# 連分數

${\displaystyle a_{0}+{\frac {1}{a_{1}+{\frac {1}{a_{2}+{\frac {1}{a_{3}+\cdots }}}}}}}$

${\displaystyle [1;1,1,1,\cdots ]=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\cdots }}}}}}}}}$

${\displaystyle [1;1,1]=1+{\frac {1}{1+{\frac {1}{1}}}}=1+{\frac {1}{2}}=3/2=1.5}$

${\displaystyle \pi }$${\displaystyle [3;7,15,1,292]=3+{\frac {1}{7+{\frac {1}{15+{\frac {1}{1+{\frac {1}{292}}}}}}}}={\frac {103993}{33102}}=3.1415926530119026407\cdots }$

## 有限連分數及簡單連分數

### 有限連分數

${\displaystyle a_{0}+{\frac {1}{a_{1}+{\frac {1}{a_{2}+{\frac {1}{\cdots +a_{n}}}}}}}}$

## 連分數轉換法

${\displaystyle t_{0}={\frac {1}{a_{1}+t_{1}}}}$同埋${\displaystyle x=[a_{0},a_{1}+t_{1}]}$

${\displaystyle t_{i}={\frac {1}{a_{i+1}+t_{i+1}}}}$同埋${\displaystyle x=[a_{0},a_{1},\cdots ,a_{i},a_{i+1}+t_{i+1}]}$

1. ${\displaystyle {\frac {5}{3}}={\frac {3+2}{3}}=1+{\frac {2}{3}}}$
2. ${\displaystyle {\frac {1}{\frac {2}{3}}}={\frac {3}{2}}={\frac {2+1}{2}}=1+{\frac {1}{2}}}$
3. ${\displaystyle {\frac {1}{\frac {1}{2}}}={\frac {2}{1}}=2}$