We are some folks working on proving the central limit theorem. The missing piece we lack is finding the limit in the following equation for $n \rightarrow \infty$. Maple is able to find a limit, but so far we haven't been succesfull in recreating this analytically by hand.
$$\left( \left( 1-p \right) {{\rm e}^{{\frac {-i \cdot\theta}{ \sqrt{n}} \sqrt{{\frac {p}{p-1}}}}}}+p{{\rm e}^{{\frac {i \cdot\theta}{ \sqrt{n}} \sqrt{{\frac {p-1}{p}}}}}} \right) ^{n} $$
If anybody has any helt with this it will be much appreciated.
HINT
Use that for $x\to 0$
$$e^x=1+x+\frac{x^2}2+o(x^2)$$
maybe also first order expansion suffices.