C\303\241lculo IILi\303\247\303\243o 16: Introdu\303\247\303\243o ao estudo de S\303\251ries Infinitas

O que significa adicionar um seq\303\274\303\252ncia de n\303\272meros? (Lembre-se de que seq\303\274\303\252ncias s\303\243o sempre infinitas. Portanto, trata-se da adi\303\247\303\243o de um conjunto infinito de n\303\272meros.) Como explicado na se\303\247\303\243o 10.2, para o c\303\241lculo da soma 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 , \303\251 preciso formar, inicialmente, a sequ\303\252ncia s de somas parciais:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRzXzFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiw2JVEkYV8xRidGNEY3Ris= , 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 , 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 , ... ,cujo termo geral \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkc19uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn = 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkYV8xRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkYV8yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkYV8zRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn + ... + LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkYV9uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn ,Calcula-se ent\303\243o o limite 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 .Este procedimento pode ser bem complicado se feito manualmente --- uma grande parte do Cap\303\255tulo 10 apresenta v\303\241rias formas de faz\303\252-lo indiretamente. Por\303\251m, este tipo de problema pode ser facilmente resolvido pelo Maple. Em princ\303\255pio, estudaremos a soma da s\303\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 .Come\303\247aremos pela defini\303\247\303\243o da sequ\303\252ncia a, de termos da s\303\251rie, e a sequ\303\252ncia s, de somas parciais. A fim de evitar trabalho de digita\303\247\303\243o desnecess\303\241rio, utilizaremos a sintaxe de fun\303\247\303\243o para definir a_n e s_n como fun\303\247\303\265es de n. Note que ambas as sequ\303\252ncias iniciam em n = 0, e n\303\243o n = 1, j\303\241 que o primeiro termo da soma \303\251 n = 0. a := proc(n) (5/9)^n end: s := proc(n) sum(a(i), i=0..n) end: Pode-se agora exibir os primeiros termos; por exemplo, a(n) $n=0..10;

Ni0iIiIjIiImIiIqIyIjRCIjIikjIiREIiIkSCgjIiREJyIlaGwjIiVESiImXCFmIyImRGMiIidUOWAjIiZEInkiKHBIeSUjIidEMVIiKUBuL1YjIihESiY+IioqWz91USMiKERjdyoiKyxXeSdbJA==

Similarmente, apresentamos a seguir as primeiras somas parciais: s(n) $n=0..10;

Ni0iIiIjIiM5IiIqIyIkXiIiIyIpIyIlJVsiIiRIKCMiJiIpUiIiJWhsIyInYSpHIiImXCFmIyIoNmk8IiInVDlgIyIpQ1NtNSIocEh5JSMiKVRvTycqIilAbi9WIyIqJXBhI3ApIioqWz91USMiK3J5MEx5IissV3knWyQ=

Note que estas duas sequ\303\252ncias s\303\243o diferentes!!! A segunda sequ\303\252ncia \303\251 formada pela soma de termos da primeira sequ\303\252ncia; por exemplo, as tr\303\252s primeiras somas parciais s\303\243o a(0); a(0) + a(1); a(0) + a(1) + a(2);

IiIi

IyIjOSIiKg==

IyIkXiIiIyIp

Para maiores esclarecimentos, exibimos tamb\303\251m os gr\303\241ficos das duas sequ\303\252ncias. Primeiramente, tem-se o gr\303\241fico da sequ\303\252ncia de termos; a seguir \303\251 apresentado o gr\303\241fico da sequ\303\252ncia de somas parciais. plot([[n,a(n)] $n=0..50], x=0..50, style=point);

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

plot([[n,s(n)] $n=0..50], x=0..50, style=point);

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NiI=

As sequ\303\252ncias de termos convergem para 0, como se pode verificar atrav\303\251s do comando limit. Isto n\303\243o significa que a s\303\251rie possui a soma igual a 0. A soma da s\303\251rie \303\251 o limite das somas parciais: limit(s(n), n=infinity);

IyIiKiIiJQ==

Pode-se ent\303\243o escrever 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 . N\303\243o \303\251 de surpreender que o Maple apresente a resposta em um \303\272nico passo. sum((5/9)^n, n=0..infinity);

IyIiKiIiJQ==

Para cada uma das s\303\251ries a seguir, definir a sequ\303\252ncia de termos e a sequ\303\252ncia de somas parciais. A seguir, encontrar a soma das s\303\251ries, tomando-se assim o limite da sequ\303\252ncia de somas parciais, se existir. (Algumas das s\303\251ries podem divergir.) Para cada exemplo, verificar o resultado atrav\303\251s do comando para o c\303\241lculo da soma das s\303\251ries.(a) 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 (b) 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 (c) 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 (d) 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(Dica para (b): se o Maple apresentar problemas no c\303\241lculo das somas parciais, utilizar fra\303\247\303\265es parciais.)Solu\303\247\303\265es. Apresentamos a seguir a primeira s\303\251rie: a := proc(n) 5/(-2)^n end: s := proc(n) sum(a(j), j=1..n) end: limit(s(n), n=infinity);

IyEiJiIiJA==

A s\303\251rie (a) converge, e sua soma \303\251 igual a -5/3. Para resolver os outros problemas, basta redefinir a; como todas as outras s\303\251ries come\303\247am por n = 1, a express\303\243o para s em termos de a deve ser a mesma em todos os casos. a := proc(n) 1/(n*(n+1)) end: limit(s(n), n=infinity);

IiIi

A s\303\251rie (b) converge, com soma igual a 1. a := proc(n) 1/n end: limit(s(n), n=infinity);

SSlpbmZpbml0eUclKnByb3RlY3RlZEc=

A resposta LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= indica que a s\303\251rie (c) diverge. a := proc(n) 1/n^2 end: limit(s(n), n=infinity);

LCQqJEkjUGlHJSpwcm90ZWN0ZWRHIiIjIyIiIiIiJw==

A s\303\251rie (d) converge, com soma igual a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSNQaUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtbkdGJDYkUSIyRidGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw==/6 . (Embora pare\303\247a estranho, a resposta est\303\241 correta: o que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn tem a ver com a soma de rec\303\255procos dos inteiros positivos ao quadrado?)