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当时i≥1，若αi=αi+k，则连分数 \[ \left[ {\alpha _0 ,\alpha _1 ,\alpha _2 , \cdots } \right] \] 称为以k为周期的周期连分数，记做 \[ \left[ {\alpha _0 ,\alpha _1 , \cdots ,\alpha _{l - 1} ,\overline {\alpha _l , \cdots ,\alpha _{l + k - 1} } } \right] \] 当l=0时，称为纯周期连分数,当l=1时，称为拟纯周期连分数．
1°实数α可展成周期连分数的充分必要条件是：α是一个有理数域上二次不可约多项式的根．
2°实二次无理数α可展成纯周期连分数的充分必要条件是：α>1且\( - 1 < α' < 0 \) ，这里α’为α的共轭实数．
3°实二次无理数α可展成拟纯周期连分数的充分必要条件是:\( \alpha ' < \left[ \alpha \right] - 1 \) ，这里α'为α的共轭实数．[α]为α的整数部分．
1°设为非完全平方数，则 \[ \sqrt d = \left[ {\left[ {\sqrt d } \right],\overline {a_1 ,a_2 , \cdots ,a_2 ,a_1 ,2\left[ {\sqrt d } \right]} } \right] \] 2°设e为自然对数的底，则 \[ e = \left[ {2,1,2,1,1,4,1,1,6,1,1, \cdots } \right] \] 式中不完全商的通式为 \[ \pi = \left[ {3,7,15,1,292,1,1,1,21,31,14,2,1,2,2,2,2,1,84, \cdots } \right] \] 它的渐近分数为 \[ \frac{3}{1},\frac{{22}}{7},\frac{{333}}{{106}},\frac{{355}}{{113}},\frac{{103993}}{{33102}},\frac{{104348}}{{33215}}, \cdots \]
1°把线段AB分成中外比（即\( AH^2 = AB \cdot HB \) ）的分割称为黄金分割.也就是求解代数方程\[ x^2 + x - 1 = 0 \] 的一个根\[ x = \frac{{\sqrt 5 - 1}}{2} \]
2°由递推关系 \[ \left\{ \begin{array}{l} \begin{array}{*{20}c} {F_{n + 2} = F_{n + 1} + F_n } & {\left( {n \ge 0} \right)} \\ \end{array} \\ \begin{array}{*{20}c} {F_0 = 0,} & {F_1 = 1} \\ \end{array} \\ \end{array} \right. \] 产生的序列 \[ 0,1,1,2,3,5,8,13,21,34, \cdots \] 称为费波那奇序列.其通项表达式为 \[ \begin{array}{*{20}c} {F_n = \frac{1}{{\sqrt 5 }}\left[ {\left( {\frac{{1 + \sqrt 5 }}{2}} \right)^n - \left( {\frac{{1 - \sqrt 5 }}{2}} \right)^n } \right]} & {\left( {n = 1,2, \cdots } \right)} \\ \end{array} \]
3°当m/n时，Fm|Fn．
4°\( \frac{{F_n }}{{F_{n + 1} }} \) 是\( \frac{{\sqrt 5 - 1}}{2} \) 的最佳渐近分数，\( \frac{{F_{n + 1} }}{{F_n }} \) 是\( \frac{{\sqrt 5 + 1}}{2} \) 的最佳渐近分数.
5°设a,b为自然数，由递推关系\[ \left\{ \begin{array}{l} \begin{array}{*{20}c} {\mathop {F_{n + 2} }\limits^ \sim = a\mathop {F_{n + 1} }\limits^ \sim + b\mathop {F_n }\limits^ \sim } & {\left( {n \ge 0} \right)} \\ \end{array} \\ \begin{array}{*{20}c} {\mathop {F_0 }\limits^ \sim = 0} & {\mathop {F_1 }\limits^ \sim = 1} \\ \end{array} \\ \end{array} \right. \] 产生的序列的通项表达式为 \[ \begin{array}{*{20}c} {\mathop {F_n }\limits^ \sim = \frac{1}{{\sqrt L }}\left[ {\left( {\frac{{a + \sqrt L }}{2}} \right)^n - \left( {\frac{{a - \sqrt L }}{2}} \right)^n } \right]} & {\left( {L = a^2 + 4b,n \ge 1} \right)} \\ \end{array} \] 并且具有性质: 当m|n时，\( \mathop {F_m }\limits^ \sim |\mathop {F_n }\limits^ \sim \) ．