Continued Fraction Expansion of Irrational Square Root/Examples/13/Convergents

Convergents to Continued Fraction Expansion of $\sqrt {13}$
The sequence of convergents to the continued fraction expansion of the square root of $13$ begins:
 * $\dfrac 3 1, \dfrac 4 1, \dfrac 7 2, \dfrac {11} 3, \dfrac {18} 5, \dfrac {119} {33}, \dfrac {137} {38}, \dfrac {256} {71}, \dfrac {393} {109}, \dfrac {649} {180}, \ldots$

Proof
Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction expansion.

Let $\dfrac {p_0} {q_0}, \dfrac {p_1} {q_1}, \dfrac {p_2} {q_2}, \ldots$ be the sequence of convergents to $\left[{a_0, a_1, a_2, \ldots}\right]$

Then from Value of Simple Continued Fraction:


 * $p_k = \begin{cases} a_0 & : k = 0 \\

a_0 a_1 + 1 & : k = 1 \\ a_k p_{k - 1} + p_{k - 2} & : k > 1\end{cases}$


 * $q_k = \begin{cases} 1 & : k = 0 \\

a_1 & : k = 1 \\ a_k q_{k - 1} + q_{k - 2} & : k > 1\end{cases}$

From Continued Fraction Expansion of $\sqrt {13}$:
 * $\sqrt {13} = \left[{3, \left \langle{1, 1, 1, 1, 6}\right \rangle}\right]$

Thus the convergents are assembled: