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

$$\begin{gathered} BC=\left[\begin{array}{cc}3& 4\\ 2& 3\end{array}\right]\left[\begin{array}{cc}3& -4\\ -2& 3\end{array}\right]\Rightarrow B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ tr\left(A\right)+tr\left(\frac{A\left(BC\right)}{2}\right)+tr\left(\frac{A{\left(BC\right)}^2}{4}\right)+tr\left(\frac{A{\left(BC\right)}^3}{8}\right)+\mathrm{.......................} \\ \Rightarrow tr\left(A\right)+\frac{1}{2}tr\left(A\right)+\frac{1}{2^2}tr\left(A\right)+\mathrm{....} \\ \Rightarrow \frac{tr\left(A\right)}{1-1/2}=2tr\left(A\right)=2(2+1)=6 \end{gathered}$$