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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 real or complex values.

A generalized continued fraction is an expression of the form

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

where the a_n (n > 0) are the partial numerators, the b_n are the partial denominators, and the leading term b₀ is called the integer part of the continued fraction.

The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas:

$x_{0}={\frac {A_{0}}{B_{0}}}=b_{0},\qquad x_{1}={\frac {A_{1}}{B_{1}}}={\frac {b_{1}b_{0}+a_{1}}{b_{1}}},\qquad x_{2}={\frac {A_{2}}{B_{2}}}={\frac {b_{2}(b_{1}b_{0}+a_{1})+a_{2}b_{0}}{b_{2}b_{1}+a_{2}}},\qquad \cdots \,$

and in general

$A_{n}=b_{n}A_{n-1}+a_{n}A_{n-2},\qquad B_{n}=b_{n}B_{n-1}+a_{n}B_{n-2},\,$

where A_n is the numerator and B_n is the denominator, called continuants, of the nth convergent.

If the sequence of convergents {x_n} 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 B_n.

History of continued fractions

The story of continued fractions begins with the Euclidean algorithm, a procedure for finding the greatest common divisor 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 devised a technique for approximating the roots of quadratic equations with continued fractions. Now the pace of development quickened. Just 24 years later Pietro Cataldi introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as

$a_{0}.\,$ &$n_{1} \over d_{1}.$ &$n_{2} \over d_{2}.$ &${n_{3} \over d_{3}},$

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 and Leibniz's calculus) had recently exploded onto the scene, and a generation of Wallis' contemporaries put the new word to use right away.

In 1748 Euler 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, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's 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 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 functions) 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:

$x=b_{0}+{\frac {a_{1}}{b_{1}+}}\,{\frac {a_{2}}{b_{2}+}}\,{\frac {a_{3}}{b_{3}+}}\cdots$

Pringsheim wrote a generalized continued fraction this way:

$x=b_{0}+{\frac {a_{1}\mid }{\mid b_{1}}}+{\frac {a_{2}\mid }{\mid b_{2}}}+{\frac {a_{3}\mid }{\mid b_{3}}}+\cdots \,$.

Carl Friedrich Gauss evoked the more familiar infinite product Π when he devised this notation:

$x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}.\,$

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 a_n+1 is zero, the infinite continued fraction

$b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,$

is really just a finite continued fraction with n fractional terms, and therefore a rational function of the first n a_i's and the first (n + 1) b_i's. 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 a_i = 0. There is no need to place this restriction on the partial denominators b_i.

The determinant formula

When the nth convergent of a continued fraction

$x_{n}=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,$

is expressed as a simple fraction x_n = A_n/B_n we can use the determinant formula

$A_{n-1}B_{n}-A_{n}B_{n-1}=(-1)^{n}a_{1}a_{2}\cdots a_{n}=\Pi _{i=1}^{n}(-a_{i})\,$

to relate the numerators and denominators of successive convergents x_n and x_n-1 to one another. Specifically, if neither B_n nor B_n-1 is zero we can express the difference between the n-1st and nth (n > 0) convergents like this:

$x_{n-1}-x_{n}={\frac {A_{n-1}}{B_{n-1}}}-{\frac {A_{n}}{B_{n}}}=(-1)^{n}{\frac {a_{1}a_{2}\cdots a_{n}}{B_{n}B_{n-1}}}={\frac {\Pi _{i=1}^{n}(-a_{i})}{B_{n}B_{n-1}}}.\,$

The equivalence transformation

If {c_i} = {c₁, c₂, c₃, ...} is any infinite sequence of non-zero complex numbers we can prove, by induction, that

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

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 a_i are zero a sequence {c_i} can be chosen to make each partial numerator a 1:

$b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{c_{i}b_{i}}}\,$

where c₁ = 1/a₁, c₂ = a₁/a₂, c₃ = a₂/(a₁a₃), and in general c_n+1 = 1/(a_n+1c_n).

Second, if none of the partial denominators b_i are zero we can use a similar procedure to choose another sequence {d_i} to make each partial denominator a 1:

$b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {d_{i}a_{i}}{1}}\,$

where d₁ = 1/b₁ and otherwise d_n+1 = 1/(b_nb_n+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

$x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,$

converges if the sequence of convergents {x_n} tends to a finite limit.

The notion of absolute convergence 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

$x={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{b_{i}}}\,$

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

$f(z)={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{z}}.\,$

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 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 of the difference

$f(z)-f_{n}(z)={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}(z)}{b_{i}(z)}}-{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}(z)}{b_{i}(z)}}\,$

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 f_n(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.)

The Śleszyński–Pringsheim theorem provides a sufficient condition for convergence.

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

$x={\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{1}}\,$

is a continued fraction, then the even part x_even and the odd part x_odd are given by

$x_{\mathrm {even} }={\cfrac {a_{1}}{1+a_{2}-{\cfrac {a_{2}a_{3}}{1+a_{3}+a_{4}-{\cfrac {a_{4}a_{5}}{1+a_{5}+a_{6}-{\cfrac {a_{6}a_{7}}{1+a_{7}+a_{8}-\ddots }}}}}}}}\,$

and

$x_{\mathrm {odd} }=a_{1}-{\cfrac {a_{1}a_{2}}{1+a_{2}+a_{3}-{\cfrac {a_{3}a_{4}}{1+a_{4}+a_{5}-{\cfrac {a_{5}a_{6}}{1+a_{6}+a_{7}-{\cfrac {a_{7}a_{8}}{1+a_{8}+a_{9}-\ddots }}}}}}}}\,$

respectively. More precisely, if the successive convergents of the continued fraction x are {x₁, x₂, x₃, ...}, then the successive convergents of x_even as written above are {x₂, x₄, x₆, ...}, and the successive convergents of x_odd are {x₁, x₃, x₅, ...}.

Conditions for irrationality

If $a_{1},a_{2},{\text{ . . .}}$ and $b_{1},b_{2},{\text{ . . .}}$ are positive integers with $a_{k}$ ≤ $b_{k}$ for all sufficiently large $k$, then

$x=b_{0}+{\underset {i=1}{\overset {\infty }{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,$

converges to an irrational limit.

Fundamental recurrence formulas

The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas:

${\begin{aligned}A_{-1}&=1&B_{-1}&=0\\A_{0}&=b_{0}&B_{0}&=1\\A_{n+1}&=b_{n+1}A_{n}+a_{n+1}A_{n-1}&B_{n+1}&=b_{n+1}B_{n}+a_{n+1}B_{n-1}\,\end{aligned}}$

The continued fraction's successive convergents are then given by

$x_{n}={\frac {A_{n}}{B_{n}}}.\,$

These recurrence relations are due to John Wallis (1616-1703) and Leonhard Euler (1707-1783).

As an example, consider the regular continued fraction in canonical form that represents the golden ratio φ:

$x=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots \,}}}}}}}}$

Applying the fundamental recurrence formulas we find that the successive numerators A_n are {1, 2, 3, 5, 8, 13, ...} and the successive denominators B_n are {1, 1, 2, 3, 5, 8, ...}, the Fibonacci numbers. Since all the partial numerators in this example are equal to one, the determinant formula assures us that the absolute value of the difference between successive convergents approaches zero quite rapidly.

Linear fractional transformations

A linear fractional transformation (LFT) is a complex function of the form

$w=f(z)={\frac {a+bz}{c+dz}},\,$

where z is a complex variable, and a, b, c, d are arbitrary complex constants. An additional restriction – that ad ≠ bc – 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.

• If d ≠ 0 the LFT has one or two fixed points. This can be seen by considering the equation

$f(z)=z\Rightarrow dz^{2}+cz=a+bz\,$

which is clearly a quadratic equation in z. The roots of this equation are the fixed points of f(z). If the discriminant (c − b)² + 4ad is zero the LFT fixes a single point; otherwise it has two fixed points.

• If ad ≠ bc the LFT is an invertible conformal mapping of the extended complex plane onto itself. In other words, this LFT has an inverse function

$z=g(w)={\frac {-a+cw}{b-dw}}\,$

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.

• The composition of two different LFTs for which ad ≠ bc is itself an LFT for which ad ≠ bc. In other words, the set of all LFTs for which ad ≠ bc 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.

• If b = 0 the LFT reduces to

$w=f(z)={\frac {a}{c+dz}},\,$

which is a very simple meromorphic function of z with one simple pole (at −c/d) and a residue equal to a/d. (See also Laurent series.)

The continued fraction as a composition of LFTs

Consider a sequence of simple linear fractional transformations

$\tau _{0}(z)=b_{0}+z,\quad \tau _{1}(z)={\frac {a_{1}}{b_{1}+z}},\quad \tau _{2}(z)={\frac {a_{2}}{b_{2}+z}},\quad \tau _{3}(z)={\frac {a_{3}}{b_{3}+z}},\quad \cdots \,$

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 Τ_n to represent the composition of n+1 little τs – that is,

${\boldsymbol {\mathrm {T} }}_{\boldsymbol {1}}(z)=\tau _{0}\circ \tau _{1}(z)=\tau _{0}(\tau _{1}(z)),\quad {\boldsymbol {\mathrm {T} }}_{\boldsymbol {2}}(z)=\tau _{0}\circ \tau _{1}\circ \tau _{2}(z)=\tau _{0}(\tau _{1}(\tau _{2}(z))),\,$

and so forth. By direct substitution from the first set of expressions into the second we see that

${\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {1}}(z)&=\tau _{0}\circ \tau _{1}(z)&=&\quad b_{0}+{\cfrac {a_{1}}{b_{1}+z}}\\{\boldsymbol {\mathrm {T} }}_{\boldsymbol {2}}(z)&=\tau _{0}\circ \tau _{1}\circ \tau _{2}(z)&=&\quad b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+z}}}}\,\end{aligned}}$

and, in general,

${\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)=\tau _{0}\circ \tau _{1}\circ \tau _{2}\circ \cdots \circ \tau _{n}(z)=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}\,$

where the last partial denominator in the finite continued fraction K is understood to be b_n + z. And, since b_n + 0 = b_n, the image of the point z = 0 under the iterated LFT Τ_n is indeed the value of the finite continued fraction with n partial numerators:

${\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(0)={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(\infty )=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}.\,$

A geometric interpretation

Defining a finite continued fraction as the image of a point under the iterated LFT Τ_n(z) leads to an intuitively appealing geometric interpretation of infinite continued fractions.

The relationship

$x_{n}=b_{0}+{\underset {i=1}{\overset {n}{\mathrm {K} }}}{\frac {a_{i}}{b_{i}}}={\frac {A_{n}}{B_{n}}}={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(0)={\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(\infty )\,$

can be understood by rewriting the LFTs Τ_n(z) and Τ_n+1(z) in terms of the fundamental recurrence formulas:

${\begin{aligned}{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)&={\frac {(b_{n}+z)A_{n-1}+a_{n}A_{n-2}}{(b_{n}+z)B_{n-1}+a_{n}B_{n-2}}}&{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)&={\frac {zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}};\\{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(z)&={\frac {(b_{n+1}+z)A_{n}+a_{n+1}A_{n-1}}{(b_{n+1}+z)B_{n}+a_{n+1}B_{n-1}}}&{\boldsymbol {\mathrm {T} }}_{\boldsymbol {n+1}}(z)&={\frac {zA_{n}+A_{n+1}}{zB_{n}+B_{n+1}}}.\,\end{aligned}}$

In the first of these equations the ratio tends toward A_n/B_n as z tends toward zero. In the second, the ratio tends toward A_n/B_n as z tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents A_n/B_n are eventually arbitrarily close together. Since the linear fractional transformation Τ_n(z) is a continuous mapping, there must be a neighborhood of z = 0 that is mapped into an arbitrarily small neighborhood of Τ_n(0) = A_n/B_n. Similarly, there must be a neighborhood of the point at infinity which is mapped into an arbitrarily small neighborhood of Τ_n(∞) = A_n-1/B_n-1. So if the continued fraction converges the transformation Τ_n(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

${\frac {A_{n-1}}{B_{n-1}}}\approx {\frac {A_{n}}{B_{n}}}\quad \Rightarrow \quad {\frac {A_{n-1}}{A_{n}}}\approx {\frac {B_{n-1}}{B_{n}}}=k\,$

where k is a constant, introduced for convenience. But then, by substituting in the expression for Τ_n(z) we obtain

${\boldsymbol {\mathrm {T} }}_{\boldsymbol {n}}(z)={\frac {zA_{n-1}+A_{n}}{zB_{n-1}+B_{n}}}={\frac {A_{n}}{B_{n}}}\left({\frac {z{\frac {A_{n-1}}{A_{n}}}+1}{z{\frac {B_{n-1}}{B_{n}}}+1}}\right)\approx {\frac {A_{n}}{B_{n}}}\left({\frac {zk+1}{zk+1}}\right)={\frac {A_{n}}{B_{n}}}\,$

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 {Τ_n} lies within the automorphism group of the extended complex plane, since each Τ_n is a linear fractional transformation for which ab ≠ cd. And every member of that automorphism group maps the extended complex plane into itself – not one of the Τ_ns can possibly map the plane into a single point. Yet in the limit the sequence {Τ_n} 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 {Τ_n} 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.

1. The two sequences {Τ_2n-1} and {Τ_2n} might themselves define two convergent continued fractions that have two different values, x_odd and x_even. In this case the continued fraction defined by the sequence {Τ_n} diverges by oscillation between two distinct limit points. And in fact this idea can be generalized – sequences {Τ_n} 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 {Τ_n} constitutes a subgroup of finite order within the group of automorphisms over the extended complex plane.
2. The sequence {Τ_n} may produce an infinite number of zero denominators B_i 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 {Τ_n} diverges by oscillation with the point at infinity in this case.
3. The sequence {Τ_n} may produce no more than a finite number of zero denominators B_i. 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.

Interesting examples of cases 1 and 3 can be constructed by studying the simple continued fraction

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

where z is any real number such that z < −¼.

Continued fractions and series

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Euler proved the following identity:

$a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdots +a_{0}a_{1}a_{2}\cdots a_{n}={\frac {a_{0}}{1-}}{\frac {a_{1}}{1+a_{1}-}}{\frac {a_{2}}{1+a_{2}-}}\cdots {\frac {a_{n}}{1+a_{n}}}.\,$

From this many other results can be derived, such as

${\frac {1}{u_{1}}}+{\frac {1}{u_{2}}}+{\frac {1}{u_{3}}}+\cdots +{\frac {1}{u_{n}}}={\frac {1}{u_{1}-}}{\frac {u_{1}^{2}}{u_{1}+u_{2}-}}{\frac {u_{2}^{2}}{u_{2}+u_{3}-}}\cdots {\frac {u_{n-1}^{2}}{u_{n-1}+u_{n}}},\,$

and

${\frac {1}{a_{0}}}+{\frac {x}{a_{0}a_{1}}}+{\frac {x^{2}}{a_{0}a_{1}a_{2}}}+\cdots +{\frac {x^{n}}{a_{0}a_{1}a_{2}\ldots a_{n}}}={\frac {1}{a_{0}-}}{\frac {a_{0}x}{a_{1}+x-}}{\frac {a_{1}x}{a_{2}+x-}}\cdots {\frac {a_{n-1}x}{a_{n}-x}}.\,$

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.

Examples

Transcendental functions and numbers

Here are two continued fractions that can be built via Euler's identity.

$e^{x}={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots =1+{\cfrac {x}{1-{\cfrac {1x}{2+x-{\cfrac {2x}{3+x-{\cfrac {3x}{4+x-\ddots }}}}}}}}$

$\log(1+x)={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots ={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+\ddots }}}}}}}}$

Here are additional generalized continued fractions:

$\tan ^{-1}{\cfrac {x}{y}}={\cfrac {xy}{1y^{2}+{\cfrac {(1xy)^{2}}{3y^{2}-1x^{2}+{\cfrac {(3xy)^{2}}{5y^{2}-3x^{2}+{\cfrac {(5xy)^{2}}{7y^{2}-5x^{2}+\ddots }}}}}}}}={\cfrac {x}{1y+{\cfrac {(1x)^{2}}{3y+{\cfrac {(2x)^{2}}{5y+{\cfrac {(3x)^{2}}{7y+\ddots }}}}}}}}$

$e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+{\cfrac {x^{2}}{22y+\ddots }}}}}}}}}}}};e^{2}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots }}}}}}}}$

$\log \left(1+{\frac {x}{y}}\right)={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}$

This last is based on an algorithm derived by Alekseĭ Nikolaevich Khovanskiĭ in the 1970's.

Example: the natural logarithm of 2 (= [0;1,2,3,1,5,2/3,7,1/2,9,2/5,...,2k-1,2/k,...] ≈ 0.693147...):

$\log 2=\log(1+1)={\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {2}{5+{\cfrac {3}{2+\ddots }}}}}}}}}}}}={\cfrac {2}{3-{\cfrac {1^{2}}{9-{\cfrac {2^{2}}{15-{\cfrac {3^{2}}{21-\ddots }}}}}}}}$

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Here are three of Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers [https://www.manateegalleries.com/?option=com_k2&view=itemlist&task=user&id=30857 house in singapore] singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.'s 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. The Leibniz formula for Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers [https://www.manateegalleries.com/?option=com_k2&view=itemlist&task=user&id=30857 house in singapore] singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.:

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+\ddots }}}}}}}}=\sum _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+-\cdots$

converges too slowly, requiring roughly 3 x 10ⁿ terms to achieve n-decimal precision. The series derived by Nilakantha Somayaji:

$\pi =3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+\ddots }}}}}}=3-\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(n+1)(2n+1)}}=3+{\frac {1}{1\cdot 2\cdot 3}}-{\frac {1}{2\cdot 3\cdot 5}}+{\frac {1}{3\cdot 4\cdot 7}}-+\cdots$

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 Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.. On the other hand:

$\pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+\ddots }}}}}}}}=4-1+{\frac {1}{6}}-{\frac {1}{34}}+{\frac {16}{3145}}-{\frac {4}{4551}}+{\frac {1}{6601}}-{\frac {1}{38341}}+-\cdots$

converges linearly to Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park., adding at least three decimals digits of precision per four terms, a pace slightly faster than the arcsine formula for Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers [https://www.manateegalleries.com/?option=com_k2&view=itemlist&task=user&id=30857 house in singapore] singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.: $\pi =6\sin ^{-1}\left({\frac {1}{2}}\right)=\sum _{n=0}^{\infty }{\frac {3\cdot {\binom {2n}{n}}}{16^{n}(2n+1)}}={\frac {3}{16^{0}\cdot 1}}+{\frac {6}{16^{1}\cdot 3}}+{\frac {18}{16^{2}\cdot 5}}+{\frac {60}{16^{3}\cdot 7}}+\cdots \!$

which adds at least three decimal digits per five terms.

Note: combining the last continued fraction with the best-known Machin-like formula provides an even more rapidly-converging expression:

$\pi =16\tan ^{-1}{\cfrac {1}{5}}-4\tan ^{-1}{\cfrac {1}{239}}={\cfrac {16}{5+{\cfrac {1^{2}}{15+{\cfrac {2^{2}}{25+{\cfrac {3^{2}}{35+\ddots }}}}}}}}-{\cfrac {4}{239+{\cfrac {1^{2}}{717+{\cfrac {2^{2}}{1195+{\cfrac {3^{2}}{1673+\ddots }}}}}}}}.$

Roots of positive numbers

The nth root of any positive number z^m can be expressed by restating z = xⁿ + y, resulting in

${\sqrt[{n}]{z^{m}}}={\sqrt[{n}]{(x^{n}+y)^{m}}}=x^{m}+{\cfrac {my}{nx^{n-m}+{\cfrac {(n-m)y}{2x^{m}+{\cfrac {(n+m)y}{3nx^{n-m}+{\cfrac {(2n-m)y}{2x^{m}+{\cfrac {(2n+m)y}{5nx^{n-m}+{\cfrac {(3n-m)y}{2x^{m}+\ddots }}}}}}}}}}}}$

which can be simplified, by folding each pair of fractions into one fraction, to

${\sqrt[{n}]{z^{m}}}=x^{m}+{\cfrac {2x^{m}\cdot my}{n(2z-y)-my-{\cfrac {(1^{2}n^{2}-m^{2})y^{2}}{3n(2z-y)-{\cfrac {(2^{2}n^{2}-m^{2})y^{2}}{5n(2z-y)-{\cfrac {(3^{2}n^{2}-m^{2})y^{2}}{7n(2z-y)-{\cfrac {(4^{2}n^{2}-m^{2})y^{2}}{9n(2z-y)-\ddots }}}}}}}}}}.$

The square root of z is a special case of this nth root algorithm (m=1, n=2):

${\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {3y}{6x+{\cfrac {3y}{2x+\ddots }}}}}}}}=x+{\cfrac {2x\cdot y}{2(2z-y)-y-{\cfrac {1\cdot 3y^{2}}{6(2z-y)-{\cfrac {3\cdot 5y^{2}}{10(2z-y)-\ddots }}}}}}$

which can be simplified by noting that 5/10 = 3/6 = 1/2:

${\sqrt {z}}={\sqrt {x^{2}+y}}=x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+{\cfrac {y}{2x+\ddots }}}}}}}}=x+{\cfrac {2x\cdot y}{2(2z-y)-y-{\cfrac {y^{2}}{2(2z-y)-{\cfrac {y^{2}}{2(2z-y)-\ddots }}}}}}.$

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 (2^1/3 or ³√2 ≈ 1.259921...):

(A) "Standard notation" of x = 1, y = 1, and 2z - y = 3:

${\sqrt[{3}]{2}}=1+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {4}{9+{\cfrac {5}{2+{\cfrac {7}{15+{\cfrac {8}{2+{\cfrac {10}{21+{\cfrac {11}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 1}{9-1-{\cfrac {2\cdot 4}{27-{\cfrac {5\cdot 7}{45-{\cfrac {8\cdot 10}{63-{\cfrac {11\cdot 13}{81-\ddots }}}}}}}}}}.$

(B) Rapid convergence with x = 5, y = 3 and 2z - y = 253:

${\sqrt[{3}]{2}}={\cfrac {5}{4}}+{\cfrac {0.5}{50+{\cfrac {2}{5+{\cfrac {4}{150+{\cfrac {5}{5+{\cfrac {7}{250+{\cfrac {8}{5+{\cfrac {10}{350+{\cfrac {11}{5+\ddots }}}}}}}}}}}}}}}}={\cfrac {5}{4}}+{\cfrac {2.5\cdot 1}{253-1-{\cfrac {2\cdot 4}{759-{\cfrac {5\cdot 7}{1265-{\cfrac {8\cdot 10}{1771-\ddots }}}}}}}}.$

Example 2

Pogson's ratio (100^1/5 or ⁵√100 ≈ 2.511886...), with x = 5, y = 75 and 2z - y = 6325:

${\sqrt[{5}]{100}}={\cfrac {5}{2}}+{\cfrac {3}{250+{\cfrac {12}{5+{\cfrac {18}{750+{\cfrac {27}{5+{\cfrac {33}{1250+{\cfrac {42}{5+\ddots }}}}}}}}}}}}={\cfrac {5}{2}}+{\cfrac {5\cdot 3}{1265-3-{\cfrac {12\cdot 18}{3795-{\cfrac {27\cdot 33}{6325-{\cfrac {42\cdot 48}{8855-\ddots }}}}}}}}.$

Example 3

The twelfth root of two (2^1/12 or ¹²√2 ≈ 1.059463...), using "standard notation":

${\sqrt[{12}]{2}}=1+{\cfrac {1}{12+{\cfrac {11}{2+{\cfrac {13}{36+{\cfrac {23}{2+{\cfrac {25}{60+{\cfrac {35}{2+{\cfrac {37}{84+{\cfrac {47}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 1}{36-1-{\cfrac {11\cdot 13}{108-{\cfrac {23\cdot 25}{180-{\cfrac {35\cdot 37}{252-{\cfrac {47\cdot 49}{324-\ddots }}}}}}}}}}.$

Example 4

Equal temperament's perfect fifth (2^7/12 or ¹²√2⁷ ≈ 1.498307...), with m=7:

(A) "Standard notation":

${\sqrt[{12}]{2^{7}}}=1+{\cfrac {7}{12+{\cfrac {5}{2+{\cfrac {19}{36+{\cfrac {17}{2+{\cfrac {31}{60+{\cfrac {29}{2+{\cfrac {43}{84+{\cfrac {41}{2+\ddots }}}}}}}}}}}}}}}}=1+{\cfrac {2\cdot 7}{36-7-{\cfrac {5\cdot 19}{108-{\cfrac {17\cdot 31}{180-{\cfrac {29\cdot 43}{252-{\cfrac {41\cdot 55}{324-\ddots }}}}}}}}}}.$

(B) Rapid convergence with x = 3, y = –7153, and 2z - y = 2¹⁹+3¹²:

${\sqrt[{12}]{2^{7}}}={\cfrac {1}{2}}{\sqrt[{12}]{3^{12}-7153}}={\cfrac {3}{2}}-{\cfrac {0.5\cdot 7153}{4\cdot 3^{12}-{\cfrac {11\cdot 7153}{6-{\cfrac {13\cdot 7153}{12\cdot 3^{12}-{\cfrac {23\cdot 7153}{6-{\cfrac {25\cdot 7153}{20\cdot 3^{12}-{\cfrac {35\cdot 7153}{6-{\cfrac {37\cdot 7153}{28\cdot 3^{12}-{\cfrac {47\cdot 7153}{6-\ddots }}}}}}}}}}}}}}}}$ ${\sqrt[{12}]{2^{7}}}={\cfrac {3}{2}}-{\cfrac {3\cdot 7153}{12(2^{19}+3^{12})+7153-{\cfrac {11\cdot 13\cdot 7153^{2}}{36(2^{19}+3^{12})-{\cfrac {23\cdot 25\cdot 7153^{2}}{60(2^{19}+3^{12})-{\cfrac {35\cdot 37\cdot 7153^{2}}{84(2^{19}+3^{12})-\ddots }}}}}}}}.$

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 monomials in several real numbers, take the logarithmic form and consider how small it can be. Another reason is to find a possible solution to Hermite's problem.

There have been numerous attempts to construct a generalized theory. Notable efforts in this direction were made by Felix Klein (the Klein polyhedron), Georges Poitou and George Szekeres.

See also

• Gauss's continued fraction
• Padé table
• Solving quadratic equations with continued fractions

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Covers both analytic theory and history).
• Lisa Lorentzen and Haakon Waadeland, Continued Fractions with Applications, North Holland, 1992. ISBN 978-0-444-89265-2. (Covers primarily analytic theory and some arithmetic theory).
• Oskar Perron, Die Lehre von den Kettenbrüchen Band I, II, B.G. Teubner, 1954.
• George Szekeres, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 13, "Multidimensional Continued Fractions", pp. 113–140, 1970.
• H.S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973. ISBN 0-8284-0207-8. (This reprint of the D. Van Nostrand edition of 1948 covers both history and analytic theory.)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Manny Sardina, General Method for Extracting Roots using (Folded) Continued Fractions, Surrey (UK), 2007.

External links

• The first twenty pages of Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, ISBN 0-521-81805-2, contains generalized continued fractions for √2 and the golden mean.
• Template:SloanesRef