C\303\241lculo IILi\303\247\303\243o 14: Introdu\303\247\303\243o ao estudo de Sequ\303\252ncias

Uma seq\303\274\303\252ncia \303\251 simplesmente uma lista infinita; ou seja, uma cole\303\247\303\243o infinita de objetos arranjados em determinada ordem. Neste curso, trataremos quase exclusivamente de seq\303\274\303\252ncias num\303\251ricas. Por exemplo,1, 2, 3, 4, 5, 6, 7, ...1, -1/3, 1/9, -1/27, 1/81, -1/243, ...1/2, 2/3, 3/4, 4/5, 5/6, 6/7, ...s\303\243o tr\303\252s seq\303\274\303\252ncias num\303\251ricas. (Note que o s\303\255mbolo ... indica que a lista continua indefinidamente.)Tipicamente, uma seq\303\274\303\252ncia \303\251 denotada por uma \303\272nica letra; os termos individuais na sequ\303\252ncia s\303\243o denotados pela mesma letra, com um \303\255ndice subscrito indicando a posi\303\247\303\243o do termo na sequ\303\252ncia. Por exemplo, se a \303\272ltima sequ\303\252ncia acima for denominada a, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRhXzFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSZtZnJhY0dGJDYoLUYjNiMtSSNtbkdGJDYkUSIxRidGPi1GIzYjLUZaNiRRIjJGJ0Y+LyUubGluZXRoaWNrbmVzc0dGZm4vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYG8vJSliZXZlbGxlZEdGQkYr , 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRhXzNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSZtZnJhY0dGJDYoLUYjNiMtSSNtbkdGJDYkUSIzRidGPi1GIzYjLUZaNiRRIjRGJ0Y+LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zhby8lKWJldmVsbGVkR0ZCRis= , e assim por diante. O termo geral, ou n-\303\251simo termo, \303\251 dado pela f\303\263rmula 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 .

Qual o termo geral na segunda sequ\303\252ncia acima?Solu\303\247\303\243o. 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 . (Por que o expoente \303\251 n + 1, e n\303\243o apenas n? )

Uma forma de gerar uma lista no Maple \303\251 atrav\303\251s do s\303\255mbolo dolar, $. n^2 $n=1..10;

NiwiIiIiIiUiIioiIzsiI0QiI08iI1wiI2siIyIpIiQrIg==

n/(n+1) $n=3..7;

NicjIiIkIiIlI0YlIiImI0YnIiInI0YpIiIoI0YrIiIp

Note o uso do termo 'lista', e n\303\243o 'sequ\303\252ncia'; o Maple gera apenas um n\303\272mero finito de termos, e n\303\243o a sequ\303\252ncia inteira. Isto \303\251 suficiente para obter-se um gr\303\241fico da sequ\303\252ncia, considerando-se o termo geral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkYV9uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn como fun\303\247\303\243o de n. Com a sintaxe correta, o comando plot do Maple \303\251 capaz de exibir o gr\303\241fico. Nos exemplos acima, foi definida inicialmente uma lista de pontos a serem tra\303\247ados (os pontos (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEkYV9uRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn)), que deve ser inserida no comando plot. a := [[n, n^2] $n=1..10];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJhR0YoNyw3JCIiIkYwNyQiIiMiIiU3JCIiJCIiKjckRjMiIzs3JCIiJiIjRDckIiInIiNPNyQiIigiI1w3JCIiKSIjazckRjYiIyIpNyQiIzUiJCsiNyNGLg==

plot(a, x=0..12, style=point);

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Ig==

O gr\303\241fico acima apresenta uma boa ilustra\303\247\303\243o da sequ\303\252ncia a, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRhX25GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJuRidGNEY3LUkjbW5HRiQ2JFEiMkYnRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGKw== . O exemplo a seguir \303\251 a sequ\303\252ncia cujo n-\303\251simo termo \303\251 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 . b := [[n, n/(n+1)] $n=1..10];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJiR0YoNyw3JCIiIiNGMCIiIzckRjIjRjIiIiQ3JEY1I0Y1IiIlNyRGOCNGOCIiJjckRjsjRjsiIic3JEY+I0Y+IiIoNyRGQSNGQSIiKTckRkQjRkQiIio3JEZHI0ZHIiM1NyRGSiNGSiIjNjcjRi4=

plot(b, x=0..12, style=point);

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Ig==

Note que as sequ\303\252ncias a e b comportam-se de maneiras diferentes quando n \303\251 grande: a sequ\303\252ncia a cresce sem limites, enquanto b aparenta aproximar-se de uma valor finito. Pode-se confirmar este fato tra\303\247ando-se o gr\303\241fico de b para mais valores de n. plot([[n, n/(n+1)] $n=1..100], x=0..100, y=0..2, style=point);

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

Est\303\241 claro que, a medida que n aumenta, os termos 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 aproximam-se de 1. (Este fato tamb\303\251m fica claro da \303\241lgebra: quando n \303\251 muito grande, o valor '+1' no denominador torna-se de pouca import\303\242ncia, de forma que a fra\303\247\303\243o \303\251 aproximadamente igual a 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 .)Em uma exemplo tal como a sequ\303\252ncia b, onde o termo geral aproxima-se de um n\303\272mero fixo e finito, L, a medida que n aumenta, diz-se que a sequ\303\252ncia converge para L, ou possui limite igual a L, e escreve-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 .Caso contr\303\241rio, diz-se que a sequ\303\252ncia diverge, ou que seu limite n\303\243o existe. Por exemplo, verificou-se acima que 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 , por\303\251m 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 n\303\243o existe.

Tra\303\247ar o gr\303\241fico das sequ\303\252ncias cujos termos gerais s\303\243o dados pelas express\303\265es a seguir, e determinar se tais sequ\303\252ncias convergem ou n\303\243o. Caso afirmativo, encontrar o limite. (Deve-se responder tais quest\303\243o antes que os gr\303\241ficos sejam exibidos; nem todos os casos s\303\243o \303\263bvios.)(a) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiQtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSy1JI21uR0YkNiRRIjFGJ0Y3RjctSSNtaUdGJDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvRjhRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== (b) 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 (c) 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 (d) 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(e) 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 (f) 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 (g) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJuRicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0YrSolu\303\247\303\265es. plot([[n, (-1)^n] $n=1..20], x=0..20, style=point);

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Ig==

A sequ\303\252ncia (a) n\303\243o converge: os termos oscilam entre 1 e - 1, e n\303\243o aproximam-se de nenhum n\303\272mero fixo. plot([[n, (1 + 1/n)^n] $n=1..30], x=0..30, style=point);

NistJSdDVVJWRVNHNiQ3QDckJCIjNSEiIiQiIz8hIiI3JCQiIz8hIiIkIiREIyEiIzckJCIjSSEiIiQiMi1QcS5QcS5QIyEjOzckJCIjUyEiIiQiKkQxOVcjISIpNyQkIiNdISIiJCInSylbIyEiJjckJCIjZyEiIiQiMkU2VTxQRTtfIyEjOzckJCIjcSEiIiQiMTgyLyhwKlxZRCEjOjckJCIjISkhIiIkIjFbLiZSXiV5bEQhIzo3JCQiIyEqISIiJCIxKD44PHp1NmUjISM6NyQkIiQrIiEiIiQiLCxZVVBmIyEjNTckJCIkNSIhIiIkIjFKdio9LCo+L0UhIzo3JCQiJD8iISIiJCIxeVlBIUhOSWgjISM6NyQkIiRJIiEiIiQiMUtkKXkpM2c/RSEjOjckJCIkUyIhIiIkIjIlcDNJYzo6RkUhIzs3JCQiJF0iISIiJCIxPnpzPCh5R2ojISM6NyQkIiRnIiEiIiQiMiYqZm10XEd6aiMhIzs3JCQiJHEiISIiJCIyJjRKPXZWVFVFISM7NyQkIiQhPSEiIiQiMmNvKDRAZVVZRSEjOzckJCIkIT4hIiIkIjFYL2tFVi5dRSEjOjckJCIkKyMhIiIkIjBVV15xKEhgRSEjOTckJCIkNSMhIiIkIjEwaCNSQGppbCMhIzo3JCQiJD8jISIiJCIyJCl5UCZlKXAqZUUhIzs3JCQiJEkjISIiJCIxI3lRJz0sWGhFISM6NyQkIiRTIyEiIiQiMSVmbyFlN3RqRSEjOjckJCIkXSMhIiIkIjIuVShbSmokZW0jISM7NyQkIiRnIyEiIiQiMVRQYG1ceW5FISM6NyQkIiRxIyEiIiQiMXZEInkoUmZwRSEjOjckJCIkIUchIiIkIjIoUjNXYHlGckUhIzs3JCQiJCFIISIiJCIyOCEzKVI5XEduIyEjOzckJCIkKyQhIiIkIjJaSHFleD1WbiMhIzstJSZDT0xPUkc2JyUkUkdCRyQiKUMpZXElISIpJCIiISEiIiQiKCc+IVwmISIpLSUrX0FUVFJJQlVURUc2Iy8lJ3NvdXJjZUclLG1hdGhkZWZhdWx0Ry0lJVZJRVdHNiQ7JCIiISEiIiQiJCskISIiJShERUZBVUxURy0mJSZfQVhJU0c2IyIiIjYmLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpMSU5FU1RZTEVHNiMiIiEtJSpUSElDS05FU1NHNiMiIiEtJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSYlJl9BWElTRzYjIiIjNiYtJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKkxJTkVTVFlMRUc2IyIiIS0lKlRISUNLTkVTU0c2IyIiIS0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJStBWEVTTEFCRUxTRzYkLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEieDYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTM2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiUSE2Ii0lJlNUWUxFRzYjJSZQT0lOVEctJSlfVklTSUJMRUc2IyIiIi0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJFMmISIiLSUpQk9VTkRTX1lHNiMkIiQrIiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJTVMISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJSFRJCEiIi0lKUNISUxEUkVORzYiLSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZRzYjJCIiISEiIi0lLUJPVU5EU19XSURUSEc2IyQiJStTISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJStTISIiLSUpQ0hJTERSRU5HNiI=NiI=

A sequ\303\252ncia (b) aparenta convergir para um limite que \303\251 aproximadamente igual a 2.7. (Na verdade, esta sequ\303\252ncia converge para o n\303\272mero e, que \303\251 aproximadamente igual a 2.7182818285.) plot([[n, (1 + 2/n)^n] $n=1..30], x=0..30, style=point);

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Ig==

Esta sequ\303\252ncia aparenta convergir para um limite aproximadamente igual a 7. (O limite exato \303\251, na verdade, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw==.) plot([[n, (1 - 1/n)^n] $n=1..30], x=0..30, style=point);

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Ig==

Novamente, a sequ\303\252ncia (d) aparenta convergir. (Neste caso, o limite exato \303\251 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=; come\303\247a a surgir um padr\303\243o!) plot([[n, sqrt(n+1) - sqrt(n)] $n=1..30], x=0..30, style=point);

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

A sequ\303\252ncia (e) converge para 0. Este \303\251 um exemplo interessante: o termo geral da sequ\303\252ncia \303\251 a diferen\303\247a de dois n\303\272meros muito grandes (se n \303\251 grande). No entanto, n\303\243o s\303\243o os n\303\272meros LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiRRIjFGJ0Y+ ou LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkmbXNxcnRHRiQ2Iy1JI21pR0YkNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= , individualmente, que contam, mas sim sua diferen\303\247a; e sua diferen\303\247a tende a 0. O pr\303\263ximo exemplo mostra que nem sempre \303\251 evidente quando tal diferen\303\247a tende a zero. plot([[n, sqrt(n^2 + n) - n] $n=1..30], x=0..30, style=point);

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Ig==

Neste caso, tem-se novamente uma diferen\303\247a entre dois n\303\272meros grandes. Uma interpreta\303\247\303\243o para este limite poderia sugerir que, quando n \303\251 grande, ent\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw== \303\251 muito maior que n, de forma que o termo n na raiz quadrada poderia ser ignorado. Isto significa que o termo geral (para n grande) deveria assemelhar-se a 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. Como pode-se observar no gr\303\241fico, este argumento est\303\241 errado! Na verdade, esta sequ\303\252ncia converge para 1/2. ( A explica\303\247\303\243o para este fato encontra-se na Se\303\247\303\243o 10.11 de Stewart: A S\303\251rie Binomial.) plot([[n, sin(n)] $n=1..30], x=0..30, style=point);

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NiI=

Esta \303\251 a sequ\303\252ncia que apresenta maior dificuldade de compreens\303\243o, embora seja poss\303\255vel observar no gr\303\241fico que esta n\303\243o converge. (Esta \303\251, de fato, a resposta correta.) No entanto, n\303\243o parece existir um padr\303\243o que torne a diverg\303\252ncia \303\263bvia. Pode-se tra\303\247ar mais alguns termos da sequ\303\252ncia: plot([[n, sin(n)] $n=1..100], x=0..100, style=point);

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

Embora mais pontos tenham sido exibidos, ainda n\303\243o se verifica nenhum padr\303\243o \303\263bvio que confirme a diverg\303\252ncia. Na verdade, esta sequ\303\252ncia nunca exibir\303\241 qualquer comportamento "simples", tal como oscilar entre dois valores: n\303\243o importa quantos pontos sejam tra\303\247ados, sempre ocorrer\303\241 algo semelhante ao que mostra o gr\303\241fico acima.

Obviamente, o Maple pode calcular os limites algebricamente: limit(n/(n+1), n=infinity);

IiIi

Utilizar o comando limit para verificar os resultados da Quest\303\243o 2. Aplicar o limite em cada sequ\303\252ncia, mesmo que esta aparente divergir. Com isso, pode-se verificar como o Maple indica que um determinado limite n\303\243o existe.Solu\303\247\303\265es. Apresentamos a seguir os c\303\241lculos do Maple para os sete limites, em ordem. limit((-1)^n, n=infinity);

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Verificou-se na Quest\303\243o 2 que esta sequ\303\252ncia oscila entre 1 e -1 e, portanto, n\303\243o possui limite. O Maple apresenta o "limite" como um ingervalo: entre 1 e -1. Como o Maple n\303\243o fonece um \303\272nico n\303\272mero como limite, deve-se concluir que este n\303\243o existe. limit((1 + 1/n)^n, n=infinity);

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Nota-se que o Maple tem conhecimento da rela\303\247\303\243o entre esta sequ\303\252ncia e o n\303\272mero e, assim como suas outras varia\303\247\303\265es. limit((1 + 2/n)^n, n=infinity);

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limit((1 - 1/n)^n, n=infinity);

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Hmmm... e quanto ao padr\303\243o geral? (Caso algum valor tenha sido atribu\303\255do a x em qualquer parte da folha de trabalho, deve-se remover o conte\303\272do desta vari\303\241vel.) x := 'x';

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limit((1 + x/n)^n, n=infinity);

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Interessante! Aparentemente, a fun\303\247\303\243o exponencial pode ser calculada como o limite de uma sequ\303\252ncia. limit(sqrt(n+1) - sqrt(n), n=infinity);

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limit(sqrt(n^2 + n) - n, n=infinity);

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Este dois limites est\303\243o consistentes com o que foi observado na Quest\303\243o 2. limit(sin(n), n=infinity);

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O Maple apresenta o resultado para o limite desta sequ\303\252ncia como um intervalo, indicando assim que a sequ\303\252ncia n\303\243o converge, assim como na primeira sequ\303\252ncia estudada.