Find the characteristic function of a random variable X such that P({X=k}) = $2^{-k}$, $k =1,2,3,4,5, \ldots$
What I was doing is: $$ \phi_x(t) = E(e^{itx}) = \sum^{\infty}_{k=1} 2^{-k} * e^{itk} = \sum^{\infty}_{k=0} \left(\frac{e^{it}}{2}\right)^k -1 $$
From that, I don't know what else to do.
We have $|e^{it}/2|<1$. So $\varphi_X(t)=e^{it}/(2-e^{it})$. Also $X$ is $\sim\textrm{Geometric}(1/2)$ over $\mathbb{N}$.