7の平方根
{\displaystyle [2;1,1,1,4,1,1,1,4,\ldots ]=2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+\dots }}}}}}}}}}.} オンライン整数列大辞典の数列 A010121
The successive partial evaluations of the continued fraction, which are called its convergents, approach 7 {\displaystyle {\sqrt {7}}} :
2 1 , 3 1 , 5 2 , 8 3 , 37 14 , 45 17 , 82 31 , 127 48 , 590 223 , 717 271 , ... {\displaystyle {\frac {2}{1}},{\frac {3}{1}},{\frac {5}{2}},{\frac {8}{3}},{\frac {37}{14}},{\frac {45}{17}},{\frac {82}{31}},{\frac {127}{48}},{\frac {590}{223}},{\frac {717}{271}},\dots }
Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257...オンライン整数列大辞典の数列 A041008 , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,...オンライン整数列大辞典の数列 A041009.