Continued Fraction Expansion of Irrational Square Root/Examples/2/Convergents

Convergents to Continued Fraction Expansion of $\sqrt 2$
The sequence of convergents to the continued fraction expansion of the square root of $2$ begins:
 * $\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \dfrac {1393} {985}, \dfrac {3363} {2378}, \ldots$

Proof
Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be its continued fraction expansion.

Let $\sequence {p_n}_{n \mathop \ge 0}$ and $\sequence {q_n}_{n \mathop \ge 0}$ be its numerators and denominators.

Then the $n$th convergent is $\dfrac {p_n} {q_n}$.

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 $\sqrt 2$:
 * $\sqrt 2 = \sqbrk {1, \sequence 2}$

Thus the convergents are assembled:

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