dados por:
,
ou seja, ![c_0 = [a_0], c_1 = [a_0; a_1], c_2 = [a_0; a_1, a_2], \cdots,
c_n = [a_0; a_1, a_2, \cdots, a_n], \cdots](https://upload.wikimedia.org/math/4/d/8/4d8b47d02ebdf53bbc8f5dcd21697c08.png)
Vamos denotar cada $c_i por \dfrac{p_i}{q_i}$
$c_0$ = $a_0$. Logo, $p_0 = a_0$ e $q_0 = 1$
$c_1$ = $a_0 + \dfrac{1}{a_1}$ = $\dfrac{a_0a_1 + 1}{a_1}$. Logo, $p_1 = a_0a_1 + 1$ e $q_1 = a_1$
$c_2$ = $a_0 + \cfrac{1}{ a_1 + \cfrac{1}{a_2}}$ = $\displaystyle \dfrac{a_0a_1a_2 + a_2 + a_0}{a_1a_2 + 1}$ =
$\dfrac{a_2(a_0a_1 + 1) + a_0}{a_1a_2 + 1}$ = $\dfrac{a_2p_1 + p_0}{q_1a_2 + q_0}$
Teorema: Seja $c_i = \dfrac{p_i}{q_i}$ a i-ésima fração parcial da fração contínua $[a_0,a_1,\ldots,a_n]$. Então o numerador $p_i$ e o denominador $q_i$ de $c_i$ satisfazem as seguintes relações: $p_i = a_ip_{i-1} + p_{i-2}$, $q_i = a_iq_{i-1} + q_{i-2}$, para i=2,3,$\ldots$,n sendo que $p_0 = a_0$, $q_0=1$, $p_1 = a_0a_1 + 1$ e $q_1 = a_1$
$\dfrac{a_2(a_0a_1 + 1) + a_0}{a_1a_2 + 1}$ = $\dfrac{a_2p_1 + p_0}{q_1a_2 + q_0}$
Teorema: Seja $c_i = \dfrac{p_i}{q_i}$ a i-ésima fração parcial da fração contínua $[a_0,a_1,\ldots,a_n]$. Então o numerador $p_i$ e o denominador $q_i$ de $c_i$ satisfazem as seguintes relações: $p_i = a_ip_{i-1} + p_{i-2}$, $q_i = a_iq_{i-1} + q_{i-2}$, para i=2,3,$\ldots$,n sendo que $p_0 = a_0$, $q_0=1$, $p_1 = a_0a_1 + 1$ e $q_1 = a_1$
1/1 1.000000000000000000 3/2 1.500000000000000000 7/5 1.399999999999999911 17/12 1.416666666666666741 41/29 1.413793103448275801 99/70 1.414285714285714368 239/169 1.414201183431952558 577/408 1.414215686274509887 1393/985 1.414213197969543145 3363/2378 1.414213624894869570 8119/5741 1.414213551646054778 19601/13860 1.414213564213564256 47321/33461 1.414213562057320406 114243/80782 1.414213562427273363 275807/195025 1.414213562363799470 665857/470832 1.414213562374689870 1607521/1136689 1.414213562372821364 3880899/2744210 1.414213562373141997 9369319/6625109 1.414213562373086930 22619537/15994428 1.414213562373096478 54608393/38613965 1.414213562373094701 131836323/93222358 1.414213562373095145 318281039/225058681 1.414213562373095145 768398401/543339720 1.414213562373095145 1855077841/1311738121 1.414213562373095145 4478554083/3166815962 1.414213562373095145 10812186007/7645370045 1.414213562373095145 26102926097/18457556052 1.414213562373095145 63018038201/44560482149 1.414213562373095145 152139002499/107578520350 1.414213562373095145 367296043199/259717522849 1.414213562373095145 886731088897/627013566048 1.414213562373095145 2140758220993/1513744654945 1.414213562373095145 5168247530883/3654502875938 1.414213562373095145 12477253282759/8822750406821 1.414213562373095145 30122754096401/21300003689580 1.414213562373095145 72722761475561/51422757785981 1.414213562373095145 175568277047523/124145519261542 1.414213562373095145 423859315570607/299713796309065 1.414213562373095145 1023286908188737/723573111879672 1.414213562373095145 2470433131948081/1746860020068409 1.414213562373095145 5964153172084899/4217293152016490 1.414213562373095145 14398739476117879/10181446324101389 1.414213562373095145 34761632124320657/24580185800219268 1.414213562373095145 83922003724759193/59341817924539925 1.414213562373095145 202605639573839043/143263821649299118 1.414213562373095145 489133282872437279/345869461223138161 1.414213562373095145 1180872205318713601/835002744095575440 1.414213562373095145 2850877693509864481/2015874949414289041 1.414213562373095145 6882627592338442563/4866752642924153522 1.414213562373095145
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