Let $X_n$ be a birth-death process, with birth rates $\lambda_n$ and death rates $\mu_n$ (with $\mu_o=0$ and $\lambda_{-1}=0$). How do you show that the invariant distribution $\pi_i$ is:
$\pi_0=\Big[ 1+ \sum_{k=0}^\infty \frac{\lambda_k\lambda_{k-1}\dots\lambda_0}{\mu_{k+1}\mu_k\dots \mu_1}\Big]^{-1}$
and
$\pi_{n+1}=\frac{\lambda_n}{\mu_{n+1}}\pi_n$?
I used the definition of invariant distribution, and arrived at the formula
$\pi_i=\frac{\pi_{i-1}\lambda_{i-1}+\pi_{i+1}\mu_{i+1}}{\lambda_i+\mu_i}$,
but I have no idea how to use this to prove what I'm being asked to prove.