Properties of Convergents of Continued Fractions

Theorem

Let $n \in \N \cup \set \infty$ be an extended natural number.

Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $\R$ of length $n$.

Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators.

Let $C_0, C_1, C_2, \ldots$ be the convergents of $\sqbrk {a_0, a_1, a_2, \ldots}$.

Then the following results apply:

Then for $k \ge 1$:

$p_k q_{k - 1} - p_{k - 1} q_k = \paren {-1}^{k + 1}$

That is:

$C_k - C_{k - 1} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 1} } {q_{k - 1} } = \dfrac {\paren {-1}^{k + 1} } {q_k q_{k - 1} }$

Difference between Adjacent Convergents But One

For $k \ge 2$:

$p_k q_{k - 2} - p_{k - 2} q_k = \paren {-1}^k a_k$

That is:

$C_k - C_{k-2} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 2} } {q_{k - 2} } = \dfrac {\paren {-1}^k a_k} {q_k q_{k - 2} }$

Simple continued fractions

Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a simple continued fraction in $\R$ of length $n$.

Convergents are Rationals in Canonical Form

For all $k \ge 1$, $\dfrac {p_k} {q_k}$ is in canonical form:

$p_k$ and $q_k$ are coprime
$q_k > 0$.

Even convergents are strictly increasing

The even convergents satisfy $C_0 < C_2 < C_4 \cdots$.

Odd convergents are strictly decreasing

The odd convergents satisfy $C_1 > C_3 > C_5 > \cdots$

Even convergents are smaller than odd convergents

Every even convergent is strictly smaller than every odd convergent.