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Mathematica > Continued Fractions & Rational Approximations >

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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: