# Definition:Convergent of Continued Fraction

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

## Definition

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.

### 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 $q_k$:

- $C_k = \dfrac {p_k} {q_k}$

## Even and odd convergents

### Even Convergent

The **even convergents** of $\sqbrk {a_0, a_1, a_2, \ldots}$ are the convergents $C_0, C_2, C_4, \ldots$, that is, those with an even subscript.

### Odd Convergent

The **odd convergents** of $\sqbrk {a_0, a_1, a_2, \ldots}$ are the convergents $C_1, C_3, C_5, \ldots$, that is, those with an odd subscript.

## Also see

- Equivalence of Definitions of Convergent of Continued Fraction
- Properties of Convergents of Continued Fractions
- Irrational Number is Limit of Unique Simple Infinite Continued Fraction, where it is shown that it is possible to talk directly about
**the convergents**to any irrational number $x$.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**convergents**