How did the authors expand $\displaystyle \phi(\omega)$ below? (Related to the characteristic function derived from the moment generating function)

I'm self studying this, and not sure what area of math covers these topics? I know how to take a Taylor expansion of a typical function, but what about this improper integral? Or the sum? I'm not sure what they did.

EDIT: $\displaystyle \phi_X(\omega)=M_X(i\omega)=E(e^{i\omega X})$

Ah, nevermind, I figured it out -- they used several theorems relating to expectation and started from the E(e^{i\omega X}) instead of the integral or sum.

But how do they know the integral converges? What test did they use?