以下是解析的 LaTeX 代码：

```latex
\documentclass{article}
\usepackage{amsmath}

\begin{document}

\section*{Number of Convergents for the Desired Accuracy}

An interesting real number \(\alpha\) is given by the infinite continued fraction expansion:

\[
\alpha = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, \dots]
\]

To determine how many terms of the expansion are required to approximate \(\alpha\) with three decimal digits accurately, we need to ensure that:

\[
\left| \alpha - \frac{p_k}{q_k} \right| < 0.0005
\]

where \(\frac{p_k}{q_k}\) is the \(k\)-th convergent. The error can be bounded by:

\[
\left| \alpha - \frac{p_k}{q_k} \right| < \frac{1}{q_k^2}
\]

Thus, we need:

\[
\frac{1}{q_k^2} < 0.0005 \quad \Rightarrow \quad q_k^2 > 2000
\]

Now, we calculate the convergents:

\[
\begin{align*}
\frac{p_0}{q_0} &= \frac{2}{1} \\
\frac{p_1}{q_1} &= \frac{3}{1} \\
\frac{p_2}{q_2} &= \frac{8}{3} \\
\frac{p_3}{q_3} &= \frac{11}{4} \\
\frac{p_4}{q_4} &= \frac{19}{7} \\
\frac{p_5}{q_5} &= \frac{87}{32} \\
\frac{p_6}{q_6} &= \frac{106}{39} \\
\frac{p_7}{q_7} &= \frac{193}{71} \\
\frac{p_8}{q_8} &= \frac{1264}{465} \\
\frac{p_9}{q_9} &= \frac{1457}{536} \\
\frac{p_{10}}{q_{10}} &= \frac{2721}{1001} \\
\frac{p_{11}}{q_{11}} &= \frac{23625}{8693} \\
\frac{p_{12}}{q_{12}} &= \frac{26346}{9694} \\
\frac{p_{13}}{q_{13}} &= \frac{49971}{18387} \\
\end{align*}
\]

Checking the inequality for \(q_{12}\):

\[
q_{12}^2 = 9694^2 = 93908236 > 2000
\]

Thus, the smallest \(q_k\) satisfying \(q_k^2 > 2000\) corresponds to \(k = 12\). Therefore, the number of terms required for the desired accuracy is 12.

\end{document}
```

该代码可以直接使用 LaTeX 编译器编译生成 PDF 文件。请问还有其他问题或需要进一步解释的地方吗？  
以下是一些相关问题：

1. 什么是连分数展开？
2. 连分数的收敛子是什么意思？
3. 如何计算连分数的收敛子？
4. 误差界限与收敛子的关系是什么？
5. 如何判断连分数展开的准确性？
6. 为什么选择误差界限为 0.0005？
7. 连分数与其他数值表示方法有什么不同？
8. 计算连分数的收敛子时，q\_k 和 p\_k 的作用分别是什么？

**小提示**: 了解连分数的性质可以帮助你理解很多数论中的概念。

Explore how to calculate convergents and error bounds in the continued fraction expansion of a real number. This problem demonstrates the application of continued fractions to approximate \(\alpha\) with three decimal digits accuracy. Suitable for advanced mathematics enthusiasts and students.

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