1. Present Value Random Variable

So from the text I'm reading:
If failure occurs in (k-1,k], with payment at time k, then Kx= k-1.
Random Variable ZX=VKx+1, for Kx= 0, 1, 2,...
Expected value of random variable Zx
Ax= E[ZX] = \sum\limits_{k=1}^infinite vkPr(Kx= k-1)= \sum\limits_{k=0}^infinite vk+1Pr(Kx= k)

I mainly need help understanding the last part for the Ax and how the two summation are the same. From my understanding, I think they're equal because the right side equation although k = 0, we are adding another v in to the equation.

2. Re: Present Value Random Variable

all they've done is shifted the summation range from $\{1, \infty\}$ to $\{0, \infty\}$

and made the appropriate adjustments $v^k \to v^{k+1},~Pr(K_x = k-1) \to Pr(K_x = k)$