Periodic continued fraction

In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a k + 1 a k + 1 + 1 ⋱ a k + m − 1 + 1 a k + m + 1 a k + 1 + 1 a k + 2 + ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}} where the initial block [ a 0 ; a 1 , … , a k ] {\displaystyle [a_{0};a_{1},\dots ,a_{k}]} of k+1 partial denominators is followed by a block [ a k + 1 , a k + 2 , … , a k + m ] {\displaystyle [a_{k+1},a_{k+2},\dots ,a_{k+m}]} of m partial denominators that repeats ad infinitum. For example, 2 {\displaystyle {\sqrt {2}}} can be expanded to the periodic continued fraction [ 1 ; 2 , 2 , 2 , .

Source: Wikipedia — Periodic continued fraction (CC BY-SA 4.0)

Periodic continued fraction

In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a k + 1 a k + 1 + 1 ⋱ a k + m − 1 + 1 a k + m + 1 a k + 1 + 1 a k + 2 + ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}} where the initial block [ a 0 ; a 1 , … , a k ] {\displaystyle [a_{0};a_{1},\dots ,a_{k}]} of k+1 partial denominators is followed by a block [ a k + 1 , a k + 2 , … , a k + m ] {\displaystyle [a_{k+1},a_{k+2},\dots ,a_{k+m}]} of m partial denominators that repeats ad infinitum. For example, 2 {\displaystyle {\sqrt {2}}} can be expanded to the periodic continued fraction [ 1 ; 2 , 2 , 2 , .

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Source: Wikipedia "Periodic continued fraction" · CC BY-SA 4.0

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