C\303\241lculo IILi\303\247\303\243o 15: Converg\303\252ncia de Seq\303\274\303\252ncias

J\303\241 foram estudadas v\303\241rias sequ\303\252ncias anteriormente. Uma delas \303\251 a sequ\303\252ncia de valores que aproximam a solu\303\247\303\243o de uma equa\303\247\303\243o, gerada pelo m\303\251todo de Newton. A seq\303\274\303\252ncia de valores que aproximam a integral de uma fun\303\247\303\243o sobre um intervalo, obtida pela subdivis\303\243o deste intervalo em mais subintervalos, \303\251 outra. Estes s\303\243o exemplos de sequ\303\252ncias potencialmente infinitas. Espera-se que estas sequ\303\252ncias convirjam para a resposta desejada, seja a solu\303\247\303\243o de uma equa\303\247\303\243o ou o valor de uma integral.Formalmente, uma sequ\303\252ncia de n\303\272meros \303\251 definida como uma fun\303\247\303\243o f, cujo dom\303\255nio s\303\243o os inteiros positivos. Os termos da sequ\303\252ncia s\303\243o os valores da fun\303\247\303\243o. Por exemplo, o d\303\251cimo termo da sequ\303\252ncia f \303\251 f(10). A sequ\303\252ncia f converge para um limite L se cada intervalo contendo L cont\303\251m todos os termos da sequ\303\252ncia, a menos de um n\303\272mero finito. Neste caso, escreve-se 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. O comando do Maple limit pode ser usado para calcular v\303\241rios limites de sequ\303\252ncias de forma bastante simples. Por exemplo, limit((1+1/n)^n,n=infinity);

LUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMiIiI=

Os teoremas a seguir resumem muitas das propriedades da converg\303\252ncia de sequ\303\252ncias.Teorema: Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMDZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD \303\251 uma sequ\303\252ncia crescrente (ie, 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 para todo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMTBGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjlRJ25vcm1hbEYn), ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMTJGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD ser\303\241 convergente se existir um limite superior para os termos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMTRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD.Teorema: Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMThGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD converge para L e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMjFGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD converge para M, ent\303\243o 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 converge para LM, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1JJW1zdWJHRiQ2JS1GLDYnUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzI1NSwwLDBdRicvJTBmb250X3N0eWxlX25hbWVHUTJDaGFyYWN0ZXJTdHlsZTMyN0YnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNidRIm5GJ0Y3RjpGPUZAL0ZBUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2L1EiK0YnRjpGPUZILyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZTLyUpc3RyZXRjaHlHRlMvJSpzeW1tZXRyaWNHRlMvJShsYXJnZW9wR0ZTLyUubW92YWJsZWxpbWl0c0dGUy8lJ2FjY2VudEdGUy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlxvLUYyNiUtRiw2J1EiYkYnRjdGOkY9RkBGQ0ZKRjpGPUZIRitGSA== converge para L + M, e 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 converge para 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 . Al\303\251m disso, se 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, ent\303\243o 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 converge para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KS1GIzYlLUkjbWlHRiQ2J1EiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzMzhGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGNy9GPlEnbm9ybWFsRictRiM2JS1GMTYnUSJNRidGNEY3RjpGPUY3RkAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkwvJSliZXZlbGxlZEdRJmZhbHNlRidGN0ZA. Teorema: Se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzNDJGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD \303\251 uma sequ\303\252ncia de n\303\272meros positivos e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzNDRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD uma sequ\303\252ncia que converge para 0, ent\303\243o se 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 para todo n, tem-se que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzNDlGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD converge para 0.Teorema: Se f \303\251 cont\303\255nua em x = L, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlsyNTUsMCwwXUYnLyUwZm9udF9zdHlsZV9uYW1lR1EyQ2hhcmFjdGVyU3R5bGUzNTZGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYnUSJuRidGMkY1RjhGOy9GPFEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZD \303\251 uma sequ\303\252ncia que converge para L, ent\303\243o a seq\303\274\303\252ncia 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 converge para 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.

Seja f uma fun\303\247\303\243o, e seja a um ponto no dom\303\255nio de f. Se cada valor de f encontra-se no dom\303\255nio de f, pode-se gerar a sequ\303\252ncia de itera\303\247\303\265es de a em f da seguite forma: 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, 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 e, de forma geral, 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 para todo inteiro positivo n. Se n \303\251 um inteiro positivo tal que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEibkYnRjdGOi9GO1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0ZCLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZNLyUpc3RyZXRjaHlHRk0vJSpzeW1tZXRyaWNHRk0vJShsYXJnZW9wR0ZNLyUubW92YWJsZWxpbWl0c0dGTS8lJ2FjY2VudEdGTS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmZuRjRGQkYrRkI=, e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtRiw2JVEiaUYnRjdGOi9GO1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RJSZuZTtGJ0ZCLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZNLyUpc3RyZXRjaHlHRk0vJSpzeW1tZXRyaWNHRk0vJShsYXJnZW9wR0ZNLyUubW92YWJsZWxpbWl0c0dGTS8lJ2FjY2VudEdGTS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmZuRjRGQkYrRkI= para todos os inteiros positivos i < n, ent\303\243o a \303\251 um ponto peri\303\263dico de per\303\255odo n para f.Pontos de per\303\255odo igual a um s\303\243o chamados pontos fixos. Pode-se encontrar os pontos fixos de uma fun\303\247\303\243o observando-se os pontos onde os gr\303\241ficos de 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 e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUYsNiVRInhGJ0Y0RjdGPkYrRj4= interceptam-se. Por exemplo, a fun\303\247\303\243o coseno possui um \303\272nico ponto fixo, como se pode observar no gr\303\241fico abaixo.plot({cos(x),x},x=-Pi..Pi);

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Para localizar o ponto fixo com maior precis\303\243o, deve-se utilizar fsolve. fix := fsolve(cos(x)=x,x,0..Pi);

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Um ponto fixo de atra\303\247\303\243o \303\251 um ponto fixo, a, com a propriedade de que, para pontos b pr\303\263ximos de a, limit(b[n],n=infinity) = a;

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onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmduRjRGQ0YrRkM= , e 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 para 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 ...Um ponto fixo de repuls\303\243o \303\251 um ponto fixo, a, com a propriedade de que, para pontos b pr\303\263ximos (mas n\303\243o iguais) a a,limit(b[n],n=infinity) <> a;

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onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JJW1zdWJHRiQ2JS1GLDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRkMvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0ZDLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZOLyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmduRjRGQ0YrRkM= , e 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 para 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, ... .Apresentamos a seguir um procedimento simples para o estudo de pontos peri\303\263dicos e pontos fixos de uma fun\303\247\303\243o.iterate := proc(f,n,x) local a,i,s; a := evalf(x); s := a; for i to n do a := f(a); s := s,a od end: Por exemplo, para verificar se o ponto fixo da fun\303\247\303\243o coseno \303\251 atrator ou n\303\243o, deve-se iterar a fun\303\247\303\243o em um ponto pr\303\263ximo ao ponto fixo. Utilizando o ponto fixo da fun\303\247\303\243o cos, tem-se fix := fsolve(cos(x)=x,x);

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iterate(cos,10,fix-1),fix;

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Neste caso, o ponto fixo \303\251 um ponto atrator. Por outro lado, ao observarmos os pontos fixos de 2x(1-x), f := x-> 2*x*(1-x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCQqJjkkIiIiLCZGN0Y3RjYhIiJGNyIiI0YoRihGKDcjRi4=

fix := fsolve(f(x)=x,x);

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iterate(f,10,fix[1]-.2);

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iterate(f,10,fix[1]+.1);

Ni0kIiIiISIiJCIjPSEiIyQiJV9IISIlJCIpI1I2OyUhIikkIis3REVmWyEjNSQiKyNmUWcqXEYxJCIraW8qKioqXEYxJCIrKysrK11GMUY2RjZGNg==

iterate(f,10,fix[2]+.4);

Ni0kIisrKysrISohIzUkIisrKysrPUYlJCIrKysrX0hGJSQiKysjUjY7JUYlJCIrN0RFZltGJSQiKyNmUWcqXEYlJCIraW8qKioqXEYlJCIrKysrK11GJUYyRjJGMg==

verifica-se que 0 \303\251 um ponto fixo repulsor e .5 e um ponto fixo atrator. Definiremos agora um procedimento para a visualiza\303\247\303\243o de cada itera\303\247\303\243o. Deve ser indicado o dom\303\255nio e o intervalo para que se possa determinar a janela de vis\303\243o. viterate := proc(f,n,start,domain,range) local a, i, s, gra, gpl, fpl, ipl; a := evalf(start); gra := [a,f(a)]; for i to n do a := f(a); gra := gra,[a,a],[a,f(a)]; od: gpl := plot([gra],color=red); fpl := plot(f,domain,color=black); ipl := plot(x->x,domain,color=blue); print(plots[display]([gpl,fpl,ipl],view=[domain,range])); end: viterate(x->2*x*(1-x),10,.8,-1..1 ,-1..1);

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Esta \303\251 uma boa ferramenta visual para o estudo de pontos fixos e pontos peri\303\263dicos de fun\303\247\303\265es.

Exerc\303\255cio: Utilizar iterate ou viterate para verificar se existem mais pontos pr\303\263ximos ao ponto fixo de cos. Verifique se, de fato, este \303\251 um ponto fixo de repuls\303\243o. fx := fsolve(cos(x)=x,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmeEdGKCQiK0s4JjNSKCEjNTcjRi4=

viterate(cos,10,.1,-1..2,-1..1);

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2. Encontrar os pontos pontos peri\303\263dicos de 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, com per\303\255odo 2. (Dica: os pontos peri\303\263dicos de ordem 2 de f devem ser os pontos fixos de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0ZBRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEjQEBGJ0ZB. Classifique-os como repulsores, atratores, ou nenhum dos dois. Exerc\303\255cio: Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== uma sequ\303\252ncia de n\303\272meros positivos que convergem para 0. Imagine uma pessoa inicialmente parada na origem. Em seguida, esta pessoa percorre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== milhas a leste, voltando-se ent\303\243o para o norte e percorrendo mais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== milhas. Percorre mais LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== milhas a oeste, e assim por diante. Forma-se ent\303\243o um ciclo, de acordo com a sequ\303\252ncia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. (1) Onde esta pessoa estar\303\241 no final deste ciclo? (2) Qual a dist\303\242ncia percorrida? Deve-se utilizar as sequ\303\252ncias LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0LUYjNiQtSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnRjQvJS5saW5ldGhpY2tuZXNzR0YzLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkYvJSliZXZlbGxlZEdRJmZhbHNlRidGNA== e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYkLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y0LUYjNiQtSSVtc3VwR0YkNiUtRjE2JFEiMkYnRjQtSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjVRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y0LyUubGluZXRoaWNrbmVzc0dGMy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZPLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjQ=. Solu\303\247\303\243o:Chamaremos o ponto final de [x,y]. Portanto, x = 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 ..., a soma da s\303\251rie alternada com termos \303\255mpares da sequ\303\252ncia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRIm5GJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==, e y \303\251 a soma da s\303\251rie alternada com termos pares da sequ\303\252ncia. Assim, para a sequ\303\252ncia a := n-> 1/n;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoZio2I0kibkdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQ5JCEiIkYoRihGKDcjRi4=

x = sum((-1)^n*a(2*n+1),n=0..infinity);

L0kieEc2IiwkSSNQaUclKnByb3RlY3RlZEcjIiIiIiIl

y = sum((-1)^n*a(2*(n+1)),n=0..infinity);

L0kieUc2IiwkLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyIiIyMiIiJGLA==

e para a sequ\303\252ncia a := n-> 1/2^n;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoZio2I0kibkdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQpIiIjOSQhIiJGKEYoRig3I0Yu

x = sum((-1)^n*a(2*n+1),n=0..infinity);

L0kieEc2IiMiIiMiIiY=

y = sum((-1)^n*a(2*(n+1)),n=0..infinity);

L0kieUc2IiMiIiIiIiY=

(2) A dist\303\242ncia total do caminho percorrido \303\251 a soma de todas as dist\303\242ncias percorridas. Portanto, para a sequ\303\252ncia a := n->1/n;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoZio2I0kibkdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQ5JCEiIkYoRihGKDcjRi4=

'distance' = sum(a(n),n=1..infinity);

L0kpZGlzdGFuY2VHNiJJKWluZmluaXR5RyUqcHJvdGVjdGVkRw==

a dist\303\242ncia \303\251 infinita! Isto \303\251 um tanto inesperado, j\303\241 que cada vez em que ocorre uma mudan\303\247a de dire\303\247\303\243o percorre-se uma dist\303\242ncia menor que da vez anterior. Por outro lado, para a sequ\303\252ncia, a := n->1/2^n;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoZio2I0kibkdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoKiQpIiIjOSQhIiJGKEYoRig3I0Yu

'distance' = sum(a(n),n=1..infinity);

L0kpZGlzdGFuY2VHNiIiIiI=

a dist\303\242ncia percorrida \303\251 de apenas 1 unidade. Exerc\303\255cio: Descrever os caminhos percorridos no problema acima. Apresentamos a seguir um procedimento para tal. Deve-se utiliz\303\241-lo para as sequ\303\252ncias 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, e 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. cycle := proc(a,m) local path,i, dir,x,y,pt,ed; x := evalf(sum((-1)^n*a(2*n+1),n=0..infinity)); y := evalf(sum((-1)^n*a(2*(n+1)),n=0..infinity)); path := [0,0]; dir := 1,0; for i from 1 to m do path := path,[path[i][1]+dir[1]*a(i), path[i][2]+dir[2]*a(i)]; dir := -dir[2],dir[1]; od; pt := plot([path],scaling=constrained,color=red,thickness=2); ed := plot({[x,y]},style=point,symbol=box): plots[display]([pt,ed],title=cat("final em ",convert([x,y],string))); end: Solu\303\247\303\243o:Para a sequ\303\252ncia 1/n, cycle(n->1/n,15);

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cycle(n->1/2^n,15);

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