Definition:Convergent Continued Fraction

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:

$(1): \quad$ For all natural numbers $n \in \N_{\ge 0}$, the $n$th denominator is nonzero

$(2): \quad$ The sequence of convergents $\sequence {C_n}_{n \mathop \ge 0}$ converges to $x$.

Also known as

A convergent continued fraction can also be referred to as a convergent fraction.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): convergent fraction
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convergent fraction