Continued fraction

${\displaystyle b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}}$

An infinite continued fraction is defined by the sequences ${\displaystyle \{a_{i}\},\{b_{i}\}}$, for ${\displaystyle i=0,1,2,\ldots }$, with ${\displaystyle a_{0}=0}$.

A continued fraction is a mathematical expression that can be writen as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, the continued fraction is finite or infinite.

Different fields of mathematics have different terminology and notation for continued fraction. In number theory the standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article Simple continued fraction. The present article treats the case where numerators and denominators are sequences ${\displaystyle \{a_{i}\},\{b_{i}\}}$ of constants or functions. From the perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis, however, they are just standard, and in the present article they will simply be called "continued fraction".