Example 6 Find the simplest fraction (with the smallest denominator) that approximates $\sqrt{8}$, with an error $\leqslant 10^{-6}$.
In Example 4, we have found $\sqrt{8}=\langle 2,1,4,1,4,1,4, \cdots\rangle$. We first list to find $h_{n}$, $k_{n}$, and then estimate the error according to formula (16).
\begin{tabular}{|c|rrrrrrrrrr|}
\hline$n$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline$a_{n}$ & 2 & 1 & 4 & 1 & 4 & 1 & 4 & 1 & 4 & 1 \\
$h_{n}$ & 2 & 3 & 14 & 17 & 82 & 99 & 478 & 577 & 2786 & 3363 \\
$k_{n}$ & 1 & 1 & 5 & 6 & 29 & 35 & 169 & 204 & 985 & 1189 \\
\hline
\end{tabular}

From the table above, when $n=8,7$, by formula (16) we get
$$\begin{aligned}
\left|\sqrt{8}-\frac{h_{8}}{k_{8}}\right| & =\left(\sqrt{8}-\frac{2786}{985}\right)\frac{1}{204(204+985)} \\
& =\frac{1}{242556}>10^{-6}
\end{aligned}$$

Therefore, the required convergent fraction is $h_{8} / k_{8}=2786 / 985$.