For step 2, notice that when k = 0 then anything times that is 0 - so basically you have 0*blah + Sigma(k=1,infinity) = Sigma(k=1,infinity) blah.

For step 3 we note that k! = k*(k-1)! (property of factorials) and that a^(b+1) = a*a^b. Re-arrange those for the results in this step.

Last step uses the fact that the sum of all probabilities is equal to 1.