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

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?