Equivalence of Definitions of Convergent of Continued Fraction
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Theorem
Let $F$ be a field, such as the field of real numbers.
Let $n \in \N \cup \set \infty$ be an extended natural number.
Let $C = \sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $F$ of length $n$.
Let $k \le n$ be a natural number.
The following definitions of the concept of Convergent of Continued Fraction are equivalent:
Definition 1
The $k$th convergent $C_k$ of $C$ is the value of the finite continued fraction:
- $C_k = \sqbrk {a_0, a_1, \ldots, a_k}$
Definition 2
The $k$th convergent $C_k$ of $C$ is the quotient of the $k$th numerator $p_k$ by the $k$th denominator $q_k$:
- $C_k = \dfrac {p_k} {q_k}$
Proof
This follows immediately from Value of Finite Continued Fraction equals Numerator Divided by Denominator.
$\blacksquare$