${\displaystyle b_0+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\ddots }}}}$

An infinite continued fraction is defined by the sequences${\displaystyle \{a_i\},\{b_i\}}$, for${\displaystyle i=0,1,2,\dots }$, with${\displaystyle a_0=0}$.

Acontinued fraction is amathematical expression written as afraction whosedenominator contains a sum involving another fraction, which may itself be a simple or a continued fraction.^[1] If thisiteration (repetitive process) terminates with a simple fraction, the result is afinite continued fraction; if it continues indefinitely, the result is aninfinite continued fraction. The special case in which all numerators are equal to one is referred to as asimple (or regular) continued fraction. Anyrational number can be expressed as a finite simple continued fraction, and anyirrational number can be expressed as an infinite simple continued fraction.

Different areas ofmathematics use different terminology andnotation for continued fractions. Innumber theory, the unqualified termcontinued fraction usually refers to simple continued fractions, whereas the general case is referred to asgeneralized continued fractions. Incomplex analysis andnumerical analysis, the general case is usually referred to by the unqualified termcontinued fraction.

The numerators and denominators of continued fractions can besequences${\displaystyle \{a_i\},\{b_i\}}$ ofconstants orfunctions.

A continued fraction is an expression of the form

${\displaystyle x=b_0+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ddots }}}}}$

where thea_n (n > 0) are thepartial numerators, theb_n are thepartial denominators, and the leading termb₀ is called theinteger part of the continued fraction.

The successiveconvergents of the continued fraction are formed by applying thefundamental recurrence formulas:

whereA_n is the numerator andB_n is the denominator, calledcontinuants,^[2][3] of thenth convergent. They are given by thethree-term recurrence relation^[4]

with initial values

If the sequence of convergents{x_n} approaches alimit, 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 denominatorsB_n.

The story of continued fractions begins with theEuclidean algorithm,^[5] a procedure for finding thegreatest common divisor of two natural numbersm andn. That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder repeatedly.

Nearly two thousand years passed beforeBombelli (1579) devised atechnique for approximating the roots of quadratic equations with continued fractions in the mid-sixteenth century. Now the pace of development quickened. Just 24 years later, in 1613,Pietro Cataldi introduced the first formal notation for the generalized continued fraction.^[6] Cataldi represented a continued fraction as

with the dots indicating where the next fraction goes, and each& representing a modern plus sign.

Late in the seventeenth centuryJohn Wallis introduced the term "continued fraction" into mathematical literature.^[7] New techniques for mathematical analysis (Newton's andLeibniz'scalculus) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use.

In 1748Euler published a theorem showing that a particular kind of continued fraction is equivalent to a certain very generalinfinite series.^[8]Euler's continued fraction formula is still the basis of many modern proofs ofconvergence of continued fractions.

In 1761,Johann Heinrich Lambert gave the firstproof thatπ is irrational, by using the following continued fraction fortanx:^[9]

${\displaystyle \tan (x)=\frac{x}{1+\frac{-x^2}{3+\frac{-x^2}{5+\frac{-x^2}{7+\ddots }}}}}$

Continued fractions can also be applied to problems innumber theory, and are especially useful in the study ofDiophantine equations. In the late eighteenth centuryLagrange used continued fractions to construct the general solution ofPell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years.^[10] Lagrange's discovery implies that the canonical continued fraction expansion of thesquare root of every non-square integer is periodic and that, if the period is of lengthp > 1, it contains apalindromic string of lengthp − 1.

In 1813Gauss derived from complex-valuedhypergeometric functions what are now calledGauss's continued fractions.^[11] They can be used to express many elementary functions and some more advanced functions (such as theBessel functions), as continued fractions that are rapidly convergent almost everywhere in the complex plane.

The long continued fraction expression displayed in the introduction is easy for an unfamiliar reader to interpret. However, it takes up a lot of space and can be difficult to typeset. So mathematicians have devised several alternative notations. One convenient way to express a generalized continued fraction sets each nested fraction on the same line, indicating the nesting by dangling plus signs in the denominators:

${\displaystyle x=b_0+\frac{a_1}{b_1+}\frac{a_2}{b_2+}\frac{a_3}{b_3+\cdots }}$

Sometimes the plus signs are typeset to vertically align with the denominators but not under the fraction bars:

${\displaystyle x=b_0+\frac{a_1}{b_1}\genfrac{}{}{0ex}{}{}{+}\frac{a_2}{b_2}\genfrac{}{}{0ex}{}{}{+}\frac{a_3}{b_3}\genfrac{}{}{0ex}{}{}{+\cdots }}$

Pringsheim wrote a generalized continued fraction this way:

${\displaystyle x=b_0+\genfrac{}{}{0ex}{}{}{|}\frac{a_1}{b_1}\genfrac{}{}{0ex}{}{|}{}+\genfrac{}{}{0ex}{}{}{|}\frac{a_2}{b_2}\genfrac{}{}{0ex}{}{|}{}+\genfrac{}{}{0ex}{}{}{|}\frac{a_3}{b_3}\genfrac{}{}{0ex}{}{|}{}+\cdots }$

Carl Friedrich Gauss evoked the more familiarinfinite productΠ when he devised this notation:

${\displaystyle x=b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{b_i}.}$

Here the "K" stands forKettenbruch, 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.

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

If one of the partial numeratorsa_n+1 is zero, the infinite continued fraction

${\displaystyle b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{b_i}}$

is really just a finite continued fraction withn fractional terms, and therefore arational function ofa₁ toa_n andb₀ tob_n+1. Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that alla_i ≠ 0. There is no need to place this restriction on the partial denominatorsb_i.

When thenth convergent of a continued fraction

${\displaystyle x_n=b_0+\underset{i=1}{\stackrel{n}{\mathrm{K}}}\frac{a_i}{b_i}}$

is expressed as a simple fractionx_n =⁠A_n/B_n⁠ we can use thedeterminant formula

${\displaystyle A_{n-1}{B}_n-A_n{B}_{n-1}={\left(-1\right)}^n{a}_1{a}_2\cdots a_n=\prod _{i=1}^n(-a_i)}$

1

to relate the numerators and denominators of successive convergentsx_n andx_{n − 1} to one another. The proof for this can be easily seen byinduction.

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

${\displaystyle b_0+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\frac{a_4}{b_4+\ddots }}}}=b_0+\frac{c_1{a}_1}{c_1{b}_1+\frac{c_1{c}_2{a}_2}{c_2{b}_2+\frac{c_2{c}_3{a}_3}{c_3{b}_3+\frac{c_3{c}_4{a}_4}{c_4{b}_4+\ddots }}}}}$

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

${\displaystyle b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{b_i}=b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{1}{c_i{b}_i}}$

wherec₁ =⁠1/a₁⁠,c₂ =⁠a₁/a₂⁠,c₃ =⁠a₂/a₁a₃⁠, and in generalc_n+1 =⁠1/a_n+1c_n⁠.

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

${\displaystyle b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{b_i}=b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{d_i{a}_i}{1}}$

whered₁ =⁠1/b₁⁠ and otherwised_n+1 =⁠1/b_nb_n+1⁠.

These two special cases of the equivalence transformation are enormously useful when the generalconvergence problem is analyzed.

As mentioned in the introduction, the continued fraction

${\displaystyle x=b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{b_i}}$

converges if the sequence of convergents{x_n} tends to a finite limit. This notion of convergence is very natural, but it is sometimes too restrictive. It is therefore useful to introduce the notion of general convergence of a continued fraction. Roughly speaking, this consists in replacing the${\displaystyle {\mathrm{K}}_{i=n}^{\mathrm{\infty }}\frac{a_i}{b_i}}$ part of the fraction byw_n, instead of by 0, to compute the convergents. The convergents thus obtained are calledmodified convergents. We say that the continued fractionconverges generally if there exists a sequence${\displaystyle \{w_n^{\ast }\}}$ such that the sequence of modified convergents converges for all${\displaystyle \{w_n\}}$ sufficiently distinct from${\displaystyle \{w_n^{\ast }\}}$. The sequence${\displaystyle \{w_n^{\ast }\}}$ is then called anexceptional sequence for the continued fraction. See Chapter 2 ofLorentzen & Waadeland (1992) for a rigorous definition.

There also exists a notion ofabsolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to beabsolutely convergent when the series

${\displaystyle f=\sum _n\left(f_n-f_{n-1}\right),}$

where${\displaystyle f_n={\mathrm{K}}_{i=1}^n\frac{a_i}{b_i}}$ are the convergents of the continued fraction,converges absolutely.^[12] TheŚleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence.

Finally, a continued fraction of one or more complex variables isuniformly convergent in anopen neighborhoodΩ when its convergentsconverge uniformly onΩ; that is, when for everyε > 0 there existsM such that for alln >M, for all${\displaystyle z\in \mathrm{\Omega }}$,

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 pointsp andq, 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 top, and the other converging toq.

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

${\displaystyle x=\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{1}}$

is a continued fraction, then the even partx_even and the odd partx_odd are given by

${\displaystyle x_{\text{even}}=\frac{a_1}{1+a_2-\frac{a_2{a}_3}{1+a_3+a_4-\frac{a_4{a}_5}{1+a_5+a_6-\frac{a_6{a}_7}{1+a_7+a_8-\ddots }}}}}$

and

${\displaystyle x_{\text{odd}}=a_1-\frac{a_1{a}_2}{1+a_2+a_3-\frac{a_3{a}_4}{1+a_4+a_5-\frac{a_5{a}_6}{1+a_6+a_7-\frac{a_7{a}_8}{1+a_8+a_9-\ddots }}}}}$

respectively. More precisely, if the successive convergents of the continued fractionx are{x₁,x₂,x₃, ...}, then the successive convergents ofx_even as written above are{x₂,x₄,x₆, ...}, and the successive convergents ofx_odd are{x₁,x₃,x₅, ...}.^[13]

Ifa₁,a₂,... andb₁,b₂,... are positive integers witha_k ≤b_k for all sufficiently largek, then

${\displaystyle x=b_0+\underset{i=1}{\stackrel{\mathrm{\infty }}{\mathrm{K}}}\frac{a_i}{b_i}}$

converges to an irrational limit.^[14]

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

The continued fraction's successive convergents are then given by

${\displaystyle x_n=\frac{A_n}{B_n}.}$

These recurrence relations are due toJohn Wallis (1616–1703) andLeonhard Euler (1707–1783).^[15]These recurrence relations are simply a different notation for the relations obtained by Pietro Antonio Cataldi (1548-1626).

As an example, consider the simple continued fraction in canonical form that represents thegolden ratioφ:

${\displaystyle \phi =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots }}}}}$

Applying the fundamental recurrence formulas we find that the successive numeratorsA_n are{1, 2, 3, 5, 8, 13, ...} and the successive denominatorsB_n are{1, 1, 2, 3, 5, 8, ...}, theFibonacci 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.

Alinear fractional transformation (LFT) is acomplex function of the form

${\displaystyle w=f(z)=\frac{az+b}{cz+d},}$

wherez is a complex variable, anda,b,c,d are arbitrary complex constants such thatcz +d ≠ 0. An additional restriction thatad ≠bc is customarily imposed, to rule out the cases in whichw =f(z) is a constant. The linear fractional transformation, also known as aMöbius transformation, has many fascinating properties. Four of these are of primary importance in developing the analytic theory of continued fractions.

• Ifc ≠ 0 the LFT has one or twofixed points. This can be seen by considering the equation

${\displaystyle f(z)=z\Rightarrow az+b=c{z}^2+dz\Rightarrow c{z}^2+(d-a)z-b=0,}$

which is clearly aquadratic equation inz. The roots of this equation are the fixed points off(z). If thediscriminant(d −a)² + 4bc is zero the LFT fixes a single point; otherwise it has two fixed points.

• Ifad ≠bc the LFT is aninvertibleconformal mapping of theextended complex plane onto itself. In other words, this LFT has an inverse function

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

such thatf(g(z)) =g(f(z)) =z for every pointz in the extended complex plane, and bothf andg preserve angles and shapes at vanishingly small scales. From the form ofz =g(w) we see thatg is also an LFT.

• Thecomposition of two different LFTs for whichad ≠bc is itself an LFT for whichad ≠bc. In other words, the set of all LFTs for whichad ≠bc is closed under composition of functions. The collection of all such LFTs, together with the "group operation" composition of functions, is known as theautomorphism group of the extended complex plane.
• Ifa = 0 the LFT reduces to

${\displaystyle w=f(z)=\frac{b}{cz+d},}$

which is a very simplemeromorphic function ofz with onesimple pole (at−⁠d/c⁠) and aresidue equal to⁠b/c⁠. (See alsoLaurent series.)

Consider a sequence of simple linear fractional transformations

Here we useτ to represent each simple LFT, and we adopt the conventional circle notation for composition of functions. We also introduce a new symbolΤ_n to represent the composition ofn + 1 transformationsτ_i; that is,

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

and, in general,

${\displaystyle {\mathbf{T}}_{\mathit{n}}(z)={\tau }_0\circ {\tau }_1\circ {\tau }_2\circ \cdots \circ {\tau }_n(z)=b_0+\underset{i=1}{\stackrel{n}{\mathrm{K}}}\frac{a_i}{b_i}}$

where the last partial denominator in the finite continued fractionK is understood to beb_n +z. And, sinceb_n + 0 =b_n, the image of the pointz = 0 under the iterated LFTΤ_n is indeed the value of the finite continued fraction withn partial numerators:

${\displaystyle {\mathbf{T}}_{\mathit{n}}(0)={\mathbf{T}}_{\mathit{n}\mathbf{+}\mathbf{1}}(\mathrm{\infty })=b_0+\underset{i=1}{\stackrel{n}{\mathrm{K}}}\frac{a_i}{b_i}.}$

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

The relationship

${\displaystyle x_n=b_0+\underset{i=1}{\stackrel{n}{\mathrm{K}}}\frac{a_i}{b_i}=\frac{A_n}{B_n}={\mathbf{T}}_{\mathit{n}}(0)={\mathbf{T}}_{\mathit{n}\mathbf{+}\mathbf{1}}(\mathrm{\infty })}$

can be understood by rewritingΤ_n(z) andΤ_n+1(z) in terms of thefundamental recurrence formulas:

In the first of these equations the ratio tends toward⁠A_n/B_n⁠ asz tends toward zero. In the second, the ratio tends toward⁠A_n/B_n⁠ asz tends to infinity. This leads us to our first geometric interpretation. If the continued fraction converges, the successive convergents⁠A_n/B_n⁠ are eventuallyarbitrarily close together. Since the linear fractional transformationΤ_n(z) is acontinuous mapping, there must be a neighborhood ofz = 0 that is mapped into an arbitrarily small neighborhood ofΤ_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 ofΤ_n(∞) =⁠A_n−1/B_n−1⁠. So if the continued fraction converges the transformationΤ_n(z) maps both very smallz and very largez into an arbitrarily small neighborhood ofx, the value of the continued fraction, asn gets larger and larger.

For intermediate values ofz, since the successive convergents are getting closer together we must have

${\displaystyle \frac{A_{n-1}}{B_{n-1}}\approx \frac{A_n}{B_n}\Rightarrow \frac{A_{n-1}}{A_n}\approx \frac{B_{n-1}}{B_n}=k}$

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

${\displaystyle {\mathbf{T}}_{\mathit{n}}(z)=\frac{z{A}_{n-1}+A_n}{z{B}_{n-1}+B_n}=\frac{A_n}{B_n}\left(\frac{z\frac{A_{n-1}}{A_n}+1}{z\frac{B_{n-1}}{B_n}+1}\right)\approx \frac{A_n}{B_n}\left(\frac{zk+1}{zk+1}\right)=\frac{A_n}{B_n}}$

so that even the intermediate values ofz (except whenz ≈ −k⁻¹) are mapped into an arbitrarily small neighborhood ofx, the value of the continued fraction, asn gets larger and larger. Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point.^[16]

Notice that the sequence{Τ_n} lies within theautomorphism group of the extended complex plane, since eachΤ_n is a linear fractional transformation for whichab ≠cd. And every member of that automorphism group maps the extended complex plane into itself: not one of theΤ_n can possibly map the plane into a single point. Yet in the limit the sequence{Τ_n} defines an infinite continued fraction which (if it converges) represents a single point in the complex plane.

When an infinite continued fraction converges, the corresponding sequence{Τ_n} of LFTs "focuses" the plane in the direction ofx, 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 ofx, 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.^[17]

For divergent continued fractions, we can 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 andx_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 asubgroup 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 denominatorsB_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.^[18]
3. The sequence{Τ_n} may produce no more than a finite number of zero denominatorsB_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

${\displaystyle x=1+\frac{z}{1+\frac{z}{1+\frac{z}{1+\frac{z}{1+\ddots }}}}}$

wherez is any real number such thatz < −⁠1/4⁠.^[19]

Euler proved the following identity:^[8]

${\displaystyle a_0+a_0{a}_1+a_0{a}_1{a}_2+\cdots +a_0{a}_1{a}_2\cdots a_n=\frac{a_0}{1-\frac{a_1}{1+a_1-\frac{a_2}{1+a_2-\cdots \frac{a_n}{1+a_n}}}}.}$

From this many other results can be derived, such as

${\displaystyle \frac{1}{u_1}+\frac{1}{u_2}+\frac{1}{u_3}+\cdots +\frac{1}{u_n}=\frac{1}{u_1-\frac{u_1^2}{u_1+u_2-\frac{u_2^2}{u_2+u_3-\cdots \frac{u_{n-1}^2}{u_{n-1}+u_n}}}},}$

and

${\displaystyle \frac{1}{a_0}+\frac{x}{a_0{a}_1}+\frac{x^2}{a_0{a}_1{a}_2}+\cdots +\frac{x^n}{a_0{a}_1{a}_2\dots a_n}=\frac{1}{a_0-\frac{a_{0}x}{a_1+x-\frac{a_{1}x}{a_2+x-\cdots \frac{a_{n-1}x}{a_n+x}}}}.}$

Euler's formula connecting continued fractions and series is the motivation for thefundamental inequalities^{[link or clarification needed]}, and also the basis of elementary approaches to theconvergence problem.

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

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

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

Here are additional generalized continued fractions:

${\displaystyle \arctan \frac{x}{y}=\frac{xy}{1y^2+\frac{(1xy{)}^2}{3y^2-1x^2+\frac{(3xy{)}^2}{5y^2-3x^2+\frac{(5xy{)}^2}{7y^2-5x^2+\ddots }}}}=\frac{x}{1y+\frac{(1x{)}^2}{3y+\frac{(2x{)}^2}{5y+\frac{(3x{)}^2}{7y+\ddots }}}}}$

${\displaystyle e^{\frac{x}{y}}=1+\frac{2x}{2y-x+\frac{x^2}{6y+\frac{x^2}{10y+\frac{x^2}{14y+\frac{x^2}{18y+\ddots }}}}}\Rightarrow e^2=7+\frac{2}{5+\frac{1}{7+\frac{1}{9+\frac{1}{11+\ddots }}}}}$

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

This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s.^[20]

Example: thenatural logarithm of 2 (=[0; 1, 2, 3, 1, 5,⁠2/3⁠, 7,⁠1/2⁠, 9,⁠2/5⁠,..., 2k − 1,⁠2/k⁠,...] ≈ 0.693147...):^[21]

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

Here are three ofπ's best-known generalized continued fractions, the first and third of which are derived from their respectivearctangent formulas above by settingx =y = 1 and multiplying by 4. TheLeibniz formula forπ:

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

converges too slowly, requiring roughly3 × 10ⁿ terms to achieven correct decimal places. The series derived byNilakantha Somayaji:

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

is a much more obvious^[why?] expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six. Both convergesublinearly toπ. On the other hand:

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

convergeslinearly toπ, adding at least three digits of precision per four terms, a pace slightly faster than thearcsine formula forπ:

${\displaystyle \pi =6{\sin }^{-1}\left(\frac{1}{2}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{3\cdot (\genfrac{}{}{0ex}{}{2n}{n})}{{16}^n(2n+1)}=\frac{3}{{16}^0\cdot 1}+\frac{6}{{16}^1\cdot 3}+\frac{18}{{16}^2\cdot 5}+\frac{60}{{16}^3\cdot 7}+\cdots }$

which adds at least three decimal digits per five terms.^[22]

• Note: this continued fraction'srate of convergenceμ tends to3 −√8 ≈ 0.1715729, hence⁠1/μ⁠ tends to3 +√8 ≈ 5.828427, whosecommon logarithm is0.7655... ≈⁠13/17⁠ >⁠3/4⁠. The same⁠1/μ⁠ = 3 +√8 (thesilver ratio squared) also is observed in theunfolded general continued fractions of both thenatural logarithm of 2 and thenth root of 2 (which works for any integern > 1) if calculated using2 = 1 + 1. For thefolded general continued fractions of both expressions, the rate convergenceμ = (3 −√8)² = 17 −√288 ≈ 0.02943725, hence⁠1/μ⁠ = (3 +√8)² = 17 +√288 ≈ 33.97056, whose common logarithm is1.531... ≈⁠26/17⁠ >⁠3/2⁠, thus adding at least three digits per two terms. This is because the folded GCF folds each pair of fractions from the unfolded GCF into one fraction, thus doubling the convergence pace. The Manny Sardina reference further explains "folded" continued fractions.
• Note: Using the continued fraction forarctan⁠x/y⁠ cited above with the best-knownMachin-like formula provides an even more rapidly, although still linearly, converging expression:

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

Thenth root of any positive numberz^m can be expressed by restatingz =xⁿ +y, resulting in

${\displaystyle \sqrt[n]{z^m}=\sqrt[n]{{\left(x^n+y\right)}^m}=x^m+\frac{my}{n{x}^{n-m}+\frac{(n-m)y}{2x^m+\frac{(n+m)y}{3n{x}^{n-m}+\frac{(2n-m)y}{2x^m+\frac{(2n+m)y}{5n{x}^{n-m}+\frac{(3n-m)y}{2x^m+\ddots }}}}}}}$

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

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

Thesquare root ofz is a special case withm = 1 andn = 2:

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

which can be simplified by noting that⁠5/10⁠ =⁠3/6⁠ =⁠1/2⁠:

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