C\303\241lculo II

Li\303\247\303\243o 17: Converg\303\252ncia de S\303\251ries Infinitas

Defini\303\247\303\243o de s\303\251ries infinitas. Defini\303\247\303\243o da soma de uma s\303\251rie infinita. Teste da integral, da raiz e da raz\303\243o para converg\303\252ncia. Raio de converg\303\252ncia. Estimativa do erro de truncamento. Aplica\303\247\303\243o de s\303\251ries em problemas de integra\303\247\303\243o.

Seja LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkobWZlbmNlZEdGJDYmLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjc2JVEibkYnRjpGPS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y+USdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRictRiM2JS1GNzYjUSFGJy1GIzYlRkItSSNtb0dGJDYtUSI9RidGSC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGZ24vJSlzdHJldGNoeUdGZ24vJSpzeW1tZXRyaWNHRmduLyUobGFyZ2VvcEdGZ24vJS5tb3ZhYmxlbGltaXRzR0Znbi8lJ2FjY2VudEdGZ24vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0Zmby1JI21uR0YkNiRGR0ZIRlItRi82JC1GIzYlRkJGVy1GNzYlUSgmaW5maW47RicvRjtGZ25GSEZILyUxc3VwZXJzY3JpcHRzaGlmdEdGR0ZF uma sequ\303\252ncia de termos. Ent\303\243o, dada a s\303\251rie infinita, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2KS1JJW1zdWJHRiQ2JS1GLDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtSSNtbkdGJDYkUSIwRicvRjtRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHRkItSSNtb0dGJDYtUSIrRidGQy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zmbi1GMjYlRjQtRiM2Iy1GQDYkUSIxRidGQ0ZFRkctRjI2JUY0LUYjNiMtRkA2JFEiMkYnRkNGRUZHLUYyNiVGNC1GIzYjLUZANiRRIjNGJ0ZDRkVGKw==+... tem-se a sequ\303\252ncia LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRic= onde 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 . O soma da s\303\251rie \303\251, por defini\303\247\303\243o, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JJ211bmRlckdGJDYlLUkjbW9HRiQ2L1EkbGltRicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdGQy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkMvJS5tb3ZhYmxlbGltaXRzR1EldHJ1ZUYnLyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHRk4vRjxRJ2l0YWxpY0YnLUY1Ni1RJyZyYXJyO0YnRjtGQUZERkZGSEZKL0ZNRkNGT0ZRL0ZVRlMtRiw2JVEoJmluZmluO0YnL0ZnbkZDRjsvJSxhY2NlbnR1bmRlckdGQ0YrLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHUSYwLjRlbUYnLyUmZGVwdGhHRmpvLyUqbGluZWJyZWFrR1ElYXV0b0YnLUklbXN1YkdGJDYlLUYsNiVRIlNGJ0ZmbkZobi1GIzYjRlkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Yr . Algumas quest\303\265es cr\303\255ticas sobre s\303\251ries infinitas est\303\243o relacionadas ao estudo de converg\303\252ncia, velocidade de converg\303\252ncia e raio de converg\303\252ncia.

Ex. 1

Consideremos a fun\303\247\303\243o arctan, que possui uma representa\303\247\303\243o em s\303\251rie bastante conhecida:

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 +...

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

Aplicando-se a defini\303\247\303\243o da soma de uma s\303\251rie neste caso, quando 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 obt\303\251m-se LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbWZyYWNHRiQ2KC1GIzYjLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiMtSSNtbkdGJDYkUSI2RidGNy8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGRS8lKWJldmVsbGVkR0Y2 . Considerando x = p, calcularemos a soma dos primeiros n termos. Em seguida, basta encontrar o limite quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEoJmluZmluO0YnL0Y1RkJGPkYr, para que se tenha a soma desta s\303\251rie.

S:=(n,p)->sum((-1)^(k+1)*p^(2*k-1)/(2*k-1),k=1..n);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJTR0YoZio2JEkibkdGKEkicEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkkc3VtR0YlNiQqKCkhIiIsJkkia0dGKCIiIkY9Rj1GPSk5JSwmRjwiIiNGOkY9Rj1GQEY6L0Y8O0Y9OSRGKEYoRig3I0Yu

evalf(limit(S(n,sqrt(3)/3),n=infinity));

JCIrY3gpZkImISM1

Esta pequena componente imagin\303\241ria \303\251, claramente, um artefato introduzido pelo algoritmo para o c\303\241lculo da s\303\251rie. Em primeiro lugar, como o conjunto dos n\303\272meros reais \303\251 completo, n\303\243o \303\251 poss\303\255vel, para qualquer s\303\251rie de n\303\272remos reais, que se obtenha uma soma imagin\303\241ria. Em segundo lugar, a componente imagin\303\241ria \303\251 extremamente pequena comparada a componente real. Portanto, pode-se desconsiderar a componente imagin\303\241ria.

6*Re(%);

JCIrYUVmVEohIio=

O Maple pode gerar expans\303\243o em s\303\251ries para uma grande variedade de fun\303\247\303\265es. Lamentavelmente, estas apresentam coeficientes n\303\272mericos, em vez de simb\303\263licos. Quando os coeficientes simb\303\263licos s\303\243o conhecidos, ou quando podem ser deduzidos de forma num\303\251rica, \303\251 poss\303\255vel aplicar uma grande variedade de m\303\251todos para o estudo de quest\303\265es de converg\303\252ncia.

Teste da Integral

O teste da integral n\303\243o apenas informa se uma s\303\251rie converge ou n\303\243o, mas tamb\303\251m pode ser usado para fornecer uma estimativa do erro produzido pelo trucamento de uma s\303\251rie com n termos. O teste da integral usa a integral 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 como um limite superior para a soma da \303\241rea dos ret\303\242ngulos de base unit\303\241ria e altura 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

Mais especificamente, seja f uma fun\303\247\303\243o positiva, decrescente e cont\303\255nua. Ent\303\243o, {f(n)} \303\251 positiva e decrescente, e {LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==} converge se e somente se 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 existe.

Ex. 2

Seja 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 . Verificar a converg\303\252ncia de 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.

restart:

a:=(.5)^(2*k-1)/(2*k-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoKiYpJCIiJiEiIiwmSSJrR0YoIiIjRjIiIiJGNkYzRjI3I0Yu

evalf(int(a,k=1..infinity));

JCIrOl9OJCo9ISM1

Neste caso, o uso de evalf invoca procedimentos num\303\251ricos do Maple para o c\303\241lculo da integral impr\303\263pria. Como a integral existe, sabe-se que a s\303\251rie converge. De fato:

evalf(limit(sum(a, k=1..n), n=infinity));

JCIrVjkxJFwmISM1

Note que o valor da integral n\303\243o est\303\241 relacionado com o valor da soma. Na verdade, a integral pode ser tomada em qualquer ponto da soma com igual efic\303\241cia, pois apenas os termos finais da s\303\251rie podem determinar sua converg\303\252ncia (assumindo que os termos individuais s\303\243o finitos). No entanto, \303\251 preciso ter cautela. A soma acima \303\251 igual a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSNsbkYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjNGJ0Y3RjdGKw==. Esta foi tomada da s\303\251rie,

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 ... = 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

Ao aplicar-se o teste da integral em 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 , deve-se integrar em rela\303\247\303\243o a vari\303\241vel k, e n\303\243o x. No exemplo, foi usado x = \302\275 . Estudaremos, brevemente, algumas formas de determinar-se os valores de x para os quais a s\303\251rie converge. Antes disso, avaliaremos erro obtido pelo truncamento da s\303\251rie em determinado termo, digamos o d\303\251cimo.

Seja 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 . Ent\303\243o 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 <=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZRLUklbXN1YkdGJDYlRjEtRiM2Iy1GLDYlUSJuRidGNEY3LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw==<= 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 , i.e., o erro obtido pelo truncamento do n-\303\251simo termo \303\251 limitado por estas duas integrais.

lower_error_limit:=evalf(int(a,k=11..infinity));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STJsb3dlcl9lcnJvcl9saW1pdEdGKCQiK1IpUSRROiEjPDcjRi4=

upper_error_limit:=evalf(int(a,k=10..infinity));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STJ1cHBlcl9lcnJvcl9saW1pdEdGKCQiKz5bRGduISM8NyNGLg==

O erro obtido com o truncamento do LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEjMTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtaUdGJDYlUSJvRicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== termo \303\251:

evalf(limit(sum(a,k=1..n), n=infinity)-sum(a,k=1..10));

JCIkJUghIzU=

Teste da Raz\303\243o

Utilizaremos a nota\303\247\303\243o adotada no primeiro par\303\241grafo. Ou seja, 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 representa uma sequ\303\252ncia, enquanto uma s\303\251rie infinita \303\251 representada por LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRic=, onde 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. Se existir um inteiro positivo, K, e um n\303\272mero positivo r < 1 tal que, para n < K, 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 , ent\303\243o a s\303\251rie LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRic= converge absolutamente. O teste da raz\303\243o consiste em apresentar um valor para r e K.

Ex. 3

Consideremos, novamente, a sequ\303\252ncia do Ex. 2, 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 . Do teste da raz\303\243o, resulta:

restart:

a[k]:=(1/2)^(2*k-1)/(2*k-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjSSJrR0YoKiYpIyIiIiIiIywmRjBGNSEiIkY0RjRGNkY3NyNGMQ==

a[k+1]:=(1/2)^(2*(k+1)-1)/(2*(k+1)-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjLCZJImtHRigiIiJGMkYyKiYpI0YyIiIjLCZGMUY2RjJGMkYyRjchIiI3I0Yz

ratio:=simplify(a[k+1]/a[k]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyYXRpb0dGKComLCZJImtHRigiIiMhIiIiIiJGMywmRjAiIikiIiVGM0YyNyNGLg==

Esta raz\303\243o \303\251 < \302\274 para todos os valores de k > 0, confirmando mais uma vez a converg\303\252ncia desta s\303\251rie. Utilizando o termo geral:

a[k]:=(x)^(2*k-1)/(2*k-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjSSJrR0YoKiYpSSJ4R0YoLCZGMCIiIyEiIiIiIkY3RjRGNjcjRjE=

a[k+1]:=(x)^(2*(k+1)-1)/(2*(k+1)-1);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjLCZJImtHRigiIiJGMkYyKiYpSSJ4R0YoLCZGMSIiI0YyRjJGMkY2ISIiNyNGMw==

ratio:=simplify(a[k+1]/a[k]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZyYXRpb0dGKCooSSJ4R0YoIiIjLCZJImtHRihGMCIiIkYzISIiLCZGMkYwRjRGM0YzNyNGLg==

verifica-se que a s\303\251rie converge para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw== < 1 ( r = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw== , K=0). O raio de converg\303\252ncia \303\251 (-1..1).

O raio de converg\303\252ncia \303\251 o dom\303\255nio da vari\303\241vel independente sobre o qual uma sequ\303\252ncia de fun\303\247\303\265es converge. O teste da raz\303\243o n\303\243o apenas representa um teste de converg\303\252ncia, mas tamb\303\251m \303\251 capaz de medir o raio de converg\303\252ncia.

Al\303\251m disso, se as condi\303\247\303\265es para o teste da raz\303\243o forem asseguradas, 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 <1, ent\303\243o, como 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 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JJW1zdXBHRiQ2JS1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRImtGJ0Y3RjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LlExJkludmlzaWJsZVRpbWVzO0YnLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk4vJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZ24tSShtZmVuY2VkR0YkNigtRiM2Iy1JJW1zdWJHRiQ2JS1GLDYlUSJhRidGN0Y6LUYjNiMtRiw2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R0ZCRkovSSttc2VtYW50aWNzR0YkUSRhYnNGJy8lJW9wZW5HUSkmdmVyYmFyO0YnLyUmY2xvc2VHRmFwRlxwRis= :

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYoLUYjNiUtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibUYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGTS8lKXN0cmV0Y2h5R0ZNLyUqc3ltbWV0cmljR0ZNLyUobGFyZ2VvcEdGTS8lLm1vdmFibGVsaW1pdHNHRk0vJSdhY2NlbnRHRk0vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zmbi1GMTYlRjMtRiM2Iy1GNDYlUSJuRidGN0Y6RkJGSS9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGZW9GYG8= = |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 + ...+ LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm1GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== | <= 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 + ... + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYoLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibUYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGTEZH <

(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 + ... LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm1GJ0YyRjUtSSNtb0dGJDYtUSgmbWludXM7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtRi82JVEibkYnRjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYoLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGTEZH = 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYoLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGTEZH

Como r<1, 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 = 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYoLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGTEZH . Ou seja, o erro devido ao truncamento do n-\303\251simo termo \303\251 menor ou igual a,

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYoLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJGFic0YnLyUlb3BlbkdRKSZ2ZXJiYXI7RicvJSZjbG9zZUdGTEZH

Neste caso, sendo r < \302\274 e x = \302\275 , o erro de truncamento \303\251 menor que 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.

err:=(1/2)^(2*n-1)/(3*(2*n-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcnJHRigqJikjIiIiIiIjLCZJIm5HRihGMiEiIkYxRjEsJkY0IiInISIkRjFGNTcjRi4=

Considerando-se um n\303\272mero de termos n, e um erro de .000001, obt\303\251m-se

solve(err=.000001,n);

JCIreHQjW3UoISIq

Conclu\303\255mos que oito termos s\303\243o necess\303\241rios para garantir uma precis\303\243o inferior a .000001.

Teste da Raiz

Utilizamos aqui a mesma nota\303\247\303\243o anterior. Se existe um inteiro, K, e um n\303\272mero r <1, tal que 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 < r para n > K, ent\303\243o a s\303\251rie LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRic= \303\251 convergente e, ao contr\303\241rio, se existe um inteiro, K, e um n\303\272mero r > 1 tal que 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 > r para n > K, ent\303\243o a s\303\251rie LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiMtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0Y7USdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRic= \303\251 divergente.

Isto segue do teste da compara\303\247\303\243o, com a s\303\251rie 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 , que \303\251 convergente para r <1, e \303\251 divergente para r > 1.

Ex. 4

Continuando com a sequ\303\252ncia do Ex. 3,

restart:with(plots):

a[n]:=((1/2)^(2*n-1)/(2*n-1));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiYUdGKDYjSSJuR0YoKiYpIyIiIiIiIywmRjBGNSEiIkY0RjRGNkY3NyNGMQ==

limit(a[n]^(1/n), n=infinity);

IyIiIiIiJQ==

A quantidade LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2Iy1JJm1mcmFjR0YkNigtRiM2Iy1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRidGOC8lLmxpbmV0aGlja25lc3NHRkovJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR1EmZmFsc2VGJ0ZLLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZZ \303\251 sempre menor que \302\274 . Isto pode ser verificado no gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2Iy1JJm1mcmFjR0YkNigtRiM2Iy1JI21uR0YkNiRRIjFGJy9GNlEnbm9ybWFsRidGOC8lLmxpbmV0aGlja25lc3NHRkovJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUS8lKWJldmVsbGVkR1EmZmFsc2VGJ0ZLLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZZ.

a:=[[n,evalf(a[n]^(1/n))] $n=1..100]:

pointplot(a);

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Ig==

Quando r = \302\274 , as condi\303\247\303\265es de converg\303\252ncia para o teste da raiz s\303\243o cumpridas. Como no teste da raz\303\243o, deve-se obter uma estimativa do limite superior para o erro de truncamento. Se as condi\303\247\303\265es para o teste da raiz forem satisfeitas, ent\303\243o,

(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Jy1JJW1zdXBHRiQ2JS1GLDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEibkYnRjdGOi1JI21vR0YkNi1RIitGJy9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkovJSlzdHJldGNoeUdGSi8lKnN5bW1ldHJpY0dGSi8lKGxhcmdlb3BHRkovJS5tb3ZhYmxlbGltaXRzR0ZKLyUnYWNjZW50R0ZKLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGWS1JI21uR0YkNiRRIjFGJ0ZGLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZCLUYyNiVGNC1GIzYlRj9GQi1GZ242JFEiMkYnRkZGam5GQi1GMjYlRjQtRiM2JUY/RkItRmduNiRRIjNGJ0ZGRmpuRis= + ... LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYlUSJtRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw==) = 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 = 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

Utilizamos agora esta estimativa para encontrar o n\303\272mero de termos que garante um erro <.000001.

restart:

r:=1/4:

err:=(r)^(n+1)/(1-r);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcnJHRigsJCkjIiIiIiIlLCZJIm5HRihGMUYxRjEjRjIiIiQ3I0Yu

fsolve(err=.000001,n);

JCIrTUlJdCIqISIq

De acordo com este resultado, 10 termos garantem uma precis\303\243o de .000001. Qual estimativa \303\251 mais pr\303\263xima? Para responder a esta pergunta, calculamos os erros de truncamento da s\303\251rie com 8, 9 e 10 termos.

seq(evalf(ln(3)/2-sum((1/2)^(2*k-1)/(2*k-1),k=1..p)),p=5..10);

NigkIidEUmMhIzUkIidMKzdGJSQiJkxoI0YlJCIlKXkmRiUkIiUrOEYlJCIkJ0hGJQ==

Verifica-se que oito termos s\303\243o suficientes. Neste caso, o teste da raz\303\243o fornece uma estimativa mais pr\303\263xima.

Aplica\303\247\303\265es de s\303\251ries s\303\243o encontradas n\303\243o somente em ferramentas computacionais, mas tamb\303\251m no c\303\241lculo de certas integrais e na solu\303\247\303\243o de equa\303\247\303\265es diferenciais.

Ex. 5

Resolver a integral 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

restart:

F:=x^sin(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJGR0YoKUkieEdGKC1JJHNpbkdGJTYjRi83I0Yu

int(F,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRpbnRHRiU2JClJInhHRigtSSRzaW5HRiU2I0YwRjBGKw==

Observe que o resultado n\303\243o foi avaliado. Pela expans\303\243o em s\303\251ries, seguida pela convers\303\243o em um polin\303\264mio e a integra\303\247\303\243o termo a termo, pode-se obter uma boa fun\303\247\303\243o de aproxima\303\247\303\243o.

f:=evalf(series(F,x=1/2,20));

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

g:=int(f,x):

g:=convert(g,polynom);

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

A fun\303\247\303\243o foi expandida em torno do ponto x = 0.5, que encontra-se no centro do intervalo de integra\303\247\303\243o. Em geral, isto pode melhorar as caracter\303\255sticas de converg\303\252ncia da s\303\251rie sobre o intervalo de interesse. A an\303\241lise dos primeiros 20 termos sugere que a converg\303\252ncia da s\303\251rie ocorre no intervalo (-1.5..1.5). A partir de ent\303\243o, torna-se simples a obten\303\247\303\243o do resultado desejado.

G:=unapply(g,x):

plot(G,-0.2..1.4);

6*-%'CURVESG6$7hw7$$!"#!""$!1"p`nKSZO)!#;7$$!0*zf>$ye">!#:$!1"zOEX\8%y!#;7$$!1#Qw_\%oU=!#;$!1(ev,89QW(!#;7$$!2#=Os%Qq.w"!#<$!1.&p49(=^q!#;7$$!2/29Gk5vn"!#<$!1f#3\M"H0n!#;7$$!0(Rz=Z/&f"!#:$!1G#p36CCS'!#;7$$!29Fa3D)e=:!#<$!1P(=8pAG:'!#;7$$!27<Mo+A%R9!#<$!1fPb`o@@f!#;7$$!215?S;[vN"!#<$!1;+/k_51d!#;7$$!2<He;*o$fF"!#<$!1[Ob%zIE^&!#;7$$!2&oOtm'*)>>"!#<$!1)zoL#\3K`!#;7$$!2LjE`!*[!=6!#<$!1:*QjQwi=&!#;7$$!2.-/374[."!#<$!1\)oVqjZ.&!#;7$$!1MkGdaF7&*!#<$!2O$*o9XVT*[!#<7$$!1:Fa3@"oq)!#<$!1/C2vz%zw%!#;7$$!0e:JAu`(z!#;$!21X=+!\<gY!#<7$$!0yb6$Qi0r!#;$!2vL\$eQJRX!#<7$$!1(4>Q;J)oj!#<$!2Nu$)>"HJUW!#<7$$!1rLnMhw6b!#<$!1"o'p^Y&[L%!#;7$$!2&***)zf$=Jv%!#=$!1)[QIKaRC%!#;7$$!1"zd:^h2#R!#<$!2%y6Gd5?[T!#<7$$!1@Ryc4;GJ!#<$!1CC)Q<E/1%!#;7$$!1zc8Fa;,B!#<$!1o%Hf$)f>(R!#;7$$!2WFa3PD<a"!#=$!1\ExP!4K*Q!#;7$$!1o$oOtmcA(!#=$!0OS=&Qk5Q!#:7$$"2o?['H**3$G"!#>$!2kyQIH;ss$!#<7$$"1A/4=w)**o)!#=$!2b@$ex2LcO!#<7$$"2()4?S+k*o;!#=$!2tr'\x?T"e$!#<7$$"2:RzeP1a\#!#=$!2/RrY6Nc]$!#<7$$"1r],.!>RI$!#<$!1YX%\D\HV$!#;7$$"1jNrUl>'3%!#<$!1p_/pN'QO$!#;7$$"2NRzeP#ya\!#=$!12'eU6i%)G$!#;7$$"1mW*)y`CNd!#<$!1bhg-dy@K!#;7$$"1Eb5@ibol!#<$!2Y+LTbD;:$!#<7$$"1dC\)pfOK(!#<$!2v[yYl**))3$!#<7$$"1-7C[s=\")!#<$!2Z*Rw0#y6-$!#<7$$"1N%)oPV%f#*)!#<$!1.=y))R@eH!#;7$$"12Fa3$zyt*!#<$!2w[F'pc7$*G!#<7$$"2iPv]TpJ0"!#<$!29Wkxge,$G!#<7$$"2&3=OsWFO6!#<$!2m\>Lg'*[w#!#<7$$"1&**)z>XJ;7!#;$!1'f93hMEq#!#;7$$"1=PuGq;)H"!#;$!1>(=(\J_RE!#;7$$"2'Rzed<Mz8!#<$!21M%o@gXxD!#<7$$"2Ed9H!H$RX"!#<$!1$p#Q(e]3_#!#;7$$"2lKlIxA%R:!#<$!21'3(*pXWcC!#<7$$"23Ig?$f)eh"!#<$!27$GAelB*R#!#<7$$"22>Qwg6up"!#<$!2)[h_MiiQB!#<7$$"2'R!3;3Uax"!#<$!2l6>@;i4G#!#<7$$"2=Pu[*[3i=!#<$!2BjuhC/t@#!#<7$$"2'>T#[?6r$>!#<$!2d2C=WuC;#!#<7$$"2=W)oPnEA?!#<$!2GZ\L-[05#!#<7$$"2Y.29ww)*4#!#<$!1'>w(o=PW?!#;7$$"1hAXI*pZ=#!#;$!2/mgKO*=$)>!#<7$$"2lX"Hy$=!eA!#<$!23\&zb`gI>!#<7$$"2dD^-")R9M#!#<$!1ltXvC$4(=!#;7$$"2)ye<&z??U#!#<$!2_QB'o/\8=!#<7$$"1iC\Q"\D]#!#;$!1'\;LTkiv"!#;7$$"2O"GcK7y#e#!#<$!2l%=Cx-T*p"!#<7$$"1MpQd"f)fE!#;$!2t5fssG\k"!#<7$$"2O)oPNC=VF!#<$!//cn*ohe"!#97$$"2Y-05c&fAG!#<$!2Vo(GxTGI:!#<7$$"2kS"GOT<1H!#<$!1i4I#4x:Z"!#;7$$"2$Ge;tG$=)H!#<$!2o,S2;<&=9!#<7$$"2;_/4%*\a1$!#<$!2$eVf'fb*f8!#<7$$"1C]+@0aXJ!#;$!1EzJ(eHRI"!#;7$$"2%f>RehVDK!#<$!2N'H/O>4[7!#<7$$"1LlIhu))3L!#;$!2GM4Q>7)*="!#<7$$"2P%*)yd"\dQ$!#<$!2x[A\Kih8"!#<7$$"1:Kk[gYkM!#;$!2(G"H#yZB"3"!#<7$$"1OsW\)y8b$!#;$!2o$ojijf?5!#<7$$"1)yd:Fr+j$!#;$!1)QTZy(*ol*!#<7$$"2dKlI,\0r$!#<$!1u%*pElB&4*!#<7$$"03;K!GR#z$!#:$!1_f-)3%zB&)!#<7$$"2lJjE<Bw'Q!#<$!1b>F<$=#)*z!#<7$$"1f<N5j"y%R!#;$!1xNp*)RbPu!#<7$$"1;Lm7!*RFS!#;$!1'zjJrS1)o!#<7$$"2.<MowFD6%!#<$!18pkDXC%G'!#<7$$"19Ji/-l(=%!#;$!2O7@%o$*Gdd!#=7$$"1wb6.fKuU!#;$!1f#*>@GU[^!#<7$$"1jHfQSa_V!#;$!1E9ZC23)f%!#<7$$"1ze<&zO*HW!#;$!1v&)e>qj_S!#<7$$"2uxb6()eJ^%!#<$!1,TsjS,lM!#<7$$"1Rze(f<nf%!#;$!1*GYm[!ytG!#<7$$"2ulJjmVFn%!#<$!1-b.>QpMB!#<7$$"1d8Fu*zMv%!#;$!1Lb]!GF4w"!#<7$$"*sqA$[!"*$!20ukhpR'*>"!#=7$$"1))zf>2o<\!#;$!13[p=k2'*e!#=7$$"2P*)yd6%e"*\!#<$!0<LCK:b.'!#=7$$"1E`1`[Yw]!#;$"1&[\78-?\&!#=7$$"0@U%3sDc^!#:$"1">B*3z$R7"!#<7$$"1;Moc)y_B&!#;$"2awJU7W[p"!#=7$$"1zhBnLO9`!#;$"1P')fy)))zE#!#<7$$"1\+,iX-&R&!#;$"1)4=HbrW&G!#<7$$"1Hf=xS_![&!#;$"1%RD?!>KyM!#<7$$"1/5?g(=*eb!#;$"1f*y$>MQ_S!#<7$$"1mMpyVIOc!#;$"1$)=L4v/@Y!#<7$$"0Gc7LG!>d!#:$"1vImY)*>J_!#<7$$"1D]+,Ad,e!#;$"1p;n!yDC%e!#<7$$"1\'HfAgc(e!#;$"1`(HxNUJR'!#<7$$"1`3<MGUjf!#;$"1k"pE8f"[q!#<7$$"1s[(\^eo.'!#;$"1Ld!y,>&)f(!#<7$$"1-3;#z\F7'!#;$"1x(yP-F\C)!#<7$$"1'QxapCZ?'!#;$"1TLN<ykk))!#<7$$"1$**)z>&=zF'!#;$"1GWBiNM?%*!#<7$$"1f<NIEBgj!#;$"29!y'y[+[+"!#<7$$"108EsB4Vk!#;$"2[di6v!Go5!#<7$$"119G'Heb_'!#;$"1'e?2Lj<8"!#;7$$"106AkZ,-m!#;$"2LDnn*o*3>"!#<7$$"117C35="o'!#;$"2N.(f=LT_7!#<7$$"1v_0^[0jn!#;$"1")3rStM;8!#;7$$"1&3<M7mY%o!#;$"22fq#e.S!Q"!#<7$$"13<M[LhGp!#;$"2)e3L38jY9!#<7$$"1U([(4Tb-q!#;$"2e^%RaEE0:!#<7$$"1b8F%*Qz&3(!#;$"2G_B)RZgr:!#<7$$"1LnMpaPpr!#;$"2e6/()=&eQ;!#<7$$"1/6A/=#*\s!#;$"1j<#Gw'[.<!#;7$$"1=Qw#flIK(!#;$"2lN1S]HFw"!#<7$$"1)zf>jS+T(!#;$"1;'>vUgN$=!#;7$$"1mMp)*)>P[(!#;$"2K7!Rvm*Q*=!#<7$$"1R!3;SE%pv!#;$"1_%R![JZk>!#;7$$"1xa4z6HXw!#;$"2-AE'4=IF?!#<7$$"1'f>R'o_Gx!#;$"2)*oJ)pii'4#!#<7$$"1%)pR>py2y!#;$"1QVD$=EI;#!#;7$$"13=Osk[!*y!#;$"2QVeS!frKA!#<7$$"0&**)zZIk'z!#:$"2<@"GHc3(H#!#<7$$"1$pQxMY$[!)!#;$"2.6sO6BpO#!#<7$$"1f=P9RVL")!#;$"2WL_.M<*RC!#<7$$"1#Gc7*G]2#)!#;$"1RN]L*RQ]#!#;7$$"1'QxaT*\(G)!#;$"1LJ%)==GtD!#;7$$"1;Lm_O9q$)!#;$"2:-;LIoak#!#<7$$"1%)oP:\*4X)!#;$"2an%ovM`;F!#<7$$"1KnMpEAH&)!#;$"2:GD)Hpr&y#!#<7$$"1<Lm_73;')!#;$"1m4$QjGI'G!#;7$$"1B['HbFTp)!#;$"1vxSVo%H$H!#;7$$"1Hf=P'eux)!#;$"2bC-_'e23I!#<7$$"1BY#\)*oH&))!#;$"1=1K0RewI!#;7$$"1(\**)R<_N*)!#;$"2Yh!GJY&>:$!#<7$$"1@U%)[u>8!*!#;$"1:YP=#HLA$!#;7$$"1\'Hf%4R%4*!#;$"1d?GR[T)H$!#;7$$"1`2:ICxt"*!#;$"1p]asGIsL!#;7$$"1&=Pu[xoD*!#;$"2O<(4z(o,X$!#<7$$"1sV([`<pL*!#;$"1**RZ[JmDN!#;7$$"1%4>Q/q(=%*!#;$"1a*4-!)zLg$!#;7$$"1;LmsZ%**\*!#;$"0^B%Q)p4o$!#:7$$"1[**)z"f`u&*!#;$"1tTd#zCFv$!#;7$$"1.29)yD+m*!#;$"2b)G$))e2b$Q!#<7$$"0oNr%*)[O(*!#:$"2l?!*)eZ/5R!#<7$$"1nNrAY,=)*!#;$"1ke:Y^.!*R!#;7$$"1<Mo'4Xg*)*!#;$"2vGQuF*4nS!#<7$$"1[(\*4zo#)**!#;$"2F?*H$>ZK:%!#<7$$"2'\**)>Urd+"!#;$"2BFM$4lLGU!#<7$$"1#Qw_(pG95!#:$"2.6?yk9TJ%!#<7$$"1T#[w(z/A5!#:$"1EjN**))z#R%!#;7$$"2Nw_XHP0."!#;$"21YV&pMTzW!#<7$$"2MoO$R@'y."!#;$"22f0O@(faX!#<7$$"2KmKDG/i/"!#;$"2nEE!3erSY!#<7$$"2b7D5QiU0"!#;$"2;!HTH.QCZ!#<7$$"2Ryc`@:B1"!#;$"29)QfY;V3[!#<7$$"2)=QwC%Q.2"!#;$"10fRVdd#*[!#;7$$"1JiC<i/y5!#:$"20'f@0[ut\!#<7$$"2hAX]ayj3"!#;$"0`TF@#zh]!#:7$$"2-/3w&)>V4"!#;$"0m.%Hx"f9&!#:7$$"2%yc8:xn-6!#;$"1m#4pYjXB&!#;7$$"2/7C))eV-6"!#;$"1@;)R))zZJ&!#;7$$"2(*)zf&H0'=6!#;$"1hIS"fBKS&!#;7$$"2-/3ON9m7"!#;$"1#[kzO6v[&!#;7$$"2PtYt"RgM6!#;$"1Z%4W%f!4d&!#;7$$"22>Qw/\H9"!#;$"1$eGN$>"pl&!#;7$$"1KkG<_j]6!#:$"1tqE"QCZt&!#;7$$"2#f=P1p]e6!#;$"1KAml'>D"e!#;7$$"26E_k=)>n6!#;$"1\"=(\<W&*e!#;7$$"2lJj'Gu1v6!#;$"1'*\]Bw)p'f!#;7$$"2.29G?:J="!#;$"1K&)\ZDlNg!#;7$$"2c9H=e*H">"!#;$"1i(Qp*p^*4'!#;7$$"2$pQx=E#))>"!#;$"1<)\Z2D::'!#;7$$"2NrUD$>%o?"!#;$"1>7>"=8")>'!#;7$$"2%pQx--![@"!#;$"1J.!eQ:JB'!#;7$$"2Z&4>yIJB7!#;$"1A)ezV)3bi!#;7$$"1\)p>KD3B"!#:$"1&e"p")z/ei!#;7$$"2b4>=*G\R7!#;$"1h?G^AsPi!#;7$$"1MoO0ZJZ7!#:$"1-"))=RcD>'!#;7$$"2c7D5)R0b7!#;$"1vFFL;:<h!#;7$$"2c6B')=wLE"!#;$"1'*z?h,[%*f!#;7$$"2;Lm71K<F"!#;$"1:<;9@L=e!#;7$$"2MqS"oYLz7!#;$"1dv()[F%>g&!#;7$$"2LnM*)H3uG"!#;$"1ASF(G>7I&!#;7$$"2u`2NP(G&H"!#;$"21(eA[?!H#\!#<7$$"2mLnMPGQI"!#;$"2j7/**z(=)R%!#<7$$"1Fa38(=7J"!#:$"1FuzrTcGQ!#;7$$"2-29oy1(>8!#;$"2u)3l`v!\,$!#<7$$"2)e<NAgoF8!#;$"2v(*GuP6R1#!#<7$$"2&ze<(=)eN8!#;$"1t<sw#e!*3*!#<7$$"2d:J#Qm\V8!#;$!2;'4RiU_-]!#=7$$"2Da3xvi:N"!#;$!2iE(*\M(G[A!#<7$$"208E#4F6g8!#;$!1HjYD$30^%!#;7$$"1Qw_x@&zO"!#:$!1YgF$zje-(!#;7$$"27E_%e9#=P"!#;$!1Kx<'zvnW)!#;7$$"2U)oPR2pv8!#;$!1j*\&3^&*****!#;7$$"2^-0q$p#)z8!#;$!20*)=V\E>="!#;7$$"2e;LY8jRQ"!#;$!2c(\d\5u"Q"!#;7$$"2Fg?"G.4)Q"!#;$!2Ep"300b+;!#;7$$"0/3;_<AR"!#9$!2xx\#4e]S=!#;7$$"0-/3w3hR"!#9$!2LEnCT)z(3#!#;7$$"#9!""$!1L*[%)fksN#!#:-%&COLORG6'%$RGBG$")C)eq%!")$""!!""$"('>!\&!")-%+_ATTRIBUTEG6#/%'sourceG%,mathdefaultG-%%VIEWG6$;$!"#!""$"#9!""%(DEFAULTG-&%&_AXISG6#"""6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-&%&_AXISG6#""#6&-%&COLORG6&%$RGBG$""!!""$""!!""$""!!""-%*LINESTYLEG6#""!-%*THICKNESSG6#""!-%-TRANSPARENCYG6#$""!!""-%+AXESLABELSG6$Q!6"Q!6"-%)_VISIBLEG6#"""-%%ROOTG6'-%)BOUNDS_XG6#$"$g#!""-%)BOUNDS_YG6#$"#]!""-%-BOUNDS_WIDTHG6#$"%]N!""-%.BOUNDS_HEIGHTG6#$"%!)Q!""-%)CHILDRENG6"-%+ANNOTATIONG6'-%)BOUNDS_XG6#$""!!""-%)BOUNDS_YG6#$""!!""-%-BOUNDS_WIDTHG6#$"%+S!""-%.BOUNDS_HEIGHTG6#$"%+S!""-%)CHILDRENG6"NiI=

O Maple implementa este m\303\251todo pela expans\303\243o da integral indefinida com o comando series.

int(F,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMtSSRpbnRHRiU2JClJInhHRigtSSRzaW5HRiU2I0YwRjBGKw==

series(%,x,20):

g:=convert(g,polynom):

g:=unapply(g,x);

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

plot(g,-.2..1.4);

Obviamente, o Maple pode gerar um resultado num\303\251rico para uma integral definida com

int(F,x=.1..0.9);

JCIrR29XJDMnISM1

Contudo, h\303\241 ocasi\303\265es em que uma aproxima\303\247\303\243o funcional expl\303\255cita para a integral \303\251 necess\303\241ria, como para gerar gr\303\241ficos.

Aplica\303\247\303\265es

1. Testar a converg\303\252ncia de cada uma das s\303\251ries a seguir e, se a s\303\251rie convergir, encontrar o raio de converg\303\252ncia.

a) 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 b) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1JK211bmRlcm92ZXJHRiQ2Jy1JI21vR0YkNi9RJiZTdW07RicvJStmb3JlZ3JvdW5kR1EuWzE0NCwxNDQsMTQ0XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRJmluZXJ0RicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkMvJSlzdHJldGNoeUdRJXRydWVGJy8lKnN5bW1ldHJpY0dGQy8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZDLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTY2NjY2N2VtRictRiM2JS1GLDYlUSJuRicvJSdpdGFsaWNHRkgvRjxRJ2l0YWxpY0YnLUY1Ni1RIj1GJ0Y7RkFGRC9GR0ZDRkkvRkxGQy9GTkZDRk8vRlJRLDAuMjc3Nzc3OGVtRicvRlVGYW8tSSNtbkdGJDYkUSIwRidGOy1GLDYlUSgmaW5maW47RicvRmduRkNGO0ZPLyUsYWNjZW50dW5kZXJHRkNGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EkNS4wRicvJSZkZXB0aEdGYnAvJSpsaW5lYnJlYWtHUSVhdXRvRictSSZtZnJhY0dGJDYoLUYjNiMtSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGZm5GaG4tRjU2LVEiK0YnRjtGQUZERl1vRklGXm9GX29GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZfci1GZG82JFEiM0YnRjtGOy1GIzYlRlktRjU2LVEoJm1pbnVzO0YnRjtGQUZERl1vRklGXm9GX29GT0ZeckZgci1GZG82JFEiMUYnRjsvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zmby1GIzYlRistRiM2JC1GZHE2JC1GIzYlRllGW3JGaXJGOy1GNTYtUSIhRidGO0ZBRkRGXW9GSUZeb0Zfb0ZPL0ZSUSwwLjExMTExMTFlbUYnL0ZVRmpzRisvJS5saW5ldGhpY2tuZXNzR0Zbcy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZgdC8lKWJldmVsbGVkR0ZDRis= c) 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

d) 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

2. Calcular cada uma das integrais a seguir por meio de s\303\251ries.

a) Encontrar uma s\303\251rie para o integrando e integrar termo a termo.

b) Resolver a integral por meio do comando series.

c) Verificar os resultados, for\303\247ando uma resolu\303\247\303\243o num\303\251rica da integral.

d) Tra\303\247ar o gr\303\241fico de cada fun\303\247\303\243o, juntamente com sua representa\303\247\303\243o por s\303\251ries. Procure expandir em torno de diferentes pontos e observe o efeito em rela\303\247\303\243o a aproxima\303\247\303\243o.

i) 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 ii) 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 iii) 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