Continued Fraction Expansion of Irrational Square Root/Examples/61/Convergents
== Convergents to Continued Fraction Expansion of $\sqrt {61}$ ==  The sequence of convergents to the continued fraction expansion of the square root of $61$ begins: :$\dfrac 7 1, \dfrac {8} 1, \dfrac {39} 5, \dfrac {125} {16}, \dfrac {164} {21}, \dfrac {453} {58}, \dfrac {1070} {137}, \dfrac {1523} {195}, \dfrac {5639} {722}, \dfrac {24079} {3083}, \ldots$  {{OEIS-Numerators|A041106}} {{OEIS-Denominators|A041107}} == Proof == Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be its continued fraction expansion. Let $\sequence {p_n}_{n \ge \mathop 0}$ and $\sequence {q_n}_{n \ge \mathop 0}$ 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 $\sqrt {61}$: :$\sqrt {61} = \sqbrk {7, \sequence {1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14} }$ Thus the convergents are assembled: {{ConvergentCalculator-Start | a0 = 7 | a1 = 1}} {{ConvergentCalculator | k =  2 | ak =  4 | pk-1 =          8 | pk-2 =          7 | qk-1 =         1 | qk-2 =         1}} {{ConvergentCalculator | k =  3 | ak =  3 | pk-1 =         39 | pk-2 =          8 | qk-1 =         5 | qk-2 =         1}} {{ConvergentCalculator | k =  4 | ak =  1 | pk-1 =        125 | pk-2 =         39 | qk-1 =        16 | qk-2 =         5}} {{ConvergentCalculator | k =  5 | ak =  2 | pk-1 =        164 | pk-2 =        125 | qk-1 =        21 | qk-2 =        16}} {{ConvergentCalculator | k =  6 | ak =  2 | pk-1 =        453 | pk-2 =        164 | qk-1 =        58 | qk-2 =        21}} {{ConvergentCalculator | k =  7 | ak =  1 | pk-1 =       1070 | pk-2 =        453 | qk-1 =       137 | qk-2 =        58}} {{ConvergentCalculator | k =  8 | ak =  3 | pk-1 =       1523 | pk-2 =       1070 | qk-1 =       195 | qk-2 =       137}} {{ConvergentCalculator | k =  9 | ak =  4 | pk-1 =       5639 | pk-2 =       1523 | qk-1 =       722 | qk-2 =       195}} {{ConvergentCalculator | k = 10 | ak =  1 | pk-1 =      24079 | pk-2 =       5639 | qk-1 =      3083 | qk-2 =       722}} {{ConvergentCalculator | k = 11 | ak = 14 | pk-1 =      29718 | pk-2 =      24079 | qk-1 =      3805 | qk-2 =      3083}} {{ConvergentCalculator | k = 12 | ak =  1 | pk-1 =     440131 | pk-2 =      29718 | qk-1 =     56353 | qk-2 =      3805}} {{ConvergentCalculator | k = 13 | ak =  4 | pk-1 =     469849 | pk-2 =     440131 | qk-1 =     60158 | qk-2 =     56353}} {{ConvergentCalculator | k = 14 | ak =  3 | pk-1 =    2319527 | pk-2 =     469849 | qk-1 =    296985 | qk-2 =     60158}} {{ConvergentCalculator | k = 15 | ak =  1 | pk-1 =    7428430 | pk-2 =    2319527 | qk-1 =    951113 | qk-2 =    296985}} {{ConvergentCalculator | k = 16 | ak =  2 | pk-1 =    9747957 | pk-2 =    7428430 | qk-1 =   1248098 | qk-2 =    951113}} {{ConvergentCalculator | k = 17 | ak =  2 | pk-1 =   26924344 | pk-2 =    9747957 | qk-1 =   3447309 | qk-2 =   1248098}} {{ConvergentCalculator | k = 18 | ak =  1 | pk-1 =   63596645 | pk-2 =   26924344 | qk-1 =   8142716 | qk-2 =   3447309}} {{ConvergentCalculator | k = 19 | ak =  3 | pk-1 =   90520989 | pk-2 =   63596645 | qk-1 =  11590025 | qk-2 =   8142716}} {{ConvergentCalculator | k = 20 | ak =  4 | pk-1 =  335159612 | pk-2 =   90520989 | qk-1 =  42912791 | qk-2 =  11590025}} {{ConvergentCalculator | k = 21 | ak =  1 | pk-1 = 1431159437 | pk-2 =  335159612 | qk-1 = 183241189 | qk-2 =  42912791}} |}