Simple Continued Fraction

A simple continued fraction is a special case of a generalized continued fraction for which the partial numerators are equal to unity, i.e., for all , 2, .... A simple continued fraction is therefore an expression of the form

(1)

When used without qualification, the term "continued fraction" is often used to mean "simple continued fraction" or, more specifically, regular (i.e., a simple continued fraction whose initial term is an integer and whose partial denominators , , ... are positive integers; Rockett and Szüsz 1992, p. 3). Care must therefore be taken to identify the intended meaning based on the context in which such terminology is encountered.

A simple continued fraction can be written in a compact abbreviated notation as

(2)

or

(3)

where may be finite (for a finite continued fraction) or (for an infinite continued fraction). In contexts where only simple continued fractions are considered, the partial denominators are often denoted instead of (e.g., Rockett and Szüsz 1992, p. 3), a practice which unfortunately conflicts with the common notation for generalized continued fractions in which denotes a partial numerator.

Further care is needed when encountering bracket notation for simple continued fractions since some authors replace the semicolon with a normal comma and begin indexing the terms at instead of , writing instead of or , causing ambiguity in the meaning of the initial term and resulting in the parity of certain fundamental results in continued fraction theory to be reversed. To complicate matters a bit further, Gaussian brackets use the notation to denote a different (but closely related) combination of partial denominators.

The partial quotients through of the regular continued fraction of a nonnegative real number can be computed in the Wolfram Language using the command ContinuedFraction[x, n]. For , this command returns , whereas the regular continued fraction is . For a positive integer , the th convergent of a simple continued fraction with partial denominators is obtained by adding to ContinuedFractionK[b[k], k, 1, n]. Setting to Infinity gives the infinite simple continued fraction when it converges.