Euler's continued fraction formula

English

Template:Lb A formula connecting a finite sum of products to a finite continued fraction: $a_{0} + a_{0} a_{1} + a_{0} a_{1} a_{2} + \dots + a_{0} a_{1} a_{2} \dots a_{n} = \frac{a_{0}}{1 - \frac{a_{1}}{1 + a_{1} - \frac{a_{2}}{1 + a_{2} - \frac{⋱}{⋱ \frac{a_{n - 1}}{1 + a_{n - 1} - \frac{a_{n}}{1 + a_{n}}}}}}} .$

It also applies in the infinite case:
$1 + \sum_{i = 1}^{\infty} (\prod_{j = 1}^{i} r_{j}) = \frac{1}{1 - \frac{r_{1}}{1 + r_{1} - \frac{r_{2}}{1 + r_{2} - \frac{r_{3}}{1 + r_{3} - ⋱}}}} .$