I am really stuck as to how I find this infinite sum:
$$\sum_{n=0}^\infty \left[ 1-p(1-q)^{n-1} \right]$$
The restrictions on $p$ and $q$ is that they both must be less than $1$ but greater than $0$, as this is the context of a probability question. The context is trying to find a stationary distribution, and one of the equations I end up with is: $$2\pi_1 + \pi_1\sum_{n=0}^\infty \left[ 1-p(1-q)^{n-1} \right] = 1$$
Thank you.
The sum diverges. I'll proove it by the integral criterium. $$\int_0^\infty1-p(1-q)^{x-1}dx=\left[x-\frac{p(1-q)^{x-1}}{\ln(1-q)}\right]_0^\infty=\infty$$ This integral diverges, so the sum diverges too.