sexta-feira, 14 de junho de 2013

Frações contínuas II


Seja um fração contínua x = $[a_1;a_2,a_3,\ldots,a_n]$, chamamos de convergentes ou frações parciais a sequência de números racionais c_0, c_1, c_2, \ldots  dados por:
c_0 = a_0, 

c_1 = a_0+\frac{1}{a_1}, 

c_2 = a_0 + \frac{1}{a_1+\frac{1}{a_2}}, \cdots, 

c_n = a_0 + \frac{1}{a_1+\frac{1}{\cdots +\frac{1}{a_n}}}, \cdots ,
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

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 $q_1 = a_1$





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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