What is the mgf of the geometric random variable with pmf of f(w) = p(1-p)^(w-1), where p = 1/8? Show how to derive it and explain any bounds on the values of t for which it is valid.
So far, I have gotten it simplified to:
M(t) = (1/7) * Sigma( ((7/8)*e^t)^w ) where the sum is from w=0 to infinity.
I don't really know where to go from here to simplify it, or what bounds t could possibly have.
Okay, so the sum of a geometric sequence is a/(1-r) where a is the first term and r is the ratio, correct? Wouldn't the first term be when w=0, so ((7/8)e^t)^0 = 1 (ignoring for the moment that I moved the 1/7 outside the sum, of course)? And since ((7/8)e^t) is to the power of w, wouldn't the ratio be (7/8)e^t?