Convergence problem

In the analytic theory of continued fractions, the convergence problem is the determination of conditions on the partial numerators a_i and partial denominators b_i that are sufficient to guarantee the convergence of the continued fraction

$x = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots}}}}.\,$

This convergence problem for continued fractions is inherently more difficult than the corresponding convergence problem for infinite series.

Elementary results

When the elements of an infinite continued fraction consist entirely of positive real numbers, the determinant formula can easily be applied to demonstrate when the continued fraction converges. Since the denominators B_n cannot be zero in this simple case, the problem boils down to showing that the product of successive denominators B_nB_n+1 grows more quickly than the product of the partial numerators a₁a₂a₃...a_n+1. The convergence problem is much more difficult when the elements of the continued fraction are complex numbers.

Periodic continued fractions

An infinite periodic continued fraction is a continued fraction of the form

$x = \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{\ddots}{\quad\ddots\quad b_{k-1} + \cfrac{a_k}{b_k + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \ddots}}}}}}\,$

where k ≥ 1, the sequence of partial numerators {a₁, a₂, a₃, ..., a_k} contains no values equal to zero, and the partial numerators {a₁, a₂, a₃, ..., a_k} and partial denominators {b₁, b₂, b₃, ..., b_k} repeat over and over again, ad infinitum.

By applying the theory of linear fractional transformations to

$s(w) = \frac{A_{k-1}w + A_k}{B_{k-1}w + B_k}\,$

where A_k-1, B_k-1, A_k, and B_k are the numerators and denominators of the k-1st and kth convergents of the infinite periodic continued fraction x, it can be shown that x converges to one of the fixed points of s(w) if it converges at all. Specifically, let r₁ and r₂ be the roots of the quadratic equation

$B_{k-1}w^2 + (B_k - A_{k-1})w - A_k = 0.\,$

These roots are the fixed points of s(w). If r₁ and r₂ are finite then the infinite periodic continued fraction x converges if and only if

1. the two roots are equal; or
2. the k-1st convergent is closer to r₁ than it is to r₂, and none of the first k convergents equal r₂.

If the denominator B_k-1 is equal to zero then an infinite number of the denominators B_nk-1 also vanish, and the continued fraction does not converge to a finite value. And when the two roots r₁ and r₂ are equidistant from the k-1st convergent – or when r₁ is closer to the k-1st convergent than r₂ is, but one of the first k convergents equals r₂ – the continued fraction x diverges by oscillation.^[1][2][3]

The special case when period k = 1

If the period of a continued fraction is 1; that is, if

$x = \underset{1}{\overset{\infty}{\mathrm K}} \frac{a}{b},\,$

where b ≠ 0, we can obtain a very strong result. First, by applying an equivalence transformation we see that x converges if and only if

$y = 1 + \underset{1}{\overset{\infty}{\mathrm K}} \frac{z}{1}\qquad \left(z = \frac{a}{b^2}\right)\,$

converges. Then, by applying the more general result obtained above it can be shown that

$y = 1 + \cfrac{z}{1 + \cfrac{z}{1 + \cfrac{z}{1 + \ddots}}}\,$

converges for every complex number z except when z is a negative real number and z < −¼. Moreover, this continued fraction y converges to the particular value of

$y = \frac{1}{2}\left(1 \pm \sqrt{4z + 1}\right)\,$

that has the larger absolute value (except when z is real and z < −¼, in which case the two fixed points of the LFT generating y have equal moduli and y diverges by oscillation).

By applying another equivalence transformation the condition that guarantees convergence of

$x = \underset{1}{\overset{\infty}{\mathrm K}} \frac{1}{z} = \cfrac{1}{z + \cfrac{1}{z + \cfrac{1}{z + \ddots}}}\,$

can also be determined. Since a simple equivalence transformation shows that

$x = \cfrac{z^{-1}}{1 + \cfrac{z^{-2}}{1 + \cfrac{z^{-2}}{1 + \ddots}}}\,$

whenever z ≠ 0, the preceding result for the continued fraction y can be restated for x. The infinite periodic continued fraction

$x = \underset{1}{\overset{\infty}{\mathrm K}} \frac{1}{z}$

converges if and only if z² is not a real number lying in the interval −4 < z² ≤ 0 – or, equivalently, x converges if and only if z ≠ 0 and z is not a pure imaginary number lying in the interval −2i < z < 2i.

Worpitzky's theorem

By applying the fundamental inequalities to the continued fraction

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

it can be shown that the following statements hold if |a_i| ≤ ¼ for the partial numerators a_i, i = 2, 3, 4, ...

• The continued fraction x converges to a finite value, and converges uniformly if the partial numerators a_i are complex variables.^[4]
• The value of x and of each of its convergents x_i lies in the circular domain of radius 2/3 centered on the point z = 4/3; that is, in the region defined by

$\Omega = \lbrace z: |z - 4/3| \leq 2/3 \rbrace.\,$^[5]

• The radius ¼ is the largest radius over which x can be shown to converge without exception, and the region Ω is the smallest image space that contains all possible values of the continued fraction x.^[5]

The proof of the first statement, by Julius Worpitzky in 1865, is apparently the oldest published proof that a continued fraction with complex elements actually converges.^[6]

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. If

$f(z) = \cfrac{1}{1 + \cfrac{c_2z}{1 + \cfrac{c_3z}{1 + \cfrac{c_4z}{1 + \ddots}}}}\,$

and a positive real number M exists such that |c_i| ≤ M (i = 2, 3, 4, ...), then the sequence of convergents {f_i(z)} converges uniformly when

$|z| < \frac{1}{4M}\,$

and f(z) is analytic on that open disk.

Śleszyński–Pringsheim criterion

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 $|b_n | \geq |a_n| + 1$ for $n \geq 1.$^[7]

Van Vleck's theorem

Jones and Thron attribute the following result to Van Vleck. Suppose that all the a_i are equal to 1, and all the b_i have arguments with:

$- \pi /2 + \epsilon < \arg ( b_i) < \pi / 2 - \epsilon, i \geq 1,$

with epsilon being any positive number less than $\pi/2$. In other words, all the b_i are inside a wedge which has its vertex at the origin, has an opening angle of $\pi - 2 \epsilon$, and is symmetric around the positive real axis. Then f_i, the ith convergent to the continued fraction, is finite and has an argument:

$- \pi /2 + \epsilon < \arg ( f_i ) < \pi / 2 - \epsilon, i \geq 1.$

Also, the sequence of even convergents will converge, as will the sequence of odd convergents. The continued fraction itself will converge if and only if the sum of all the |b_i| diverges.^[8]

Notes

1. 1886 Otto Stolz, Verlesungen über allgemeine Arithmetik, pp. 299-304
2. 1900 Alfred Pringsheim, Sb. München, vol. 30, "Über die Konvergenz unendlicher Kettenbrüche"
3. 1905 Oskar Perron, Sb. München, vol. 35, "Über die Konvergenz periodischer Kettenbrüche"
4. 1865 Julius Worpitzky, Jahresbericht Friedrichs-Gymnasium und Realschule, "Untersuchungen über die Entwickelung der monodromen und monogenen Functionen durch Kettenbrüche"
5. 1942 J. F. Paydon and H. S. Wall, Duke Math. Journal, vol. 9, "The continued fraction as a sequence of linear transformations"
6. 1905 Edward Burr Van Vleck, The Boston Colloquium, "Selected topics in the theory of divergent series and of continued fractions"
7. See for example Theorem 4.35 on page 92 of Jones and Thron (1980).
8. See theorem 4.29, on page 88, of Jones and Thron (1980).

References

• Jones, William B.; Thron, W. J. (1980), Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications. 11, Reading. Massachusetts: Addison-Wesley Publishing Company, ISBN 0-201-13510-8
• Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Company, New York, NY 1950.
• H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., 1948 ISBN 0-8284-0207-8