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

$x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{\ddots }}}}}}}}.\,$

This convergence problem for continued fractions is inherently more difficult (and also more interesting) 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

$x={\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {\ddots }{\quad \ddots \quad b_{k-1}+{\cfrac {a_{k}}{b_{k}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{\ddots }}}}}}}}}}}}\,$

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.

The special case when period k = 1

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

$x=K_{1}^{\infty }{\frac {a}{b}},\,$

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

$y=K_{1}^{\infty }{\frac {z}{1}}\qquad \left(z={\frac {a}{b^{2}}}\right)\,$

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

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

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).

References

• 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