Euler's Continued Fraction Formula - Euler's Formula in Modern Notation

Euler's Formula in Modern Notation

$x = \cfrac{1}{1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots}}}}\,$

is a continued fraction with complex elements and none of the denominators B_i are zero, a sequence of ratios {r_i} can be defined by

$r_i = -\frac{a_{i+1}B_{i-1}}{B_{i+1}}.\,$

For x and r_i so defined, these equalities can be proved by induction.

$x = \cfrac{1}{1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots}}}} = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\,$

$x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = 1 + \sum_{i=1}^\infty \left( \prod_{j=1}^i r_j \right)\,$

Here equality is to be understood as equivalence, in the sense that the n'th convergent of each continued fraction is equal to the n'th partial sum of the series shown above. So if the series shown is convergent – or uniformly convergent, when the a_i's and b_i's are functions of some complex variable z – then the continued fractions also converge, or converge uniformly.

Read more about this topic: Euler's Continued Fraction Formula

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