Definition:Convergent Continued Fraction

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


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.