Fractions Definitions and Basic Algebraic Properties

The nth approximant (sometimes also called the nth convergent) of a continued fraction is defined hy truncating it at the nth level, that is, by setting the partial numerator equal to 0. The value of an infinite continued fraction is defined as the limit of its sequence of approximants if this limit exists and is finite. 

The truncation at successive steps also defines the nth numerator A and the nth denominator of the continued fraction. The first few approximants are 

The quantities A and B may be obtained by means of the fundamental recurrence formulas 

From the structure of the continued fraction (2.1), we see that its value remains unchanged if a , b , a + (for all n 1) are multiphed by the same constant (other than zero). Operations of this kind do not aflfect the sequence of approximants and are called equivalence transformations. 

By use of equivalence transformations we can always transform a continue fraction K(a /b ) into an equivalent one whose partial numerators equal unity, or into an equivalent one whose partial denominators equal unity. We have, in fact. 

---