In mathematics, an infinite periodic continued fraction is a 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 + ⋱ ⋱ 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 {\ddots }{\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 of k + 1 partial denominators is followed by a block [ak+1, ak+2,... Wikipedia