循环连分数
循环连分数是一种可表示为以下形式的连分数：
前k+1个部分分母不算，后面的部分分母[ak+1, ak+2,…ak+m]会一直重复出现。例如
即可表示为循环连分数[1,2,2,2,...]。
循环连分数的部份分母{ai}可以是任何实数或虚数。
1770年，拉格朗日证明一个数字能表示成循环连分数，若且唯若此数为二次无理数。例如![\sqrt{3}=1.732\ldots=[1;1,2,1,2,1,2,\ldots]](/uploads/202502/02/68b96770cc169c17c5d88c07a6196b104640.png)
。
Periodic continued fraction
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Periodic continued fraction 循环连分数
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form
where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,…ak+m] of partial denominators that repeats over and over again, ad infinitum. For example can be expanded to a periodic continued fraction, namely as [1,2,2,2,...].
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