Chapter 9.2, Problem 54E

The transition matric for a Markov chain is

$P=\left[\begin{array}{ccc}0& .2& .8\\ .3& .3& .4\\ .6& .1& .3\end{array}\right]$

Let $m_k$ denote the maximum entry in the third column of $P^k$. Note that $m_1=.3$.

(A) Find $m_2,m_3,m_4$, and $m_5$ to three decimal places.

(B) Explain why $m_k\le m_{k+1}$ for all positive integers $k$.