Convergent (continued Fraction) - Convergents and Convergence

Convergents and Convergence

In mathematical analysis a continued fraction is usually written as

where the a_i and the b_i are integers. The a_i are the partial numerators of the continued fraction x. The b_i are the partial denominators, and the ratios a_i / b_i are the partial quotients. The convergents of this fraction can be computed by using the fundamental recurrence formulas.

An infinite continued fraction converges if the sequence of convergents approaches a limit. If the sequence of convergents does not approach a limit, the continued fraction is divergent.

Because of the way the partial denominators and partial numerators interact with each other as the successive convergents are calculated, the convergence problem for continued fractions is inherently more difficult than it is for infinite series. The Śleszyński–Pringsheim theorem provides one sufficient condition for convergence.