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:

$\dfrac 1 1, \dfrac 3 2, \dfrac 7 5, \dfrac {17} {12}, \dfrac {41} {29}, \dfrac {99} {70}, \dfrac {239} {169}, \dfrac {577} {408}, \dfrac {1393} {985}, \dfrac {3363} {2378}, \ldots$

The numerators form sequence A001333 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

The denominators form sequence A000129 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

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:

$k$	$a_k$	$p_k = a_k p_{k - 1} + p_{k - 2}$	$q_k = a_k q_{k - 1} + q_{k - 2}$	$\dfrac {p_k} {q_k}$	Decimal value
$0$	$1$	$1$	$1$	$\dfrac { 1 } 1$	$1$
$1$	$2$	$1 \times 2 + 1 = 3$	$2$	$\dfrac { 3 } { 2 }$	$1.5$
$2$	$2$	$2 \times 3 + 1 = 7$	$2 \times 2 + 1 = 5$	$\dfrac { 7 } { 5 }$	$1.4$
$3$	$2$	$2 \times 7 + 3 = 17$	$2 \times 5 + 2 = 12$	$\dfrac { 17 } { 12 }$	$1.4166666667$
$4$	$2$	$2 \times 17 + 7 = 41$	$2 \times 12 + 5 = 29$	$\dfrac { 41 } { 29 }$	$1.4137931034$
$5$	$2$	$2 \times 41 + 17 = 99$	$2 \times 29 + 12 = 70$	$\dfrac { 99 } { 70 }$	$1.4142857143$
$6$	$2$	$2 \times 99 + 41 = 239$	$2 \times 70 + 29 = 169$	$\dfrac { 239 } { 169 }$	$1.4142011834$
$7$	$2$	$2 \times 239 + 99 = 577$	$2 \times 169 + 70 = 408$	$\dfrac { 577 } { 408 }$	$1.4142156863$
$8$	$2$	$2 \times 577 + 239 = 1393$	$2 \times 408 + 169 = 985$	$\dfrac { 1393 } { 985 }$	$1.414213198$
$9$	$2$	$2 \times 1393 + 577 = 3363$	$2 \times 985 + 408 = 2378$	$\dfrac { 3363 } { 2378 }$	$1.4142136249$

$\blacksquare$