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anonymous
 one year ago
Find the power series for f(x)= (4x)/(12x3x^2).
Need a medium detailed solution.
anonymous
 one year ago
Find the power series for f(x)= (4x)/(12x3x^2). Need a medium detailed solution.

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ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4As a start, do the partial fraction decomposition of given rational function : \[f(x)=\dfrac{4x}{12x3x^2} = \dfrac{1}{13x}\dfrac{1}{1+x}\]

dan815
 one year ago
Best ResponseYou've already chosen the best response.1use that for both partial fractions

dan815
 one year ago
Best ResponseYou've already chosen the best response.1and write the formula for each a_n

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0$$f(x)= (4x)/(12x3x^2)\\(12x3x^2)f(x)=4x$$so consider the expansion $$f(x)=\sum_{n=0}^\infty a_n x^n$$ which gives: $$\sum_{n=0}^\infty a_n x^n2x\sum_{n=0}^\infty a_n x^n3x^2\sum_{n=0}^\infty a_n x^n=4x\\\sum_{n=0}^\infty a_n x^n2\sum_{n=0}^\infty a_n x^{n+1}3\sum_{n=0}^\infty a_n x^{n+2}=4x\\\sum_{n=0}^\infty a_n x^n2\sum_{n=1}^\infty a_{n1} x^n3\sum_{n=2}^\infty a_{n2} x^n=4x\\a_0+a_1 x+\sum_{n=2}^\infty a_n x^n2a_0 x2\sum_{n=2}^\infty a_{n1} x^n3\sum_{n=2}^\infty a_{n2} x^n=4x\\a_0+(2a_0+a_1)x+\sum_{n=2}^\infty (a_n2a_{n1}3a_{n2}) x^n=4x$$ so it follows: $$a_0=0\\2a_0+a_1=4\\a_n2a_{n1}3a_{n2}=0$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0now notice \(a_n2a_{n1}3a_{n2}\) has characteristic polynomial \(r^22r3\), which gives roots \(r\in\{1,3\}\) so \(a_n=c_1\cdot (1)^n+c_2\cdot 3^n\) using our initial terms we have \(2a_0+a_1=4\implies a_1=4\) so: $$a_0=0\\c_1\cdot(1)^0+c_2\cdot 3^0=0\\c_1+c_2=0$$ and $$a_1=4\\c_1\cdot(1)^1+c_2\cdot 3^1=4\\c_1+3c_2=4$$ these eliminate to give $$4c_2=4\implies c_2=1$$ and then substituting that back in gives $$c_1+1=0\implies c_1=1$$ so the power series coefficients are given by $$a_n=(1)^n+3^n=(1)^{n+1}+3^n$$ so the power series is $$f(x)=\sum_{n=0}^\infty ((1)^{n+1}+3^n)\,x^n$$

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0this is a special case of the fact that rational functions \(f\) are generating functions of linear recurrence relations

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0*sequences that solve linear recurrence relations
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