Euler's continued fraction formula

Etymology
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Proper noun
a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_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}}}}}}\,. $$ It also applies in the infinite case:
 * 1)  A formula connecting a finite sum of products to a finite continued fraction: $$
 * $$1 + \sum_{i=1}^\infty \left( \prod_{j=1}^i r_j \right) = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}.$$