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anonymous

  • one year ago

Find the power series for f(x)= (4x)/(1-2x-3x^2). Need a medium detailed solution.

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  1. dan815
    • one year ago
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    |dw:1438101987707:dw|

  2. ganeshie8
    • one year ago
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    As a start, do the partial fraction decomposition of given rational function : \[f(x)=\dfrac{4x}{1-2x-3x^2} = \dfrac{1}{1-3x}-\dfrac{1}{1+x}\]

  3. dan815
    • one year ago
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    |dw:1438102165432:dw|

  4. dan815
    • one year ago
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    there is a bound x<1

  5. dan815
    • one year ago
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    use that for both partial fractions

  6. dan815
    • one year ago
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    and write the formula for each a_n

  7. dan815
    • one year ago
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    |dw:1438102436197:dw|

  8. dan815
    • one year ago
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    |dw:1438102550876:dw|

  9. dan815
    • one year ago
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    |dw:1438102636067:dw|

  10. anonymous
    • one year ago
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    $$f(x)= (4x)/(1-2x-3x^2)\\(1-2x-3x^2)f(x)=4x$$so consider the expansion $$f(x)=\sum_{n=0}^\infty a_n x^n$$ which gives: $$\sum_{n=0}^\infty a_n x^n-2x\sum_{n=0}^\infty a_n x^n-3x^2\sum_{n=0}^\infty a_n x^n=4x\\\sum_{n=0}^\infty a_n x^n-2\sum_{n=0}^\infty a_n x^{n+1}-3\sum_{n=0}^\infty a_n x^{n+2}=4x\\\sum_{n=0}^\infty a_n x^n-2\sum_{n=1}^\infty a_{n-1} x^n-3\sum_{n=2}^\infty a_{n-2} x^n=4x\\a_0+a_1 x+\sum_{n=2}^\infty a_n x^n-2a_0 x-2\sum_{n=2}^\infty a_{n-1} x^n-3\sum_{n=2}^\infty a_{n-2} x^n=4x\\a_0+(-2a_0+a_1)x+\sum_{n=2}^\infty (a_n-2a_{n-1}-3a_{n-2}) x^n=4x$$ so it follows: $$a_0=0\\-2a_0+a_1=4\\a_n-2a_{n-1}-3a_{n-2}=0$$

  11. anonymous
    • one year ago
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    now notice \(a_n-2a_{n-1}-3a_{n-2}\) has characteristic polynomial \(r^2-2r-3\), 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 $$f(x)=\sum_{n=0}^\infty ((-1)^{n+1}+3^n)\,x^n$$

  12. anonymous
    • one year ago
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    this is a special case of the fact that rational functions \(f\) are generating functions of linear recurrence relations

  13. anonymous
    • one year ago
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    *sequences that solve linear recurrence relations

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