4. It is known that the continued fraction of $\pi$ is
$$x=[3,7,15,1,292,1,1, \ldots]$$

Try to find its first seven convergents and their approximate values.
4. Solution: By Lemma 1, we get
$$\begin{aligned}
\frac{p_{1}}{q_{1}} & =\frac{3}{1}, \frac{p_{2}}{q_{2}}=\frac{3 \times 7+1}{7} \\
& =\frac{22}{7}=3.14285714 \cdots \\
\frac{p_{3}}{q_{3}} & =\frac{22 \times 15+3}{7 \times 15+1}=\frac{333}{106}=3.141509433 \cdots \\
\frac{p_{4}}{q_{4}} & =\frac{333 \times 1+22}{106 \times 1+7}=\frac{355}{113}=3.141592920 \cdots, \\
\frac{p_{5}}{q_{5}} & =\frac{355 \times 292+333}{113 \times 292+106} \\
& =\frac{103993}{33102}=3.141592653011 \cdots \\
\frac{p_{6}}{q_{6}} & =\frac{103993 \times 1+355}{33102 \times 1+113} \\
& =\frac{104348}{33215}=3.141592653921 \cdots \\
\frac{p_{7}}{q_{7}} & =\frac{104348 \times 1+103993}{33215 \times 1+33102} \\
& =\frac{208341}{66317}=3.14159265346 \cdots
\end{aligned}$$

Therefore,
$$3.1415926534<\pi<3.1415926540$$