Definition:Value of Continued Fraction/Infinite

Not to be confused with Definition:Convergent of Continued Fraction.

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 $x \in F$ if and only if 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