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連分數

比如 : \frac{13}{17} = 0 + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}}

而我們說 \frac{13}{17} 的 continued fraction expansion 是 [0, 1, 3, 4]

convergents of the continued fraction expansion of \frac{13}{17}

\begin{aligned} c_0 = 0 = \frac{0}{1} \\ c_1 = 0 + \frac{1}{1} = \frac{1}{1} \\ c_2 = 0 + \frac{1}{1 + \frac{1}{3}} = \frac{3}{4} \\ c_3 = 0 + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}} = \frac{13}{17} \end{aligned}

就是一個不斷的逼近 \frac{13}{17} 的數列

有理數的 continued fraction expansion 包含有限的元素

無理數的 continued fraction expansion 包含無限的元素