Continued Fraction Expansion of Irrational Square Root/Examples/29/Convergents

Convergents to Continued Fraction Expansion of $\sqrt {29}$
The sequence of convergents to the continued fraction expansion of the square root of $29$ begins:
 * $\dfrac 5 1, \dfrac {11} 2, \dfrac {16} 3, \dfrac {27} 5, \dfrac {70} {13}, \dfrac {727} {135}, \dfrac {1524} {283}, \dfrac {2251} {418}, \dfrac {3775} {701}, \dfrac {9801} {1820}, \ldots$

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

Let $(p_n)_{n\geq 0}$ and $(q_n)_{n\geq 0}$ be its numerators and denominators.

Then the $n$th convergent is $p_n/q_n$.

By definition:


 * $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 $\sqrt {29}$:
 * $\sqrt {29} = \left[{5, \left \langle{2, 1, 1, 2, 10}\right \rangle}\right]$

Thus the convergents are assembled:


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