Why Does A_n Converge in This Markov Chain Problem?

[spitz]

Homework Statement

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.

The Attempt at a Solution

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?

 

[Office_Shredder]

So for example

I+P+P²+P³+P⁴+P⁵ =
[3 3]
[3 3]

and when you divide this by six you get a matrix with all 1/2s. Try adding up some more guys and see what happens