Convergent

A convergent of a continued fraction is the rational number obtained by truncating the infinite fraction at a finite depth. If the continued fraction expansion of a real number α is [a₀; a₁, a₂, a₃, ...], then the nth convergent pₙ/qₙ is the rational number represented by the finite continued fraction [a₀; a₁, ..., aₙ].

Convergents are not merely approximations. They are the best approximations: for any rational number p/q with q ≤ qₙ, the inequality |α − p/q| ≥ |α − pₙ/qₙ| holds. This property, known as the best approximation property, makes convergents the canonical sequence of rational approximations to an irrational number. The sequence of convergents alternates between being less than and greater than α, converging inward with a speed controlled by the partial quotients.

The convergents satisfy a remarkable three-term recurrence relation that connects them to the theory of linear recurrences and the arithmetic of modular arithmetic. The numerators and denominators of consecutive convergents are coprime, and the difference between successive convergents has absolute value 1/(qₙqₙ₊₁), revealing a hidden Fibonacci-like structure in the approximation process.