We're told not to use the central limit theorem to show that the normal approximation is suitable for a binomial distribution when n tends to infinity.
I've managed to show the answer, but it involves some messy infinite series.
Eg: $$M_Z(T) = M_{\frac{X-\mu}{\sigma}} = e^{-\frac{t\mu}{\sigma}}[1 + \theta(e^{\frac{t}{\sigma}} - 1]^n$$ $$ \mu = n\theta \ and \ \sigma = \sqrt{n\theta(1-\theta)}$$
Then without putting it all here, I take logarithms, use the infinite series of $ e^{\frac{t}{\sigma}}$ and then the infinite series of $ln(1+x)$, collect some terms and show that a lot tend to zero as n tends to infinity, leaving us with the Normal MGF. It's quite ugly, and I'm wondering if there is a clearer method (but still using MGFs)?