Markov Chains with No Limit: Proving Convergence for $A_n$

[Jason4]

Consider:

$P=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$

Show that $P^n$ has no limit, but that: $A_n=\frac{1}{n+1}(I+P+P^2+\ldots+P^n)$ has a limit.

I can see that $P^{EVEN}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P^{ODD}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$, so a steady state is never reached, but I can't figure out the second part.

Any suggestions?

 

[Plato2]

Consider:
$P=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$
Show that $P^n$ has no limit, but that: $A_n=\frac{1}{n+1}(I+P+P^2+\ldots+P^n)$ has a limit.

$\sum\limits_{k = 0}^N {{P^k}} = \left\{ {\begin{array}{rl}{\tfrac{1}{2}\left[ {\begin{array}{rl}{N + 2}&N\\N&{N + 2}\end{array}} \right]}&{,N\text{ even}}\\{\tfrac{{N + 1}}{2}\left[ {\begin{array}{rl}1&1\\1&1\end{array}} \right]}&{,N\text{ odd}}\end{array}} \right.$