Euler's continued fraction formula

Named after Swiss mathematician Leonhard Euler (1707–1783).

Euler's continued fraction formula

1. (mathematics) A formula connecting a finite sum of products to a finite continued fraction: ${\displaystyle a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots +a_{0}a_{1}a_{2}\cdots a_{n}={\cfrac {a_{0}}{1-{\cfrac {a_{1}}{1+a_{1}-{\cfrac {a_{2}}{1+a_{2}-{\cfrac {\ddots }{\ddots {\cfrac {a_{n-1}}{1+a_{n-1}-{\cfrac {a_{n}}{1+a_{n}}}}}}}}}}}}}\,.}$

It also applies in the infinite case:

${\displaystyle 1+\sum _{i=1}^{\infty }\left(\prod _{j=1}^{i}r_{j}\right)={\cfrac {1}{1-{\cfrac {r_{1}}{1+r_{1}-{\cfrac {r_{2}}{1+r_{2}-{\cfrac {r_{3}}{1+r_{3}-\ddots }}}}}}}}.}$