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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.
  • 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.
Generate the first 10 convergents to the Golden Ratio:
Generate convergents from the continued fraction terms for GoldenRatio:
Generate the first 10 convergents to the Golden Ratio:
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Generate convergents from the continued fraction terms for GoldenRatio:
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Quadratic irrationals have periodic continued fractions:
Give all convergents for a rational number:
Convergents continues until the precision of the input is reached:
The convergents of a number converge to it while alternating sides:
The results from Rationalize are not always among the list of convergents:
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