Euler's continued fraction formula

In the analytic theory of continued fractions, Euler's continued fraction formula is an identity connecting a certain very general infinite series with an infinite continued fraction. First published in 1748, it was at first regarded as a simple identity connecting a finite sum with a finite continued fraction in such a way that the extension to the infinite case was immediately apparent. Today it is more fully appreciated as a useful tool in analytic attacks on the general convergence problem for infinite continued fractions with complex elements. Euler derived the formula as connecting a finite sum of products with a finite continued fraction. The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite continued fraction. This is written more compactly using generalized continued fraction notation: If ri are complex numbers and x is defined by then this equality 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 ri's are functions of some complex variable z – then the continued fractions also converge, or converge uniformly. Theorem: Let be a natural number. For complex values , and for complex values , Proof: We perform a double induction. For , we have and Now suppose both statements are true for some . We have where by applying the induction hypothesis to . But if implies implies , contradiction. Hence completing that induction. Note that for , if , then both sides are zero. Using and , and applying the induction hypothesis to the values , completing the other induction. As an example, the expression can be rearranged into a continued fraction.