Properties of Convergents of Continued Fractions

Theorem
Let $\left[{a_1, a_2, a_3, \ldots}\right]$ be a simple continued fraction.

Let $p_1, p_2, p_3, \ldots$ and $q_1, q_2, q_3, \ldots$ be its numerators and denominators.

Let $C_1, C_2, C_3, \ldots$ be the convergents of $\left[{a_1, a_2, a_3, \ldots}\right]$.

Then the following results apply: