Sum of IID random variables and MGF of normal distribution

[Luna=Luna]

If the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity, shouldn't the MGF of all random variables raised to the Nth power tend to the MGF of the normal distribution?

I tried to do this with the sum of bernouli variables and exponential variables and didn'treally get anywhere with either.

Does anyone know if this is even possible and where I can find the proof steps?

 

[Stephen Tashi]

If the distribution of a sum of N iid random variables tends to the normal distribution as n tends to infinity

You have to specify what type of convergence you're talking about when you say "tends". ( http://en.wikipedia.org/wiki/Convergence_of_random_variables)

The sum of iid random variables doesn't converge (in distribution) to a normal distribution. It's the mean of the sum that converges to a normal distribution.