Let $G \in \text{Geometric}(\theta)$ be a D4001: Geometric random positive integer.
Let $t \in \mathbb{R}$ be a D993: Real number.

Using results

(i)	R3205: Probability mass function for geometric random positive integer
(ii)	R3415: Necessary and sufficient conditions for convergence of geometric complex series

we have \begin{equation} \begin{split} \mathbb{E} e^{i t G} & = \sum_{n \in 1, 2, 3, \ldots} e^{i t n} \mathbb{P}(G = n) \\ & = \sum_{n \in 1, 2, 3, \ldots} e^{i t n} \theta (1 - \theta)^{n - 1} \\ & = \theta e^{i t} \sum_{n \in 1, 2, 3, \ldots} e^{i t (n - 1)} (1 - \theta)^{n - 1} \\ & = \theta e^{i t} \sum_{n \in \mathbb{N}} e^{i t n} (1 - \theta)^n \\ & = \theta e^{i t} \sum_{n \in \mathbb{N}} (e^{i t} (1 - \theta))^n \\ & = \theta e^{i t} \frac{1}{1 - e^{i t} (1 - \theta)} \\ & = \theta e^{i t} \frac{1}{1 - e^{i t} (1 - \theta)} \frac{e^{- i t}}{e^{- i t}} \\ & = \frac{\theta}{e^{- i t} - (1 - \theta)} \\ \end{split} \end{equation} $\square$