C\303\241lculo IILi\303\247\303\243o 23: Expans\303\243o em S\303\251rie de Pot\303\252ncias Nesta li\303\247\303\243o, exploraremos m\303\251todos de expans\303\243o de fun\303\247\303\265es em s\303\251ries de pot\303\252ncias. A id\303\251ia b\303\241sica consiste na expans\303\243o da s\303\251rie geom\303\251trica 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. No entanto, \303\251 poss\303\255vel expandir muitas outras fun\303\247\303\265es em s\303\251ries de pot\303\252ncias atrav\303\251s da diferencia\303\247\303\243o e da integra\303\247\303\243o. Al\303\251m disso, estudaremos o intervalo de converg\303\252ncia e como este \303\251 afetado pela localiza\303\247\303\243o da expans\303\243o e das caracter\303\255sticas da fun\303\247\303\243o, tais como suas ass\303\255ntotas verticais. O objetivo desta se\303\247\303\243o consiste em refor\303\247ar os m\303\251todos de expans\303\243o em s\303\251ries, sem levar em conta m\303\251todos de expans\303\243o autom\303\241ticos que podem ser gerados pelo Maple.___________________________________________________________________________________A. Expans\303\243o de Fun\303\247\303\265es em S\303\251ries___________________________________________________________________________________Uma s\303\251rie geom\303\251trica infinita pode ser simplificada como: Sum( a*r^k, k = 0..infinity): % = value(%);

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Este processo pode ser revertido. Dada uma express\303\243o, \303\251 poss\303\255vel expand\303\255-la em s\303\251rie. a/(1 - r): % = series( %, r);

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A partir de uma express\303\243o em x, pode-se calcular sua expans\303\243o em s\303\251rie. Isto pode ser feito manualmente. series( 1/(1-x), x);

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series( 1/(1-x), x, 15);

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Em geral, o Maple expande em uma s\303\251rie com certo n\303\272mero padr\303\243o de termos. No entanto, tamb\303\251m \303\251 poss\303\255vel especificar at\303\251 que pot\303\252ncia de x deseja-se expandir. O Maple utiliza o nota\303\247\303\243o de " O mai\303\272sculo " para indicar a ordem do erro, ou resto.Em todos os exemplos anteriores, as s\303\251ries de pot\303\252ncias foram expandidas em torno de x = 0. Por\303\251m, \303\251 poss\303\255vel expandir em torno de outros valores. series( 1/(1-x), x = 3, 12);

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series( 1/(2 + x^2), x = -1,12);

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___________________________________________________________________________________B. Integra\303\247\303\243o / S\303\251ries / Diferencia\303\247\303\243o___________________________________________________________________________________O m\303\251todo utilizado acima para expandir uma s\303\251rie em uma s\303\251rie geom\303\251trica funciona apenas em certos casos. Mesmo que uma express\303\243o n\303\243o se preste a este m\303\251todo, ainda existem outros truques. Um deles consiste na integra\303\247\303\243o da fun\303\247\303\243o, seguida pela expans\303\243o da anti-derivada em s\303\251rie, e a diferencia\303\247\303\243o do resultado. Por esta forma indireta, \303\251 poss\303\255vel encontrar a s\303\251rie.Consideremos o exemplo a seguir.Iniciamos com fun\303\247\303\243o a seguir, que \303\251 ent\303\243o integrada, expandida em s\303\251rie de pot\303\252ncias e, finalmente, diferenciada, de forma a obter-se a fun\303\247\303\243o original. f := x -> 3/(x-5)^2;

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F := int( f(x), x);

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series( %, x, 10);

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diff( %, x);

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Pode-se converter a nova s\303\251rie encontrada em um polin\303\264mio. Para demostrar que este possui valores pr\303\263ximos da fun\303\247\303\243o original, basta resolv\303\252-lo. Como \303\251 tomado apenas um n\303\272mero finito de termos, haver\303\241 um pequeno erro. g := unapply(convert( %, polynom), x);

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f(1); g(1); f(1) - g(1);

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IyIoWjUkPSIoRGN3Kg==

IyIkQiIiKisrRGMi

___________________________________________________________________________________C. Diferencia\303\247\303\243o / S\303\251ries / Integra\303\247\303\243o___________________________________________________________________________________O mesmo truque pode ser usado no caso inverso. Dada uma fun\303\247\303\243o, pode-se diferenci\303\241-la, expand\303\255-la em s\303\251rie de pot\303\252ncias e, finalmente, integr\303\241-la. Ao integrar-se, a constante de integra\303\247\303\243o torna-se o termo constante da s\303\251rie.Consideremos um exemplo.Iniciamos com fun\303\247\303\243o a seguir, que \303\251 ent\303\243o diferenciada, expandida em s\303\251rie de pot\303\252ncias e, finalmente, integrada, de forma a obter-se a fun\303\247\303\243o original. f := x -> arctan(x);

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diff( f(x), x);

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series( %, x, 14);

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Int( %, x) : % = int( %%, x);

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Toda integral indefinida possui uma constante de integra\303\247\303\243o. Em particular, ao intergrar-se uma s\303\251rie, n\303\243o se pode desconsiderar a constante de integra\303\247\303\243o, que representa o termo constante da s\303\251rie. Consideremos um exemplo. Dada uma fun\303\247\303\243o inicial, calcularemos sua expans\303\243o em s\303\251rie pelo m\303\251todo descrito acima. Em seguida, convertemos a s\303\251rie em um polin\303\264mio, e exibimos seu gr\303\241fico, juntamente com f(x). f := x -> 1/(1-x); c := -1;

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series( D(f)(x), x=c, 6);

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g := unapply( convert( int(%, x), polynom), x);

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plot( [f(x), g(x)], x = -5..1, y = -2..5, thickness=[3,2], color=[red,coral]);

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A curva vermelha \303\251 a fun\303\247\303\243o original, enquanto a curva em amarelo \303\251 a s\303\251rie - que representa a fun\303\247\303\243o. No entanto, como se pode observar, n\303\243o se tem uma boa representa\303\247\303\243o da fun\303\247\303\243o.No gr\303\241fico acima, faltou somente adicionar o termo constante, que \303\251 f(-1). Definiremos ent\303\243o uma fun\303\247\303\243o h(x) = g(x) + f(-1). h := unapply( g(x) + f(c), x);

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plot( [f(x), g(x), h(x), [[c,0],[c,f(c)]] ], x= -5..1, y = -2..5, thickness= [3,1,2,2], color = [red, coral, blue, green]);

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A curva em vermelho \303\251 a fun\303\247\303\243o original, enquanto a curva em amarelo \303\251 a s\303\251rie original. A curva azul \303\251 h(x), a s\303\251rie acrescida da constante de integra\303\247\303\243o. A segmento em verde \303\251, precisamente, o comprimento de f(-1), que \303\251 igual a diferen\303\247a entre as alturas de g(x) e h(x). Com a adi\303\247\303\243o deste termo constante, a s\303\251rie torna-se uma boa representa\303\247\303\243o da fun\303\247\303\243o original, num determinado intervalo.___________________________________________________________________________________D. Intervalo de Converg\303\252ncia___________________________________________________________________________________Apresentamos a seguir outro exemplo, utilizando o m\303\251todo descrito acima. Contudo, pelo gr\303\241fico da fun\303\247\303\243o e sua expans\303\243o em s\303\251rie, observa-se que tem-se uma boa representa\303\247\303\243o apenas em um determinado intervalo. Fora deste intervalo, a s\303\251rie e a fun\303\247\303\243o s\303\243o bastante diferentes. f := x -> ln(1+x);

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diff( f(x), x);

KiQsJkkieEc2IiIiIkYmRiYhIiI=

series( %, x, 7);

KzNJInhHNiIiIiIiIiEhIiJGJUYlIiIjRiciIiRGJSIiJUYnIiImRiUiIictSSJPRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNGJSIiKA==

g := unapply( int(convert( %, polynom), x), x);

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plot( [f(x), g(x), [[-1,0],[-1, g(-1)]], [[1,0],[1,g(1)]] ], x = -3..3, y = -7..3, color = [red, blue, coral, coral], thickness = [3,1,2,2] );

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Uma s\303\251rie de pot\303\252ncias n\303\243o necessariamente representa a fun\303\247\303\243o para todos os valores de x. Por exemplo, 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 apenas para | x | < 1. Toda s\303\251rie de pot\303\252ncias possui um intervalo de converg\303\252ncia - seja este intervalo apenas um n\303\272mero ou todos os n\303\272meros reais. O processo para a determina\303\247\303\243o da s\303\251rie inclui em um de seus passos a expans\303\243o de uma s\303\251rie de pot\303\252ncias. Portanto, o intervalo de converg\303\252ncia afeta o resultado final da s\303\251rie.Este conceito pode ser estudado em maiores detalhes atrav\303\251s da fun\303\247\303\243o plot modificada, abaixo. restart; with(plots): conv_plot := proc( f, g1,g2, a,b, c1, c2, eps) local x1,x2,y1,y2,delta, m, M, A, B, i, n, CL; n := 1: delta := (b-a)/n: x2 := a: M := maximize( f(x), x = a..b): m := minimize(f(x), x = a..b): for i from 1 to n do x1 := x2: x2:= x1 + delta: y1 := evalf( f(x1)): y2 := evalf( f(x2)): if( abs(evalf( y1 - g1(x1))) < eps ) then CL := maroon: else CL := black: fi: #print ( [[x1, m],[x1,y1],[x2,y2],[x2,m]]); A[i] := polygonplot( [[x1, m],[x1,y1],[x2,y2],[x2,m]], color = CL, style = patchnogrid): if( abs(evalf( y1 - g2(x1))) < eps ) then CL := coral: else CL := black: fi: B[i] := polygonplot( [[x1, M],[x1,y1],[x2,y2],[x2,M]], color = CL, style = patchnogrid): od: display( [plot([f(x), g1(x), g2(x)], x = a..b, y = n..M, thickness = [4,2,2], color = [red, blue, green] ), plot( [[[c1, m],[c1,g1(c1)]], [[c2, g2(c2)],[c2,M]] ] , x =a..b, color = [blue, green], linestyle = 3), seq( A[i], i = 1..n), seq( B[i], i = 1..n)], axes = framed ); end: Esta fun\303\247\303\243o plot especial indica onde a s\303\251rie apresenta uma pequena toler\303\242ncia em rela\303\247\303\243o a fun\303\247\303\243o original. Esta fun\303\247\303\243o nos fornece uma maneira conveniente de verificar onde a s\303\251rie converge para a fun\303\247\303\243o e onde esta n\303\243o converge. Desta forma, \303\251 poss\303\255vel comparar duas fun\303\247\303\265es aproximadas com a fun\303\247\303\243o original. f := x -> ln( 1+x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkjbG5HRiU2IywmIiIiRjg5JEY4RihGKEYoNyNGLg==

g := unapply( convert(series( f(x), x, 7), polynom), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLC45JCIiIiokRjUiIiMjISIiRjgqJEY1IiIkI0Y2RjwqJEY1IiIlI0Y6Rj8qJEY1IiImI0Y2RkIqJEY1IiInI0Y6RkVGKEYoRig3I0Yu

h := unapply( convert(series( f(x), x, 12), polynom), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJoR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLDg5JCIiIiokRjUiIiMjISIiRjgqJEY1IiIkI0Y2RjwqJEY1IiIlI0Y6Rj8qJEY1IiImI0Y2RkIqJEY1IiInI0Y6RkUqJEY1IiIoI0Y2RkgqJEY1IiIpI0Y6RksqJEY1IiIqI0Y2Rk4qJEY1IiM1I0Y6RlEqJEY1IiM2I0Y2RlRGKEYoRig3I0Yu

plot( {f(x), g(x)}, x = -3..3, y = -6..3);

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fun\303\247\303\243o original centro da expans\303\243o em s\303\251rieA fun\303\247\303\243o original, f(x0 = ln(1+x), \303\251 representada pela curva vermelha. A curva estimada por g(x) apresenta uma expans\303\243o em s\303\251rie com 7 termos. O intervalo onde g difere de f por uma valor menor que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEqJmVwc2lsb247RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== = .05 \303\251 indicado em roxo, no canto da figura. A reta pontilhada em azul indica o centro da expans\303\243o. De forma similar, a curva verde \303\251 h(x), que representa uma expans\303\243o em s\303\251rie com 12 termos. O intervalo onde esta curva encontra-se pr\303\263xima de f(x) \303\251 indicado pela barra amarela acima do gr\303\241fico. Este diagrama mostra que a s\303\251rie com mais termos representa melhor a fun\303\247\303\243o. Al\303\251m disso, observa-se que ambas as s\303\251ries convergem dentro da mesma regi\303\243o, entre -1 e +1.Analisemos mais um exemplo. Consideremos tan -1(x), e observemos sua expans\303\243o em s\303\251ries, com 3 e 15 termos. f := x -> arctan(x); c := 0;

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series( D(f)(x), x = c, 3); convert(int(%, x), polynom); g := unapply( %,x);

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series( D(f)(x), x = c, 15); convert(int(%, x), polynom); h := unapply( %,x);

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plot( [f(x),g(x), h(x) ], x = -5..5, y = -4..4, color = [red, blue, green], thickness = [3,1,2] );

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A primeira vista, o gr\303\241fico em azul (expans\303\243o em s\303\251rie com 3 termos) parece minimizar a fun\303\247\303\243o. No entanto, por uma inspe\303\247\303\243o mais cuidadosa, verifica-se que dentro do intervalo [-1, 1] a curva verde (expans\303\243o em s\303\251rie com 15 termos) aproxima melhor a fun\303\247\303\243o. Isto pode ser melhor visualizado com o comando plot especial. conv_plot( f, g, h, -3, 3, c, c, .1);

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Neste exemplo, o centro da expans\303\243o em s\303\251rie \303\251 c = 0. O mesmo pode ser feito com diferentes valores para o centro da expans\303\243o. Apresentamos a seguir um exemplo com c = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEjUGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn/3. Note que toda a regi\303\243o de converg\303\252ncia move-se com o centro da expans\303\243o para a direita. f := x -> arctan(x); c := Pi/3;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUknYXJjdGFuR0YlNiM5JEYoRihGKDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJjR0YoLCRJI1BpR0YmIyIiIiIiJDcjRi4=

series( f(x) , x = c, 3): convert( %, polynom): g := unapply( %, x): series( f(x) , x = c, 15): convert( %, polynom): h := unapply( %, x): conv_plot( f, g, h, -4, 4, c, c, .1);

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Existe mais um aspecto interessante. A fun\303\247\303\243o estudada acima, f(x) = ln(1 + x), apresenta uma ass\303\255ntota vertical em x = -1. Quando a s\303\251rie \303\251 expandida em torno de x = 0, o intervalo de converg\303\252ncia \303\251 (-1, 1). O que acontece quando a s\303\251rie \303\251 expandida em torno de outros valores de x, mais afastados desta singularidade? f := x -> ln(1+x); c1 := 0; c2 := 2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkjbG5HRiU2IywmIiIiRjg5JEY4RihGKEYoNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCIiITcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMkdGKCIiIzcjRi4=

g := unapply(convert(series( f(x), x = c1,12), polynom), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLDg5JCIiIiokRjUiIiMjISIiRjgqJEY1IiIkI0Y2RjwqJEY1IiIlI0Y6Rj8qJEY1IiImI0Y2RkIqJEY1IiInI0Y6RkUqJEY1IiIoI0Y2RkgqJEY1IiIpI0Y6RksqJEY1IiIqI0Y2Rk4qJEY1IiM1I0Y6RlEqJEY1IiM2I0Y2RlRGKEYoRig3I0Yu

h := unapply(convert(series( f(x), x = c2, 12), polynom), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJoR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLDwtSSNsbkdGJTYjIiIkIiIiOSQjRjlGOCMhIiNGOEY5KiQsJkY6RjlGPUY5IiIjIyEiIiIjPSokRj9GOCNGOSIjIikqJEY/IiIlI0ZCIiRDJCokRj8iIiYjRjkiJTo3KiRGPyIiJyNGQiIldVYqJEY/IiIoI0Y5IiY0YCIqJEY/IiIpI0ZCIiYpW18qJEY/IiIqI0Y5IidacjwqJEY/IiM1I0ZCIichXCFmKiRGPyIjNiNGOSIoPCdbPkYoRihGKDcjRi4=

plot( [ f(x), g(x), h(x) ], x = -0.85..7, y = -2..3, color = [red, blue, green], thickness = [3,1,2]);

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conv_plot( f, g, h, -0.8, 7, c1, c2, .15);

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A curva em vermelho \303\251 a fun\303\247\303\243o original, o gr\303\241fico em azul representa a expans\303\243o em s\303\251rie em torno de x = 0, e a regi\303\243o em roxo indica o intervalo de converg\303\252ncia. A gr\303\241fico verde representa a expans\303\243o em s\303\251rie em torno de x = 2, e a barra amarela indica o intervalo de converg\303\252ncia. Pelo diagrama acima, verifica-se que a expans\303\243o em torno de 2 possui intervalo de converg\303\252ncia pr\303\263ximo de (-1, 5), diferente do caso anterior, dado por (-1, 1). Observemos outro valor para a expans\303\243o. c1 := 0; c2 := 4;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMUdGKCIiITcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNjMkdGKCIiJTcjRi4=

g := unapply( convert(series( f(x), x = c1, 12), polynom), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJnR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLDg5JCIiIiokRjUiIiMjISIiRjgqJEY1IiIkI0Y2RjwqJEY1IiIlI0Y6Rj8qJEY1IiImI0Y2RkIqJEY1IiInI0Y6RkUqJEY1IiIoI0Y2RkgqJEY1IiIpI0Y6RksqJEY1IiIqI0Y2Rk4qJEY1IiM1I0Y6RlEqJEY1IiM2I0Y2RlRGKEYoRig3I0Yu

h := unapply( convert(series( f(x), x = c2, 12), polynom), x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJoR0YoZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLDwtSSNsbkdGJTYjIiImIiIiOSQjRjlGOCMhIiVGOEY5KiQsJkY6RjlGPUY5IiIjIyEiIiIjXSokRj8iIiQjRjkiJHYkKiRGPyIiJSNGQiIlK0QqJEY/RjgjRjkiJkRjIiokRj8iIicjRkIiJl1QKiokRj8iIigjRjkiJ3ZvYSokRj8iIikjRkIiKCtdNyQqJEY/IiIqI0Y5IilEInl2IiokRj8iIzUjRkIiKV1pbCgqKiRGPyIjNiNGOSIqdiQ0cmBGKEYoRig3I0Yu

conv_plot(f, g, h, -0.8, 12, c1, c2, .15.2 );

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A expans\303\243o em torno de x = 4 possui um intervalo de converg\303\252ncia pr\303\263ximo de (-1, 9). Note que \303\251 mais dif\303\255cil observar o que acontece no canto \303\240 direita. O gr\303\241fico a seguir ajuda a demonstrar este fato. plot( {f(x), g(x), h(x), [[9,0], [9,h(9)]] }, x = -.5..12, y = -1..3);

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Por este exemplo, verifica-se que a ass\303\255ntota vertical \303\251 como uma parede, limitando a converg\303\252ncia da s\303\251rie de pot\303\252ncias. Al\303\251m disso, o intervalo de converg\303\252ncia deve ser sim\303\251trico em rela\303\247\303\243o ao centro da expans\303\243o. Quando o centro da expans\303\243o \303\251 escolhido pr\303\263ximo da ass\303\255ntota vertical, estes dois fatores, juntos, contribuem para tornar os intervalos de converg\303\252ncia menores.