Generalized continued fraction

# Generalized continued fraction

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In complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...

s in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values. A generalized continued fraction is an expression of the form $x = b_0 + \cfrac\left\{a_1\right\}\left\{b_1 + \cfrac\left\{a_2\right\}\left\{b_2 + \cfrac\left\{a_3\right\}\left\{b_3 + \cfrac\left\{a_4\right\}\left\{b_4 + \ddots\,\right\}\right\}\right\}\right\}$ where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction. The successive convergents
Convergent (continued fraction)
A convergent is one of a sequence of values obtained by evaluating successive truncations of a continued fraction The nth convergent is also known as the nth approximant of a continued fraction.-Representation of real numbers:...

of the continued fraction are formed by applying the fundamental recurrence formulas
Fundamental recurrence formulas
In the theory of continued fractions, the fundamental recurrence formulas relate the partial numerators and the partial denominators with the numerators and denominators of the fraction's successive convergents...

: $x_0 = \frac\left\{A_0\right\}\left\{B_0\right\} = b_0, \qquad x_1 = \frac\left\{A_1\right\}\left\{B_1\right\} = \frac\left\{b_1b_0+a_1\right\}\left\{b_1\right\},\qquad x_2 = \frac\left\{A_2\right\}\left\{B_2\right\} = \frac\left\{b_2\left(b_1b_0+a_1\right) + a_2b_0\right\}\left\{b_2b_1 + a_2\right\},\qquad\cdots\,$ where An is the numerator and Bn is the denominator, called continuant
Continuant (mathematics)
In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.-Definition:...

s
, of the nth convergent. If the sequence of convergents {xn} approaches a limit the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators Bn.

## History of continued fractions

The story of continued fractions begins with the Euclidean algorithm
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...

, a procedure for finding the greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

of two natural numbers m and n. That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder again, and again, and again. Nearly two thousand years passed before Rafael Bombelli
Rafael Bombelli
Rafael Bombelli was an Italian mathematician.Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers....

devised a technique for approximating the roots of quadratic equations
Solving quadratic equations with continued fractions
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form isax^2+bx+c=0,\,\!where a ≠ 0.Students and teachers all over the world are familiar with the quadratic formula that can be derived by completing the square...

with continued fractions. Now the pace of development quickened. Just 24 years later Pietro Cataldi
Pietro Cataldi
Pietro Antonio Cataldi was an Italian mathematician. A citizen of Bologna, he taught mathematics and astronomy and also worked on military problems. His work included the development of continued fractions and a method for their representation. He was one of many mathematicians who attempted to...

introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as $a_0.\,$ &$n_1 \over d_1.$ &$n_2 \over d_2.$ &$\left\{n_3 \over d_3\right\},$ with the dots indicating where the next fraction goes, and each & representing a modern plus sign. Late in the seventeenth century John Wallis introduced the term "continued fraction" into the mathematical literature. New techniques for mathematical analysis (Newton's
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

and Leibniz's calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

) had recently exploded onto the scene, and a generation of Wallis' contemporaries put the new word to use right away. In 1748 Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

published a very important theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series. Euler's continued fraction theorem is still of central importance in modern attempts to whittle away at the convergence problem. Continued fractions can also be applied to problems in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, and are especially useful in the study of Diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

s. In the late eighteenth century Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

used continued fractions to construct the general solution of Pell's equation
Pell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...

, thus answering a question that had fascinated mathematicians for more than a thousand years. Amazingly, Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1, it contains a palindromic
Palindrome
A palindrome is a word, phrase, number, or other sequence of units that can be read the same way in either direction, with general allowances for adjustments to punctuation and word dividers....

string of length p - 1. In 1813 Gauss used a very clever trick with the complex-valued hypergeometric function to derive a versatile continued fraction expression that has since been named in his honor. That formula can be used to express many elementary functions (and even some more advanced functions, like the Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

s) as rapidly convergent continued fractions valid almost everywhere in the complex plane.

## Notation

The long continued fraction expression displayed in the introduction is probably the most intuitive form for the reader. Unfortunately, it takes up a lot of space in a book (and it's not easy for the typesetter, either). So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction looks like this: $x = b_0+ \frac\left\{a_1\right\}\left\{b_1+\right\}\, \frac\left\{a_2\right\}\left\{b_2+\right\}\, \frac\left\{a_3\right\}\left\{b_3+\right\}\cdots$ Pringsheim
Alfred Pringsheim
Alfred Israel Pringsheim was a German mathematician and patron of the arts. He was born in Ohlau, Prussian Silesia and died in Zürich, Switzerland.- Family and academic career :...

wrote a generalized continued fraction this way: $x = b_0 + \frac\left\{a_1 \mid\right\}\left\{\mid b_1\right\} + \frac\left\{a_2 \mid\right\}\left\{\mid b_2\right\} + \frac\left\{a_3 \mid\right\}\left\{\mid b_3\right\}+\cdots\,$. Karl Friedrich Gauss evoked the more familiar infinite product Π when he devised this notation: $x = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}.\,$ Here the K stands for Kettenbruch, the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters.

## Some elementary considerations

Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions.

### Partial numerators and denominators

If one of the partial numerators an+1 is zero, the infinite continued fraction $b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}\,$ is really just a finite continued fraction with n fractional terms, and therefore a rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

of the first n ais and the first (n + 1) bis. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that none of the ai = 0. There is no need to place this restriction on the partial denominators bi.

### The determinant formula

When the nth convergent of a continued fraction $x_n = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{n\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}\,$ is expressed as a simple fraction xn = An/Bn we can use the determinant formula $A_\left\{n-1\right\}B_n - A_nB_\left\{n-1\right\} = \left(-1\right)^na_1a_2\cdots a_n = \Pi_\left\{i=1\right\}^n \left(-a_i\right)\,$ to relate the numerators and denominators of successive convergents xn and xn-1 to one another. Specifically, if neither Bn nor Bn-1 is zero we can express the difference between the n-1st and nth (n > 0) convergents like this: $x_\left\{n-1\right\} - x_n = \frac\left\{A_\left\{n-1\right\}\right\}\left\{B_\left\{n-1\right\}\right\} - \frac\left\{A_n\right\}\left\{B_n\right\} = \left(-1\right)^n \frac\left\{a_1a_2\cdots a_n\right\}\left\{B_nB_\left\{n-1\right\}\right\} = \frac\left\{\Pi_\left\{i=1\right\}^n \left(-a_i\right)\right\}\left\{B_nB_\left\{n-1\right\}\right\}.\,$

### The equivalence transformation

If {ci} = {c1, c2, c3, ...} is any infinite sequence of non-zero complex numbers we can prove, by induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

, that $b_0 + \cfrac\left\{a_1\right\}\left\{b_1 + \cfrac\left\{a_2\right\}\left\{b_2 + \cfrac\left\{a_3\right\}\left\{b_3 + \cfrac\left\{a_4\right\}\left\{b_4 + \ddots\,\right\}\right\}\right\}\right\} = b_0 + \cfrac\left\{c_1a_1\right\}\left\{c_1b_1 + \cfrac\left\{c_1c_2a_2\right\}\left\{c_2b_2 + \cfrac\left\{c_2c_3a_3\right\}\left\{c_3b_3 + \cfrac\left\{c_3c_4a_4\right\}\left\{c_4b_4 + \ddots\,\right\}\right\}\right\}\right\}$ where equality is understood as equivalence, which is to say that the successive convergents of the continued fraction on the left are exactly the same as the convergents of the fraction on the right. The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of the ai are zero a sequence {ci} can be chosen to make each partial numerator a 1: $b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\} = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{1\right\}\left\{c_i b_i\right\}\,$ where c1 = 1/a1, c2 = a1/a2, c3 = a2/(a1a3), and in general cn+1 = 1/(an+1cn). Second, if none of the partial denominators bi are zero we can use a similar procedure to choose another sequence {di} to make each partial denominator a 1: $b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\} = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{d_i a_i\right\}\left\{1\right\}\,$ where d1 = 1/b1 and otherwise dn+1 = 1/(bnbn+1). These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.

### Simple convergence concepts

It has already been noted that the continued fraction $x = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}\,$ converges if the sequence of convergents {xn} tends to a finite limit. The notion of absolute convergence
Absolute convergence
In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

plays a central role in the theory of infinite series. No corresponding notion exists in the analytic theory of continued fractions – in other words, mathematicians do not speak of an absolutely convergent continued fraction. Sometimes the notion of absolute convergence does enter the discussion, however, especially in the study of the convergence problem. For instance, a particular continued fraction $x = \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{1\right\}\left\{b_i\right\}\,$ diverges by oscillation if the series b1 + b2 + b3 + ... is absolutely convergent. Sometimes the partial numerators and partial denominators of a continued fraction are expressed as functions of a complex variable z. For example, a relatively simple function might be defined as $f\left(z\right) = \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{1\right\}\left\{z\right\}.\,$ For a continued fraction like this one the notion of uniform convergence arises quite naturally. A continued fraction of one or more complex variables is uniformly convergent in an open neighborhood
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...

Ω if the fraction's convergents converge uniformly at every point in Ω. Or, in gory detail: if, for every ε > 0 an integer M can be found such that the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of the difference $f\left(z\right) - f_n\left(z\right) = \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\left(z\right)\right\}\left\{b_i\left(z\right)\right\} - \underset\left\{i=1\right\}\left\{\overset\left\{n\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\left(z\right)\right\}\left\{b_i\left(z\right)\right\}\,$ is less than ε for every point z in an open neighborhood Ω whenever n > M, the continued fraction defining f(z) is uniformly convergent on Ω. (Here fn(z) denotes the nth convergent of the continued fraction, evaluated at the point z inside Ω, and f(z) is the value of the infinite continued fraction at the point z.)

### Even and odd convergents

It is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points p and q, then the sequence {x0, x2, x4, ...} must converge to one of these, and {x1, x3, x5, ...} must converge to the other. In such a situation it may be convenient to express the original continued fraction as two different continued fractions, one of them converging to p, and the other converging to q. The formulas for the even and odd parts of a continued fraction can be written most compactly if the fraction has already been transformed so that all its partial denominators are unity. Specifically, if $x = \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{1\right\}\,$ is a continued fraction, then the even part xeven and the odd part xodd are given by $x_\mathrm\left\{even\right\} = \cfrac\left\{a_1\right\}\left\{1+a_2-\cfrac\left\{a_2a_3\right\} \left\{1+a_3+a_4-\cfrac\left\{a_4a_5\right\} \left\{1+a_5+a_6-\cfrac\left\{a_6a_7\right\} \left\{1+a_7+a_8-\ddots\right\}\right\}\right\}\right\}\,$ and $x_\mathrm\left\{odd\right\} = a_1 - \cfrac\left\{a_1a_2\right\}\left\{1+a_2+a_3-\cfrac\left\{a_3a_4\right\} \left\{1+a_4+a_5-\cfrac\left\{a_5a_6\right\} \left\{1+a_6+a_7-\cfrac\left\{a_7a_8\right\} \left\{1+a_8+a_9-\ddots\right\}\right\}\right\}\right\}\,$ respectively. More precisely, if the successive convergents of the continued fraction x are {x1, x2, x3, ,,,}, then the successive convergents of xeven as written above are {x2, x4, x6, ,,,}, and the successive convergents of xodd are {x1, x3, x5, ,,,}.

### Conditions for irrationality

If $a_1,a_2, \text\left\{ . . .\right\}$ and $b_1,b_2, \text\left\{ . . .\right\}$ are positive integers with $a_k$$b_k$ for all sufficiently large $k$, then$x = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{\infty\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}\,$ converges to an irrational limit.

## Linear fractional transformations

A linear fractional transformation (LFT) is a complex function of the form $w = f\left(z\right) = \frac\left\{a + bz\right\}\left\{c + dz\right\},\,$ where z is a complex variable, and a, b, c, d are arbitrary complex constants. An additional restriction – that adbc – is customarily imposed, to rule out the cases in which w = f(z) is a constant. The linear fractional transformation, also known as a Möbius transformation, has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions.NEWLINE
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• If d ≠ 0 the LFT has one or two fixed points
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

. This can be seen by considering the equation
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which is clearly a quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

in z. The roots of this equation are the fixed points of f(z). If the discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

(cb)2 + 4ad is zero the LFT fixes a single point; otherwise it has two fixed points.
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• If adbc the LFT is an invertible
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

conformal map
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

ping of the extended complex plane
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

onto itself. In other words, this LFT has an inverse function
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such that f(g(z)) = g(f(z)) = z for every point z in the extended complex plane, and both f and g preserve angles and shapes at vanishingly small scales. From the form of z = g(w) we see that g is also an LFT.
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• The composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...

of two different LFTs for which adbc is itself an LFT for which adbc. In other words, the set of all LFTs for which adbc is closed under composition of functions. The collection of all such LFTs – together with the "group operation" composition of functions – is known as the automorphism group of the extended complex plane.
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• If b = 0 the LFT reduces to
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which is a very simple meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

of z with one simple pole (at −c/d) and a residue
Residue theorem
The residue theorem, sometimes called Cauchy's Residue Theorem, in complex analysis is a powerful tool to evaluate line integrals of analytic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula...

Laurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...

.)
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### The continued fraction as a composition of LFTs

Consider a sequence of simple linear fractional transformations $\tau_0\left(z\right) = b_0 + z,\quad \tau_1\left(z\right) = \frac\left\{a_1\right\}\left\{b_1 + z\right\},\quad \tau_2\left(z\right) = \frac\left\{a_2\right\}\left\{b_2 + z\right\},\quad \tau_3\left(z\right) = \frac\left\{a_3\right\}\left\{b_3 + z\right\},\quad\cdots\,$ Here we use the Greek letter τ (tau) to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbol Τn to represent the composition of n+1 little τs – that is, $\boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{1\right\}\right\}\left(z\right) = \tau_0\circ\tau_1\left(z\right) = \tau_0\left(\tau_1\left(z\right)\right),\quad \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{2\right\}\right\}\left(z\right) = \tau_0\circ\tau_1\circ\tau_2\left(z\right) = \tau_0\left(\tau_1\left(\tau_2\left(z\right)\right)\right),\,$ and so forth. By direct substitution from the first set of expressions into the second we see that \begin\left\{align\right\} \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{1\right\}\right\}\left(z\right)& = \tau_0\circ\tau_1\left(z\right)& =&\quad b_0 + \cfrac\left\{a_1\right\}\left\{b_1 + z\right\}\\ \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{2\right\}\right\}\left(z\right)& = \tau_0\circ\tau_1\circ\tau_2\left(z\right)& =&\quad b_0 + \cfrac\left\{a_1\right\}\left\{b_1 + \cfrac\left\{a_2\right\}\left\{b_2 + z\right\}\right\}\, \end\left\{align\right\} and, in general, $\boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n\right\}\right\}\left(z\right) = \tau_0\circ\tau_1\circ\tau_2\circ\cdots\circ\tau_n\left(z\right) = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{n\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}\,$ where the last partial denominator in the finite continued fraction K is understood to be bn + z. And, since bn + 0 = bn, the image of the point z = 0 under the iterated LFT Τn is indeed the value of the finite continued fraction with n partial numerators: $\boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n\right\}\right\}\left(0\right) = \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n+1\right\}\right\}\left(\infty\right) = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{n\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\}.\,$

### A geometric interpretation

Intuition can never replace a mathematical proof. Still, intuition is a useful tool, often suggesting new lines of attack that may finally resolve a previously intractable problem. Defining a finite continued fraction as the image of a point under the iterated LFT Τn(z) leads to an intuitively appealing geometric interpretation of infinite continued fractions. Let's see how that works. The relationship $x_n = b_0 + \underset\left\{i=1\right\}\left\{\overset\left\{n\right\}\left\{\mathrm K\right\}\right\} \frac\left\{a_i\right\}\left\{b_i\right\} = \frac\left\{A_n\right\}\left\{B_n\right\} = \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n\right\}\right\}\left(0\right) = \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n+1\right\}\right\}\left(\infty\right)\,$ is probably best understood by rewriting the LFTs Τn(z) and Τn+1(z) in terms of the fundamental recurrence formulas
Fundamental recurrence formulas
In the theory of continued fractions, the fundamental recurrence formulas relate the partial numerators and the partial denominators with the numerators and denominators of the fraction's successive convergents...

: \begin\left\{align\right\} \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n\right\}\right\}\left(z\right)& = \frac\left\{\left(b_n+z\right)A_\left\{n-1\right\} + a_nA_\left\{n-2\right\}\right\}\left\{\left(b_n+z\right)B_\left\{n-1\right\} + a_nB_\left\{n-2\right\}\right\}& \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n\right\}\right\}\left(z\right)& = \frac\left\{zA_\left\{n-1\right\} + A_n\right\}\left\{zB_\left\{n-1\right\} + B_n\right\};\\ \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n+1\right\}\right\}\left(z\right)& = \frac\left\{\left(b_\left\{n+1\right\}+z\right)A_n + a_\left\{n+1\right\}A_\left\{n-1\right\}\right\}\left\{\left(b_\left\{n+1\right\}+z\right)B_n + a_\left\{n+1\right\}B_\left\{n-1\right\}\right\}& \boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n+1\right\}\right\}\left(z\right)& = \frac\left\{zA_n + A_\left\{n+1\right\}\right\} \left\{zB_n + B_\left\{n+1\right\}\right\}.\, \end\left\{align\right\} In the first of these equations the ratio tends toward An/Bn as z tends toward zero. In the second, the ratio tends toward An/Bn as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents An/Bn are eventually arbitrarily close together
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...

. Since the linear fractional transformation Τn(z) is a continuous mapping
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

, there must be a neighborhood of z = 0 that is mapped into an arbitrarily small neighborhood of Τn(0) = An/Bn. Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of Τn(∞) = An-1/Bn-1. So if the continued fraction converges the transformation Τn(z) maps both very small z and very large z into an arbitrarily small neighborhood of x, the value of the continued fraction, as n gets larger and larger. What about intermediate values of z? Well, since the successive convergents are getting closer together we must have $\frac\left\{A_\left\{n-1\right\}\right\}\left\{B_\left\{n-1\right\}\right\} \approx \frac\left\{A_n\right\}\left\{B_n\right\} \quad\Rightarrow\quad \frac\left\{A_\left\{n-1\right\}\right\}\left\{A_n\right\} \approx \frac\left\{B_\left\{n-1\right\}\right\}\left\{B_n\right\} = k\,$ where k is a constant, introduced for convenience. But then, by substituting in the expression for Τn(z) we obtain $\boldsymbol\left\{\Tau\right\}_\left\{\boldsymbol\left\{n\right\}\right\}\left(z\right) = \frac\left\{zA_\left\{n-1\right\} + A_n\right\}\left\{zB_\left\{n-1\right\} + B_n\right\} \frac\left\{A_n\right\}\left\{B_n\right\} \left\left(\frac\left\{z\frac\left\{A_\left\{n-1\right\}\right\}\left\{A_n\right\} + 1\right\}\left\{z\frac\left\{B_\left\{n-1\right\}\right\}\left\{B_n\right\} + 1\right\}\right\right) \approx \frac\left\{A_n\right\}\left\{B_n\right\} \left\left(\frac\left\{zk + 1\right\}\left\{zk + 1\right\}\right\right) = \frac\left\{A_n\right\}\left\{B_n\right\}\,$ so that even the intermediate values of z (except when z ≈ −k−1) are mapped into an arbitrarily small neighborhood of x, the value of the continued fraction, as n gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point. Notice that the sequence {Τn} lies within the automorphism group of the extended complex plane, since each Τn is a linear fractional transformation for which abcd. And every member of that automorphism group maps the extended complex plane into itself – not one of the Τns can possibly map the plane into a single point. Yet in the limit the sequence {Τn} defines an infinite continued fraction which (if i t converges) represents a single point in the complex plane. How is this possible? Think of it this way. When an infinite continued fraction converges, the corresponding sequence {Τn} of LFTs "focuses" the plane in the direction of x, the value of the continued fraction. At each stage of the process a larger and larger region of the plane is mapped into a neighborhood of x, and the smaller and smaller region of the plane that's left over is stretched out ever more thinly to cover everything outside that neighborhood. What about divergent continued fractions? Can those also be interpreted geometrically? In a word, yes. We distinguish three cases.NEWLINE
NEWLINE
1. The two sequences {Τ2n-1} and {Τ2n} might themselves define two convergent continued fractions that have two different values, xodd and xeven. In this case the continued fraction defined by the sequence {Τn} diverges by oscillation between two distinct limit points. And in fact this idea can be generalized – sequences {Τn} can be constructed that oscillate among three, or four, or indeed any number of limit points. Interesting instances of this case arise when the sequence {Τn} constitutes a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

of finite order within the group of automorphisms over the extended complex plane.
2. NEWLINE
3. The sequence {Τn} may produce an infinite number of zero denominators Bi while also producing a subsequence of finite convergents. These finite convergents may not repeat themselves or fall into a recognizable oscillating pattern. Or they may converge to a finite limit, or even oscillate among multiple finite limits. No matter how the finite convergents behave, the continued fraction defined by the sequence {Τn} diverges by oscillation with the point at infinity in this case.
4. NEWLINE
5. The sequence {Τn} may produce no more than a finite number of zero denominators Bi. while the subsequence of finite convergents dances wildly around the plane in a pattern that never repeats itself and never approaches any finite limit, either.
NEWLINE Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction $x = 1 + \cfrac\left\{z\right\}\left\{1 + \cfrac\left\{z\right\}\left\{1 + \cfrac\left\{z\right\}\left\{1 + \cfrac\left\{z\right\}\left\{1 + \ddots\right\}\right\}\right\}\right\}\,$ where z is any real number such that z < −¼.

## Continued fractions and series

{{main|Euler's continued fraction formula}} Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

proved the following identity: $a_0 + a_0a_1 + a_0a_1a_2 + \cdots + a_0a_1a_2\cdots a_n = \frac\left\{a_0\right\}\left\{1-\right\} \frac\left\{a_1\right\}\left\{1+a_1-\right\} \frac\left\{a_2\right\}\left\{1+a_2-\right\}\cdots \frac\left\{a_\left\{n\right\}\right\}\left\{1+a_n\right\}.\,$ From this many other results can be derived, such as $\frac\left\{1\right\}\left\{u_1\right\}+ \frac\left\{1\right\}\left\{u_2\right\}+ \frac\left\{1\right\}\left\{u_3\right\}+ \cdots+ \frac\left\{1\right\}\left\{u_n\right\} = \frac\left\{1\right\}\left\{u_1-\right\} \frac\left\{u_1^2\right\}\left\{u_1+u_2-\right\} \frac\left\{u_2^2\right\}\left\{u_2+u_3-\right\}\cdots \frac\left\{u_\left\{n-1\right\}^2\right\}\left\{u_\left\{n-1\right\}+u_n\right\},\,$ and $\frac\left\{1\right\}\left\{a_0\right\} + \frac\left\{x\right\}\left\{a_0a_1\right\} + \frac\left\{x^2\right\}\left\{a_0a_1a_2\right\} + \cdots + \frac\left\{x^n\right\}\left\{a_0a_1a_2 \ldots a_n\right\} = \frac\left\{1\right\}\left\{a_0-\right\} \frac\left\{a_0x\right\}\left\{a_1+x-\right\} \frac\left\{a_1x\right\}\left\{a_2+x-\right\}\cdots \frac\left\{a_\left\{n-1\right\}x\right\}\left\{a_n-x\right\}.\,$ Euler's formula connecting continued fractions and series is the motivation for the fundamental inequalities, and also the basis of elementary approaches to the convergence problem.

### Transcendental functionTranscendental functionA transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...s and numberTranscendental numberIn mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...s

Here are two continued fractions that can be built via Euler's identity. $e^x = \frac\left\{x^0\right\}\left\{0!\right\} + \frac\left\{x^1\right\}\left\{1!\right\} + \frac\left\{x^2\right\}\left\{2!\right\} + \frac\left\{x^3\right\}\left\{3!\right\} + \frac\left\{x^4\right\}\left\{4!\right\} + \cdots 1+\cfrac\left\{x\right\} \left\{1-\cfrac\left\{1x\right\} \left\{2+x-\cfrac\left\{2x\right\} \left\{3+x-\cfrac\left\{3x\right\} \left\{4+x-\ddots\right\}\right\}\right\}\right\}$ $\log\left(1+x\right) = \frac\left\{x^1\right\}\left\{1\right\} - \frac\left\{x^2\right\}\left\{2\right\} + \frac\left\{x^3\right\}\left\{3\right\} - \frac\left\{x^4\right\}\left\{4\right\} + \cdots\cfrac\left\{x\right\} \left\{1-0x+\cfrac\left\{1^2x\right\} \left\{2-1x+\cfrac\left\{2^2x\right\} \left\{3-2x+\cfrac\left\{3^2x\right\} \left\{4-3x+\ddots\right\}\right\}\right\}\right\}$ Here are additional generalized continued fractions: $\tan^\left\{-1\right\}\cfrac\left\{x\right\}\left\{y\right\}=\cfrac\left\{xy\right\} \left\{1y^2+\cfrac\left\{\left(1xy\right)^2\right\} \left\{3y^2-1x^2+\cfrac\left\{\left(3xy\right)^2\right\} \left\{5y^2-3x^2+\cfrac\left\{\left(5xy\right)^2\right\} \left\{7y^2-5x^2+\ddots\right\}\right\}\right\}\right\}\cfrac\left\{x\right\} \left\{1y+\cfrac\left\{\left(1x\right)^2\right\} \left\{3y+\cfrac\left\{\left(2x\right)^2\right\} \left\{5y+\cfrac\left\{\left(3x\right)^2\right\} \left\{7y+\ddots\right\}\right\}\right\}\right\}$ $e^\left\{2x/y\right\} = 1+\cfrac\left\{2x\right\} \left\{1y-x+\cfrac\left\{x^2\right\} \left\{3y+\cfrac\left\{x^2\right\} \left\{5y+\cfrac\left\{x^2\right\} \left\{7y+\cfrac\left\{x^2\right\} \left\{9y+\cfrac\left\{x^2\right\} \left\{11y+\ddots\right\}\right\}\right\}\right\}\right\}\right\}; e^2 = 7+\cfrac\left\{2\right\} \left\{5+\cfrac\left\{1\right\} \left\{7+\cfrac\left\{1\right\} \left\{9+\cfrac\left\{1\right\} \left\{11+\ddots\right\}\right\}\right\}\right\}$ $\log \left\left( 1+\frac\left\{2x\right\}\left\{y\right\} \right\right) = \cfrac\left\{2x\right\} \left\{y+\cfrac\left\{1x\right\} \left\{1+\cfrac\left\{1x\right\} \left\{3y+\cfrac\left\{2x\right\} \left\{1+\cfrac\left\{2x\right\} \left\{5y+\cfrac\left\{3x\right\} \left\{1+\ddots\right\}\right\}\right\}\right\}\right\}\right\} \cfrac\left\{2x\right\} \left\{y+x-\cfrac\left\{\left(1x\right)^2\right\} \left\{3\left(y+x\right)-\cfrac\left\{\left(2x\right)^2\right\} \left\{5\left(y+x\right)-\cfrac\left\{\left(3x\right)^2\right\} \left\{7\left(y+x\right)-\ddots\right\}\right\}\right\}\right\}$ Example: the natural logarithm of 2 (≈ 0.693147...): $\log 2 = \log \left(1+1\right) = \cfrac\left\{2\right\} \left\{2+\cfrac\left\{1\right\} \left\{1+\cfrac\left\{1\right\} \left\{6+\cfrac\left\{2\right\} \left\{1+\cfrac\left\{2\right\} \left\{10+\cfrac\left\{3\right\} \left\{1+\ddots\right\}\right\}\right\}\right\}\right\}\right\}$

#### PiPi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

Here are three of its best-known generalized continued fractions, the first and third of which are derived from their respective arctangent formulas above by setting x=y=1 and multiplying by four. $\pi = \sum_\left\{n=0\right\}^\infty \frac\left\{4\left(-1\right)^n\right\}\left\{2n+1\right\} \cfrac\left\{4\right\} \left\{1+\cfrac\left\{1^2\right\} \left\{2+\cfrac\left\{3^2\right\} \left\{2+\cfrac\left\{5^2\right\} \left\{2+\ddots\right\}\right\}\right\}\right\}$ converges too slowly, requiring roughly 3 x 10n terms to achieve n-decimal precision. $\pi = 3 - \sum_\left\{n=1\right\}^\infty \frac\left\{\left(-1\right)^n\right\} \left\{n \left(n+1\right) \left(2n+1\right)\right\} 3 + \cfrac\left\{1^2\right\} \left\{6+\cfrac\left\{3^2\right\} \left\{6+\cfrac\left\{5^2\right\} \left\{6+\ddots\right\}\right\}\right\}$ is a much more obvious expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six. Both converge sublinearly to π. On the other hand: $\pi = \cfrac\left\{4\right\} \left\{1+\cfrac\left\{1^2\right\} \left\{3+\cfrac\left\{2^2\right\} \left\{5+\cfrac\left\{3^2\right\} \left\{7+\ddots\right\}\right\}\right\}\right\}$ converges linearly to π, adding at least three decimals digits of precision per four terms, a pace some 25% faster than Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's 1665-66 arcsine formula for π, which adds at least three decimal digits per five terms. Note: combining this last continued fraction with the best-known Machin-like formula provides an even more rapidly-converging expression: $\pi = 16 \tan^\left\{-1\right\} \cfrac\left\{1\right\}\left\{5\right\} - 4 \tan^\left\{-1\right\} \cfrac\left\{1\right\}\left\{239\right\} \cfrac\left\{16\right\} \left\{5+\cfrac\left\{1^2\right\} \left\{15+\cfrac\left\{2^2\right\} \left\{25+\cfrac\left\{3^2\right\} \left\{35+\ddots\right\}\right\}\right\}\right\} - \cfrac\left\{4\right\} \left\{239+\cfrac\left\{1^2\right\} \left\{717+\cfrac\left\{2^2\right\} \left\{1195+\cfrac\left\{3^2\right\} \left\{1673+\ddots\right\}\right\}\right\}\right\}.$

### Roots of positive numbers

The nth root
Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

of any positive number zm can be expressed by restating z = xn + y, resulting in $\sqrt\left[n\right]\left\{z^m\right\} = \sqrt\left[n\right]\left\{\left(x^n+y\right)^m\right\} = x^m+\cfrac\left\{my\right\} \left\{nx^\left\{n-m\right\}+\cfrac\left\{\left(n-m\right)y\right\} \left\{2x^m+\cfrac\left\{\left(n+m\right)y\right\} \left\{3nx^\left\{n-m\right\}+\cfrac\left\{\left(2n-m\right)y\right\} \left\{2x^m+\cfrac\left\{\left(2n+m\right)y\right\} \left\{5nx^\left\{n-m\right\}+\cfrac\left\{\left(3n-m\right)y\right\} \left\{2x^m+\ddots\right\}\right\}\right\}\right\}\right\}\right\}$ which can be simplified, by folding each pair of fractions into one fraction, to $\sqrt\left[n\right]\left\{z^m\right\} = x^m+\cfrac\left\{2x^m \cdot my\right\} \left\{n\left(2z - y\right)-my-\cfrac\left\{\left(1^2n^2-m^2\right)y^2\right\} \left\{3n\left(2z - y\right)-\cfrac\left\{\left(2^2n^2-m^2\right)y^2\right\} \left\{5n\left(2z - y\right)-\cfrac\left\{\left(3^2n^2-m^2\right)y^2\right\} \left\{7n\left(2z - y\right)-\cfrac\left\{\left(4^2n^2-m^2\right)y^2\right\} \left\{9n\left(2z - y\right)-\ddots\right\}\right\}\right\}\right\}\right\}.$ The square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of z is a special case of this nth root algorithm (m=1, n=2): $\sqrt\left\{z\right\} = \sqrt\left\{x^2+y\right\} = x+\cfrac\left\{y\right\} \left\{2x+\cfrac\left\{y\right\} \left\{2x+\cfrac\left\{3y\right\} \left\{6x+\cfrac\left\{3y\right\} \left\{2x+\ddots\right\}\right\}\right\}\right\} x+\cfrac\left\{2x \cdot y\right\} \left\{2\left(2z - y\right)-y-\cfrac\left\{1\cdot 3y^2\right\} \left\{6\left(2z - y\right)-\cfrac\left\{3\cdot 5y^2\right\} \left\{10\left(2z - y\right)-\ddots\right\}\right\}\right\}$ which can be simplified by noting that 5/10 = 3/6 = 1/2: $\sqrt\left\{z\right\} = \sqrt\left\{x^2+y\right\} = x+\cfrac\left\{y\right\} \left\{2x+\cfrac\left\{y\right\} \left\{2x+\cfrac\left\{y\right\} \left\{2x+\cfrac\left\{y\right\} \left\{2x+\ddots\right\}\right\}\right\}\right\} x+\cfrac\left\{2x \cdot y\right\} \left\{2\left(2z - y\right)-y-\cfrac\left\{y^2\right\} \left\{2\left(2z - y\right)-\cfrac\left\{y^2\right\} \left\{2\left(2z - y\right)-\ddots\right\}\right\}\right\}.$ The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper x and y.

#### Example 1

The cube root of two (21/3 or 3√2 ≈ 1.259921...): (A) "Standard notation" of x = 1, y = 1, and 2z - y = 3: $\sqrt\left[3\right]2 = 1+\cfrac\left\{1\right\} \left\{3+\cfrac\left\{2\right\} \left\{2+\cfrac\left\{4\right\} \left\{9+\cfrac\left\{5\right\} \left\{2+\cfrac\left\{7\right\} \left\{15+\cfrac\left\{8\right\} \left\{2+\cfrac\left\{10\right\} \left\{21+\cfrac\left\{11\right\} \left\{2+\ddots\right\}\right\}\right\}\right\}\right\}\right\}\right\}\right\} = 1+\cfrac\left\{2 \cdot 1\right\} \left\{9-1-\cfrac\left\{2 \cdot 4\right\} \left\{27-\cfrac\left\{5 \cdot 7\right\} \left\{45-\cfrac\left\{8 \cdot 10\right\} \left\{63-\cfrac\left\{11 \cdot 13\right\} \left\{81-\ddots\right\}\right\}\right\}\right\}\right\}.$ (B) Rapid convergence with x = 5, y = 3 and 2z - y = 253: $\sqrt\left[3\right]2 = \cfrac\left\{5\right\}\left\{4\right\}+\cfrac\left\{0.5\right\} \left\{50+\cfrac\left\{2\right\} \left\{5+\cfrac\left\{4\right\} \left\{150+\cfrac\left\{5\right\} \left\{5+\cfrac\left\{7\right\} \left\{250+\cfrac\left\{8\right\} \left\{5+\cfrac\left\{10\right\} \left\{350+\cfrac\left\{11\right\} \left\{5+\ddots\right\}\right\}\right\}\right\}\right\}\right\}\right\}\right\} = \cfrac\left\{5\right\}\left\{4\right\}+\cfrac\left\{2.5 \cdot 1\right\} \left\{253-1-\cfrac\left\{2 \cdot 4\right\} \left\{759-\cfrac\left\{5 \cdot 7\right\} \left\{1265-\cfrac\left\{8 \cdot 10\right\} \left\{1771-\ddots\right\}\right\}\right\}\right\}.$

#### Example 2

Pogson's ratio (1001/5 or 5√100 ≈ 2.511886...), with x = 5, y = 75 and 2z - y = 6325: $\sqrt\left[5\right]\left\{100\right\} = \cfrac\left\{5\right\}\left\{2\right\}+\cfrac\left\{3\right\} \left\{250+\cfrac\left\{12\right\} \left\{5+\cfrac\left\{18\right\} \left\{750+\cfrac\left\{27\right\} \left\{5+\cfrac\left\{33\right\} \left\{1250+\cfrac\left\{42\right\} \left\{5+\ddots\right\}\right\}\right\}\right\}\right\}\right\} = \cfrac\left\{5\right\}\left\{2\right\}+\cfrac\left\{5\cdot 3\right\} \left\{1265-3-\cfrac\left\{12 \cdot 18\right\} \left\{3795-\cfrac\left\{27 \cdot 33\right\} \left\{6325-\cfrac\left\{42 \cdot 48\right\} \left\{8855-\ddots\right\}\right\}\right\}\right\}.$

#### Example 3

The twelfth root of two (21/12 or 12√2 ≈ 1.059463...), using "standard notation": $\sqrt\left[12\right]2 = 1+\cfrac\left\{1\right\} \left\{12+\cfrac\left\{11\right\} \left\{2+\cfrac\left\{13\right\} \left\{36+\cfrac\left\{23\right\} \left\{2+\cfrac\left\{25\right\} \left\{60+\cfrac\left\{35\right\} \left\{2+\cfrac\left\{37\right\} \left\{84+\cfrac\left\{47\right\} \left\{2+\ddots\right\}\right\}\right\}\right\}\right\}\right\}\right\}\right\} = 1+\cfrac\left\{2 \cdot 1\right\} \left\{36-1 - \cfrac\left\{11 \cdot 13\right\} \left\{108-\cfrac\left\{23 \cdot 25\right\} \left\{180-\cfrac\left\{35 \cdot 37\right\} \left\{252-\cfrac\left\{47 \cdot 49\right\} \left\{324-\ddots\right\}\right\}\right\}\right\}\right\}.$

#### Example 4

Equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

's perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...

(27/12 or 12√27 ≈ 1.498307...), with m=7: (A) "Standard notation": $\sqrt\left[12\right]\left\{2^7\right\} = 1+\cfrac\left\{7\right\} \left\{12+\cfrac\left\{5\right\} \left\{2+\cfrac\left\{19\right\} \left\{36+\cfrac\left\{17\right\} \left\{2+\cfrac\left\{31\right\} \left\{60+\cfrac\left\{29\right\} \left\{2+\cfrac\left\{43\right\} \left\{84+\cfrac\left\{41\right\} \left\{2+\ddots\right\}\right\}\right\}\right\}\right\}\right\}\right\}\right\} = 1+\cfrac\left\{2 \cdot 7\right\} \left\{36-7 - \cfrac\left\{5 \cdot 19\right\} \left\{108-\cfrac\left\{17 \cdot 31\right\} \left\{180-\cfrac\left\{29 \cdot 43\right\} \left\{252-\cfrac\left\{41 \cdot 55\right\} \left\{324-\ddots\right\}\right\}\right\}\right\}\right\}.$ (B) Rapid convergence with x = 3, y = –7153, and 2z - y = 219+312: $\sqrt\left[12\right]\left\{2^7\right\} = \cfrac\left\{1\right\}\left\{2\right\} \sqrt\left[12\right]\left\{3^\left\{12\right\}-7153\right\} = \cfrac\left\{3\right\}\left\{2\right\} - \cfrac\left\{0.5 \cdot 7153\right\}\left\{4\cdot 3^\left\{12\right\} - \cfrac\left\{11\cdot 7153\right\}\left\{6 - \cfrac\left\{13\cdot 7153\right\}\left\{12\cdot 3^\left\{12\right\} - \cfrac\left\{23\cdot 7153\right\}\left\{6 - \cfrac\left\{25\cdot 7153\right\}\left\{20\cdot 3^\left\{12\right\} - \cfrac\left\{35\cdot 7153\right\}\left\{6 - \cfrac\left\{37\cdot 7153\right\}\left\{28\cdot 3^\left\{12\right\} - \cfrac\left\{47\cdot 7153\right\}\left\{6 - \ddots\right\}\right\}\right\}\right\}\right\}\right\}\right\}\right\}$ $\sqrt\left[12\right]\left\{2^7\right\} = \cfrac\left\{3\right\}\left\{2\right\} - \cfrac\left\{3\cdot 7153\right\}\left\{12\left(2^\left\{19\right\}+3^\left\{12\right\}\right) + 7153 - \cfrac\left\{11\cdot 13\cdot 7153^2\right\}\left\{36\left(2^\left\{19\right\}+3^\left\{12\right\}\right) - \cfrac\left\{23\cdot 25\cdot 7153^2\right\}\left\{60\left(2^\left\{19\right\}+3^\left\{12\right\}\right) - \cfrac\left\{35\cdot 37\cdot 7153^2\right\}\left\{84\left(2^\left\{19\right\}+3^\left\{12\right\}\right) - \ddots\right\}\right\}\right\}\right\}.$ More details on this technique can be found in General Method for Extracting Roots using (Folded) Continued Fractions.

## Higher dimensions

Another meaning for generalized continued fraction is a generalization to higher dimensions. For example, there is a close relationship between the simple continued fraction in canonical form for the irrational real number α, and the way lattice points in two dimensions lie to either side of the line y = αx. Generalizing this idea, one might ask about something related to lattice points in three or more dimensions. One reason to study this area is to quantify the mathematical coincidence idea; for example, for monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...

s in several real numbers, take the logarithmic form
Logarithmic form
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind....

and consider how small it can be. Another reason is to find a possible solution to Hermite's problem
Hermite's problem
Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.-Motivation:A standard way of...

. There have been numerous attempts to construct a generalized theory. Two notable efforts are those of Georges Poitou and George Szekeres
George Szekeres
George Szekeres AM was a Hungarian-Australian mathematician.-Early years:Szekeres was born in Budapest, Hungary as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked six years in Budapest as an analytical chemist. He married Esther Klein in 1936...

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• Gauss's continued fraction
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In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximantsto a given complex formal power series...

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• Solving quadratic equations with continued fractions
Solving quadratic equations with continued fractions
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form isax^2+bx+c=0,\,\!where a ≠ 0.Students and teachers all over the world are familiar with the quadratic formula that can be derived by completing the square...

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