Consider three matrices $\mathrm{A}=\left[\begin{array}{cc}2& 1\\ 4& 1\end{array}\right], \mathrm{B}=\left[\begin{array}{cc}3& 4\\ 2& 3\end{array}\right]$, and $\mathrm{C}=\left[\begin{array}{cc}3& -4\\ -2& 3\end{array}\right]$. Then the value of the sum $\mathrm{tr}\left(\mathrm{A}\right)+\mathrm{tr}\left(\frac{\mathrm{ABC}}{2}\right)+\mathrm{tr}\left(\frac{\mathrm{A}(\mathrm{BC}{)}^2}{4}\right)+\mathrm{tr}\left(\frac{\mathrm{A}(\mathrm{BC}{)}^3}{8}\right)+\cdots +\mathrm{\infty }$ is

$\begin{array}{l}\mathrm{BC}=\left[\begin{array}{ll}3& 4\\ 2& 3\end{array}\right]\left[\begin{array}{cc}3& -4\\ -2& 3\end{array}\right]\Rightarrow \mathrm{B}=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right]\\ \Rightarrow \mathrm{tr}\left(\mathrm{A}\right)+\mathrm{tr}\left(\frac{\mathrm{ABC}}{2}\right)+\mathrm{tr}\left(\frac{\mathrm{A}(\mathrm{BC}{)}^2}{4}\right)+\mathrm{tr}\left(\frac{\mathrm{A}(\mathrm{BC}{)}^3}{8}\right)+\cdots +\mathrm{\infty }\end{array}$

                 $\begin{array}{l}=\mathrm{tr}\left(\mathrm{A}\right)+\mathrm{tr}\left(\frac{\mathrm{A}}{2}\right)+\mathrm{tr}\left(\frac{\mathrm{A}}{2^2}\right)+\cdots \\ =\mathrm{tr}\left(\mathrm{A}\right)+\frac{1}{2}\mathrm{tr}\left(\mathrm{A}\right)+\frac{1}{2^2}\mathrm{tr}\left(\mathrm{A}\right)\cdots \\ =\frac{\mathrm{tr}\left(\mathrm{A}\right)}{1-(1/2)}\\ =2\mathrm{tr}\left(\mathrm{A}\right)=2(2+1)=6\end{array}$