$$sp \sum_{k=1}^\infty(s(1-p))^{k-1} = \frac{sp}{1-s(1-p)}$$
I am struggling with these series. I've learned a lot about them (E.g. Maclaurin series for e^x was used in one problem). However, for this one I simply don't know how to derive it.
Further, do you recommend any methods to solving these on my own? I saw how the series for e^x was derived and it seemed quite doable though a bit more creative than I believe myself to be :(
Thank you in advance!
P.S. This is for Stochastic Processes. I'm working on a problem for finding the mean and variance using the Probability Generating Function.