In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators ai and partial denominators bi that are sufficient to guarantee the convergence of the infinite continued fraction This convergence problem is inherently more difficult than the corresponding problem for infinite series.
The convergence problem is much more difficult when the elements of the continued fraction are complex numbers.
If r1 and r2 are finite then the infinite periodic continued fraction x converges if and only if If the denominator Bk-1 is equal to zero then an infinite number of the denominators Bnk-1 also vanish, and the continued fraction does not converge to a finite value.
By applying another equivalence transformation the condition that guarantees convergence of can also be determined.
Since a simple equivalence transformation shows that whenever z ≠ 0, the preceding result for the continued fraction y can be restated for x.
(Not including either endpoint) By applying the fundamental inequalities to the continued fraction it can be shown that the following statements hold if |ai| ≤ 1/4 for the partial numerators ai, i = 2, 3, 4, ... Because the proof of Worpitzky's theorem employs Euler's continued fraction formula to construct an infinite series that is equivalent to the continued fraction x, and the series so constructed is absolutely convergent, the Weierstrass M-test can be applied to a modified version of x.
In the late 19th century, Śleszyński and later Pringsheim showed that a continued fraction, in which the as and bs may be complex numbers, will converge to a finite value if
Suppose that all the ai are equal to 1, and all the bi have arguments with: with epsilon being any positive number less than
In other words, all the bi are inside a wedge which has its vertex at the origin, has an opening angle of