In this proof about a central limit theorem (see Theorem 2.6.2), the author writes the Taylor series for $\log(1+ix)$ and $e^{ix}$ and use it for the calculation for the complex random variable $e^{itS_n}$.
The taylor series for $\log(1+ix)$ only converges for $|x|<1$ or am I wrong?
I also found this argument in other similar proofs but in my opinion the calculation of $e^{itS_n}$ should just hold on the set where $|tS_n|<1$.