Definition:Value of Continued Fraction/Infinite

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

Definition

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

Let $C = (a_n)_{n\geq 0}$ 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 $n \in \N_{\geq 0}$, the $n$th denominator is nonzero
  2. The sequence of convergents $\sequence{C_n}_{n\geq 0}$ converges to $x$.


Also known as

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


Also see