Definition:Value of Continued Fraction/Infinite

Definition
Let $\struct {F, \norm {\,\cdot\,} }$ be a valued field.

Let $C = \sequence {a_n}_{n \mathop \ge 0}$ be a infinite continued fraction in $F$.

Then $C$ converges to its value $x \in F$ the following hold:
 * 1) For all natural numbers $n \in \N_{\ge 0}$, the $n$th denominator is nonzero
 * 2) The sequence of convergents $\sequence {C_n}_{n \mathop \ge 0}$ converges to $x$.

Also known as
The value of an infinite continued fraction is also known as its limit.

Also see

 * Irrational Number is Limit of Unique Simple Infinite Continued Fraction