# 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.

$\blacksquare$