In the analytic theory of

continued fractions A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or no ...

, 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 infinite continued fraction :

x = b_0 + \cfrac.\,

This convergence problem is inherently more difficult than the corresponding problem for

infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...

Elementary results

When the elements of an infinite continued fraction consist entirely of positive

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, 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''_''n''''B''_''n''+1 grows more quickly than the product of the partial numerators ''a''₁''a''₂''a''₃...''a''_''n''+1. The convergence problem is much more difficult when the elements of the continued fraction are

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

Periodic continued fractions

An infinite

periodic continued fraction In mathematics, an infinite periodic continued fraction is a simple continued fraction that can be placed in the form : x = a_0 + \cfrac where the initial block _0; a_1, \dots, a_kof ''k''+1 partial denominators is followed by a block linear fractional transformations to :

s(w) = \frac\,

where ''A''_''k''-1, ''B''_''k''-1, ''A''_''k'', and ''B''_''k'' are the numerators and denominators of the ''k''-1st and ''k''th 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 ''r''₁ and ''r''₂ be the roots of the quadratic equation :

B_w^2 + (B_k - A_)w - A_k = 0.\,

These roots are the

fixed points Fixed may refer to: * ''Fixed'' (EP), EP by Nine Inch Nails * ''Fixed'' (film), an upcoming animated film directed by Genndy Tartakovsky * Fixed (typeface), a collection of monospace bitmap fonts that is distributed with the X Window System * Fi ...

of ''s''(''w''). If ''r''₁ and ''r''₂ are finite then the infinite periodic continued fraction ''x'' converges if and only if # the two roots are equal; or # the ''k''-1st convergent is closer to ''r''₁ than it is to ''r''₂, and none of the first ''k'' convergents equal ''r''₂. 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 ''r''₁ and ''r''₂ are equidistant from the ''k''-1st convergent – or when ''r''₁ is closer to the ''k''-1st convergent than ''r''₂ is, but one of the first ''k'' convergents equals ''r''₂ – the continued fraction ''x'' diverges by oscillation.

The special case when period ''k'' = 1

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

x = \underset \frac,\,

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 \frac\qquad \left(z = \frac\right)\,

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

y = 1 + \cfrac\,

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

y = \frac\left(1 \pm \sqrt\right)\,

that has the larger absolute value (except when ''z'' is real and ''z'' < −, 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 \frac = \cfrac\,

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

x = \cfrac\,

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

x = \underset \frac

converges if and only if ''z''² is not a real number lying in the interval −4 < ''z''² ≤ 0 – or, equivalently, ''x'' converges if and only if ''z'' ≠ 0 and ''z'' is not a pure imaginary number with imaginary part between -2 and 2. (Not including either endpoint)

Worpitzky's theorem

By applying the fundamental inequalities to the continued fraction :

x = \cfrac\,

it can be shown that the following statements hold if , ''a''_''i'', ≤ 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. *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.\,

1942 J. F. Paydon and H. S. Wall, ''Duke Math. Journal'', vol. 9, "The continued fraction as a sequence of linear transformations" *The radius 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''. 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 In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values, and is analogous to ...

can be applied to a modified version of ''x''. If :

f(z) = \cfrac\,

and a positive real number ''M'' exists such that , ''c''_''i'', ≤ ''M'' (''i'' = 2, 3, 4, ...), then the sequence of convergents converges uniformly when :

, z,  < \frac\,

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 ''a''s and ''b''s may be complex numbers, will converge to a finite value if

, b_n ,  \geq , a_n,  + 1

for

n \geq 1.

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 An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...

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.See theorem 4.29, on page 88, of Jones and Thron (1980).

Notes

References

* *

Oskar Perron Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differentia ...

, ''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 Continued fractions Convergence (mathematics)