How did the authors expand 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:

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?