Changing order of summation in a probability generating function

Question

Let $G(s)$ to be the generating function of $X$ which takes non-negative integer values. I want to express $$\sum_{n=0}^{\infty}s^n P(x \leq n),$$ in terms of $G(s)$. We have that $$F(x) = P(x \leq n) = \sum_{k \leq n} f(x=k),$$ where $f$ is the mass function and $F$ is the distribution. Then, $$\sum_{n=0}^{\infty}s^n P(x \leq n) = \sum_{n=0}^{\infty}s^n \left[ \sum_{k \leq n} f(x=k)\right] = \sum_{n=0}^{\infty}s^{n-k} \left[ \sum_{k \leq n} f(x=k)s^k\right].$$ We now proceed to change the summation to get: $$\sum_{n=0}^{\infty}s^n P(x \leq n) = \sum_{k =0}^{\infty}s^kf(x=k)\left[\sum_{n \geq k} s^{n-k}\right] = \sum_{n \geq k} s^{n-k}G(s).$$ Is this the right approach? I am not sure if i am changing the order of summation correctly.

Answer 1

This approach can work, but it looks like you got caught up summing over $n$. We have \begin{align} \sum\limits_{n=0}^\infty P(x \leq n)s^n &= \sum\limits_{n=0}^\infty \sum\limits_{k = 0}^n f(x = k) s^n \\ &=\sum\limits_{k = 0}^\infty \sum\limits_{n = k}^\infty f(x = k)s^n \\ &=\sum\limits_{k = 0}^\infty f(x = k)\frac{s^k}{1 - s} \\ &=\frac{G(s)}{1-s}. \end{align}