I'm awful with infinite summations! No idea how they did the last step in the proof in the attached image. Can anyone help me figure it out?
And in general, does anyone know a good place to conquer my fear of infinite summations? Thanks!

In situations like this where they give you the outcome and your goal is to figure out how they did it I suggest pulling out as much from the summation as you can. For instance you there is a 4^m inside the sum but the sum is indexed by i so you can pull that term outside of the sum. Similarly m! is constant inside the sum so you can pull that outside. What else remains? See if you can pull the remaining bit out (and justify to yourself why you can do this). Once you pull the entire thing outside of the summation for the two sides to be equal you'll need the sum to add up to 1. See if you can provide a justification for why that must be. Think about probability distributions and what facts must be true about them.

I tried to pull everything out and solve the summation with partial sums but that was clearly pointless. So I set the summation of m=0 to infinity of P(x=m) = 1 since the sum of all probabilities in a PDF have to sum to 1. But that just led me to an equation with two infinite sums that I don't know what to do with ):