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Convergents

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

gives the first n convergents for a number x.

Convergents[x]
gives if possible all convergents leading to the number x.

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 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: