Example 4 Find the value of the finite continued fraction $\langle-2,1,2 / 3,2,1 / 2,3\rangle$.
We use equation (25) to calculate.
$$\begin{aligned}
\langle-2,1 & , 2 / 3,2,1 / 2,3\rangle=\langle-2,1,2 / 3,2,1 / 2+1 / 3\rangle \\
& =\langle-2,1,2 / 3,2,5 / 6\rangle=\langle-2,1,2 / 3,2+6 / 5\rangle \\
& =\langle-2,1,2 / 3,16 / 5\rangle=\langle-2,1,2 / 3+5 / 16\rangle \\
& =\langle-2,1,47 / 48\rangle=\langle-2,1+48 / 47\rangle \\
& =\langle-2,95 / 47\rangle=-2+47 / 95=-143 / 95 .
\end{aligned}$$