Then $$\mu_n\to(2,2,2,2,2,102,102,102,102,102).The exact rate of convergence will be determined by the second-largest singular value of the matrix

$$M =

\begin{bmatrix}

\tfrac13 & \tfrac13 & \tfrac13 & & & \\

\tfrac13 & \tfrac13 & \tfrac13 & & & \\

\tfrac13 & \tfrac13 & \tfrac13 & & & \\

& & & 1 & & \\

& & & & 1 & \\

& & & & & \ddots

\end{bmatrix}

\begin{bmatrix}

1 & & & & & \\

& \tfrac13 & \tfrac13 & \tfrac13 & & \\

& \tfrac13 & \tfrac13 & \tfrac13 & & \\

& \tfrac13 & \tfrac13 & \tfrac13 & & \\

& & & & 1 & \\

& & & & & \ddots

\end{bmatrix}

\begin{bmatrix}

1 & & & & & \\

& 1 & & & & \\

& & \tfrac13 & \tfrac13 & \tfrac13 & \\

& & \tfrac13 & \tfrac13 & \tfrac13 & \\

& & \tfrac13 & \tfrac13 & \tfrac13 & \\

& & & & & \ddots

\end{bmatrix}

\cdots$$

but I don't know how to compute that.