MGFs, power series, and pushing the expectation through

I got stuck on what I thought was a simple point while working through a stats textbook! I figured it out, I think, but was this the most straightforward path to understanding? You tell me! In this note, we explore some oft-used facts about moment-generating functions, and explain a connection with tail and concentration inequalities. Overview and my confusion I am currently reading through Martin Wainwright’s High-dimensional statistics: A non-asymptotic viewpoint, to fill some gaps in my stat theory knowledge.