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A 'convergent' is one of a sequence of values obtained by evaluating successive truncations of a continued fraction. The ''n''th convergent is also known as the ''n''th 'approximant' of a continued fraction.

Contents

Representation of real numbers

Convergents and convergence

See also

Representation of real numbers

Every real number can be expressed as a regular continued fraction in canonical form. Each convergent of that continued fraction is in a sense the best possible rational approximation to that real number, for a given number of digits. Such a convergent is usually about as accurate as a finite decimal expansion having as many digits as the total number of digits in the ''n''th numerator and ''n''th denominator. For example, the third convergent 333/106 for π (Pi) is roughly 3.1415094, which is not quite as accurate as the 6-digit 3.14159; the fourth convergent 355/113 = 3.14159292 is more accurate than the 6-digit decimal.
By the determinant formula it appears that the successive convergents ''A_k''/''B_k'' of a 'regular' continued fraction are connected by the formula
:$$
A_{k-1}B_K - A_kB_{k-1} = (-1)^k ,

This implies, in particular, that the greatest common divisor (''A_k'', ''B_k'') = 1; in other words, each convergent of a regular continued fraction, as given by the fundamental recurrence formulas, is automatically expressed in lowest terms.
More detailed properties of best rational approximations and convergents of π are discussed in the continued fraction article.

Convergents and convergence

In mathematical analysis a continued fraction is usually written as
:$x\; =\; b\_0\; +\; cfrac\{a\_1\}\{b\_1\; +\; cfrac\{a\_2\}\{b\_2\; +\; cfrac\{a\_3\}\{b\_3\; +\; cfrac\{a\_4\}\; \{ddots,\}\}\}\}$
where the ''a''_''i'' and the ''b''_''i'' are numbers. The ''a''_''i'' are the partial numerators of the continued fraction ''x''. The ''b''_''i'' are the partial denominators, and the ratios ''a''_''i'' / ''b''_''i'' are the partial quotients. The convergents of this fraction can be computed by using the fundamental recurrence formulas.
An infinite continued fraction converges if the sequence of convergents approaches a limit. If the sequence of convergents does not approach a limit, the continued fraction is divergent.
Because of the way the partial denominators and partial numerators interact with each other as the successive convergents are calculated, the convergence problem for continued fractions is inherently more difficult than it is for infinite series.

See also

★ Complete quotient