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.
The story of continued fractions begins with the Euclidean algorithm,[4] 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 repeatedly.
Nearly two thousand years passed before Bombelli (1579) devised a technique for approximating the roots of quadratic equations with continued fractions in the mid-sixteenth century.
Just 24 years later, in 1613, Pietro Cataldi introduced the first formal notation for the generalized continued fraction.
Late in the seventeenth century John Wallis introduced the term "continued fraction" into mathematical literature.
[6] New techniques for mathematical analysis (Newton's and Leibniz's calculus) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use.
In 1748 Euler published a theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series.
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.
The long continued fraction expression displayed in the introduction is easy for an unfamiliar reader to interpret.
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.
Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all ai ≠ 0.
Base case Inductive step If {ci} = {c1, c2, c3, ...} is any infinite sequence of non-zero complex numbers we can prove, by induction, that 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.
First, if none of the ai are zero, a sequence {ci} can be chosen to make each partial numerator a 1: where c1 = 1/a1, c2 = a1/a2, c3 = a2/a1a3, and in general cn+1 = 1/an+1cn.
These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed.
[12] If a1, a2,... and b1, b2,... are positive integers with ak ≤ bk for all sufficiently large k, then converges to an irrational limit.
[13] The partial numerators and denominators of the fraction's successive convergents are related by the fundamental recurrence formulas: The continued fraction's successive convergents are then given by 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: Applying the fundamental recurrence formulas we find that the successive numerators An are {1, 2, 3, 5, 8, 13, ...} and the successive denominators Bn 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.
An additional restriction that ad ≠ bc is customarily imposed, to rule out the cases in which w = f(z) is a constant.
By direct substitution from the first set of expressions into the second we see that and, in general, where the last partial denominator in the finite continued fraction K is understood to be bn + z.
The relationship can be understood by rewriting Τn(z) and Τn+1(z) in terms of the fundamental recurrence formulas: In the first of these equations the ratio tends toward An/Bn as z tends toward zero.
Intuitively, it is almost as if the convergent continued fraction maps the entire extended complex plane into a single point.
Yet in the limit the sequence {Τn} defines an infinite continued fraction which (if it converges) represents a single point in the complex plane.
Here are additional generalized continued fractions: This last is based on an algorithm derived by Aleksei Nikolaevich Khovansky in the 1970s.
[19] 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...):[20] Here are three of π'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 4.
The Leibniz formula for π: converges too slowly, requiring roughly 3 × 10n terms to achieve n correct decimal places.
The series derived by Nilakantha Somayaji: is a much more obvious expression but still converges quite slowly, requiring nearly 50 terms for five decimals and nearly 120 for six.
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.
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.