Example 6 Please convert $\frac{1}{17408}$ into a decimal.
Since $17408=1024 \times 17=2^{10} \cdot 17, \varphi(17)=16$. We also have
$$\begin{array}{cc}
10 \equiv 10(\bmod 17), & 10^{2} \equiv 15(\bmod 17) \\
10^{4} \equiv 4(\bmod 17), & 10^{8} \equiv 16(\bmod 17) \\
10^{16} \equiv 1(\bmod 17)
\end{array}$$

Therefore, in Lemma 5, we can take $a=1, b=17408, a=10, \beta=0$, $h=16, b_{1}=17$. By Lemma 5, we have $\frac{1}{17408}=0 . a_{1} a_{2} \cdots a_{10} a_{11}$ $a_{12} \cdots a_{26} g$. After calculation, we have
$$\frac{1}{17408}=0.0000574448 \dot{5} 29411764705882 \dot{3}$$