Convergents

Convergents[list]

gives a list of the convergents corresponding to the continued fraction terms list.

Convergents[x,n]

gives the first n convergents for a number x.

Convergents[x]

gives if possible all convergents leading to the number x.

Details

  • The convergents of the continued fraction a1+1/(a2+1/(a3+…)) are the rationals a1,a1+1/a2,a1+1/(a2+1/a3),….
  • For exact numbers, Convergents[x] can be used if x is rational or a quadratic irrational.
  • If x is a quadratic irrational or a representation of a quadratic irrational as a continued fraction, the final list element returned by Convergents[x] is the quadratic irrational represented by x.
  • For inexact numbers, Convergents[x] generates a list of all convergents that can be obtained given the precision of x.
  • Convergents[x,n] will return n convergents if possible. If x represents a rational or an inexact number, fewer than n terms may be returned.

Examples

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Basic Examples  (3)

Generate the first 10 convergents to the golden ratio:

Generate convergents from the continued fraction terms for GoldenRatio:

Scope  (4)

Quadratic irrationals have periodic continued fractions:

Give all convergents for a rational number:

Convergents continues until the precision of the input is reached:

Properties & Relations  (2)

The convergents of a number converge to it while alternating sides:

The results from Rationalize are not always among the list of convergents:

Wolfram Research (2007), Convergents, Wolfram Language function, https://reference.wolfram.com/language/ref/Convergents.html.

Text

Wolfram Research (2007), Convergents, Wolfram Language function, https://reference.wolfram.com/language/ref/Convergents.html.

CMS

Wolfram Language. 2007. "Convergents." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Convergents.html.

APA

Wolfram Language. (2007). Convergents. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Convergents.html

BibTeX

@misc{reference.wolfram_2024_convergents, author="Wolfram Research", title="{Convergents}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Convergents.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_convergents, organization={Wolfram Research}, title={Convergents}, year={2007}, url={https://reference.wolfram.com/language/ref/Convergents.html}, note=[Accessed: 22-November-2024 ]}