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

- Elementary results
- Periodic continued fractions
- The special case when period k = 1
- Worpitzky's theorem
- Śleszyński–Pringsheim criterion
- Van Vleck's theorem
- Notes
- References

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

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*_{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 numbers.

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

where *k*≥ 1, the sequence of partial numerators {*a*_{1}, *a*_{2}, *a*_{3}, ..., *a*_{k}} contains no values equal to zero, and the partial numerators {*a*_{1}, *a*_{2}, *a*_{3}, ..., *a*_{k}} and partial denominators {*b*_{1}, *b*_{2}, *b*_{3}, ..., *b*_{k}} repeat over and over again, *ad infinitum*.

By applying the theory of linear fractional transformations to

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

These roots are the fixed points 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.^{ [1] }^{ [2] }^{ [3] }

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

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

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

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

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

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*. The infinite periodic continued fraction

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)

By applying the fundamental inequalities to the continued fraction

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

^{ [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.^{[ disputed (for: Euler's continued fraction formula is older) – discuss ]}^{ [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

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

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

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 for ^{ [7] }

Jones and Thron attribute the following result to Van Vleck. Suppose that all the *a _{i}* are equal to 1, and all the

with epsilon being any positive number less than . In other words, all the *b _{i}* are inside a wedge which has its vertex at the origin, has an opening angle of , and is symmetric around the positive real axis. Then

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.

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

In mathematics, an **exponential function** is a function of the form

In mathematics, a **square root** of a number *x* is a number *y* such that *y*^{2} = *x*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *x*. For example, 4 and −4 are square roots of 16, because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *x* has a unique nonnegative square root, called the *principal square root*, which is denoted by where the symbol is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by because 3^{2} = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this case 9.

The **square root of 2**, or the **one-half power of 2**, written in mathematics as or , is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it must be called the **principal square root of 2**, to distinguish it from the negative number with the same property.

In complex analysis, a branch of mathematics, a **generalized continued fraction** is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.

Approximations for the mathematical constant pi in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes). In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

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.

In the metrical theory of regular continued fractions, the *k*th **complete quotient** ζ_{ k} is obtained by ignoring the first *k* partial denominators *a*_{i}. For example, if a regular continued fraction is given by

In mathematics, an infinite **periodic continued fraction** is a continued fraction that can be placed in the form

In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction *x* is said to be *restricted*, or composed of **restricted partial quotients**, if the sequence of denominators of its partial quotients is bounded; that is

In complex analysis, a **Padé table** is an array, possibly of infinite extent, of the rational Padé approximants

In complex analysis, **Gauss's continued fraction** is a particular class of continued fractions derived from hypergeometric functions. It was one of the first analytic continued fractions known to mathematics, and it can be used to represent several important elementary functions, as well as some of the more complicated transcendental functions.

The **square root of 5** is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the **principal square root of 5**, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

The **Rogers–Ramanujan continued fraction** is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.

In mathematics, a **rational number** is a number such as -3/7 that can be expressed as the quotient or fraction *p*/*q* of two integers, a numerator *p* and a non-zero denominator *q*. Every integer is a rational number: for example, 5 = 5/1. The set of all rational numbers, often referred to as "**the rationals**", the **field of rationals** or the **field of rational numbers** is usually denoted by a boldface **Q** ; it was thus denoted in 1895 by Giuseppe Peano after *quoziente*, Italian for "quotient".

In mathematics, an **infinite expression** is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth. A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions, but there are several instances of infinite expressions that are well-defined.

In mathematics, the **Śleszyński–Pringsheim theorem** is a statement about convergence of certain continued fractions. It was discovered by Ivan Śleszyński and Alfred Pringsheim in the late 19th century.

In mathematics, **infinite compositions of analytic functions (ICAF)** offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a *single function* see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

- 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

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