A markov chain X has the state space transition matrix P (see attached image) where q = 1-p and 0 < p < 1

Classify the states of X and give the stationary distributions.

Show that $\displaystyle P^n $ does not converge, but $\displaystyle P^{2n} $ and $\displaystyle P^{2n+1} $ do converge as n approaches infinity.