連分数の収束子の奇偶性
\[\frac{p_{2k}}{q_{2k}}<\frac{p_{2\left(k+1\right)}}{q_{2\left(k+1\right)}}\]
連分数と最大公約数
\[\gcd\left(p_{n},q_{n}\right)=1\]
無限連分数の収束条件
\[\left[a_{0};a_{1},a_{2},\cdots\right]<\infty\Leftrightarrow\sum_{k=0}^{\infty}a_{k}=\infty\]
連分数の収束子と漸化式
\[\frac{p_{n}}{q_{n}}=a_{0}+\sum_{k=1}^{n}\frac{\left(-1\right)^{k+1}}{q_{k}q_{k-1}}\prod_{j=0}^{k-1}b_{j}\]
(*)簡単な連分数展開
\[\left[x;x,x,\cdots\right]=\frac{1}{2}\left(x+\sqrt{x^{2}+4}\right)\]
連分数の性質
\[\left[\left(a_{0},b_{0}\right);\left(a_{1},b_{1}\right),\cdots,a_{n}+c\right]=\left[\left(a_{0},b_{0}\right);\left(a_{1},b_{1}\right),\cdots,\left(a_{n},b_{n}\right),\frac{b_{n}}{c}\right]\]
連分数の定義
\[\left[a_{0};a_{1},a_{2},a_{3},\cdots\right]:=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\cdots}}}\]