{\lambda}} \right \} \\
        &= \exp \left \{ -t\sqrt{\lambda}-\lambda \right \} \exp \left \{ \lambda e^{t/\sqrt{\lambda}} \right \} 
\end{align*}
$$
And by expanding the term $e^{t/\sqrt{\lambda}}$ by the Maclaurin series we have
$$
\begin{align*}
M_X (t) &= \exp \left \{ -t\sqrt{\lambda}-\lambda \right \} \exp \left \{ \lambda \sum_{i=0}^{\infty} \frac{(t/ \lambda)^i}{i!} \right \} \\ 
&= \exp \left \{ -t\sqrt{\lambda}-\lambda \right \} \exp \left \{ \lambda \left ( 1 + \frac{t}{\sqrt{\lambda}} + \frac{t^2}{2\lambda} + \frac{t^3}{6\lambda^{3/2}} + \cdots \right ) \right \} \\
&= \exp \left \{ -t\sqrt{\lambda}-\lambda \right \} \exp \left \{ \lambda + t\sqrt{\lambda} + \frac{t^2}{2} + \frac{t^3}{6 \sqrt{\lambda}} + \cdots  \right \} \\
&= \exp \left \{ \frac{t^2}{2} + \frac{t^3}{6 \sqrt{\lambda}} + \cdots  \right \} \rightarrow e^{t^2/2} \text{ as } \lambda \rightarrow \infty. 
\end{align*}
$$
Woot. 
