M.g.f for Poisson distribution

I am having trouble understanding the maths between E(\(e^{tx}\)) where I get a summation from 0 to \(\infty\) and the m.g.f for Poisson distribution. The notes I have are to break out \(e^{-\lambda}/e^{-\lambda e^t}\)
(Now, don't get me wrong - I DO get how to make this last expression easier on the eye - It's just that breaking this out from the sum makes the sum go towards 1 as x approaches \(\infty\))
I can't seem to find what is supposed to be left in the summation nor why that would behave so nicely when the whole expression looks so strange (to me). I have been to (http://mathworld.wolfram.com/PoissonDistribution.html) and other resources, but my problem is not finding the mgf or using it but understanding how to get to it myself from the probability mass function.


TS Contributor
\( X \sim \mathrm{Poisson}(\lambda) \)

\( E[e^{tX}]
= \sum_{x=0}^{+\infty} e^{tx} e^{-\lambda} \frac {\lambda^x} {x!}
= e^{-\lambda} \sum_{x=0}^{+\infty} \frac {(e^t\lambda)^x} {x!}
= e^{-\lambda} \exp\{e^t\lambda\} = \exp\{\lambda(e^t - 1)\}

It is not difficult; You just need to know the series for exponential function
(or just the kernel of the Poisson distribution)