Euler's Formula in Modern Notation
If
is a continued fraction with complex elements and none of the denominators Bi are zero, a sequence of ratios {ri} can be defined by
For x and ri so defined, these equalities can be proved by induction.
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 ai's and bi'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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